title: Spectral Invariance of Dense Subalgebras of Operator Algebras

abstract:
  We define the notion of strong spectral invariance for a dense Frechet
subalgebra A of a Banach algebra B. We show that if A is strongly spectral
invariant in a C*-algebra B, and G is a compactly generated polynomial growth
Type R Lie group, not necessarily connected, then the smooth crossed product
G\rtimes A is spectral invariant in the C*-crossed product G\rtimes B. Examples
of such groups are given by finitely generated polynomial growth discrete
groups, compact or connected nilpotent Lie groups, the group of Euclidean
motions on the plane, the Mautner group, or any closed subgroup of one of
these. Our theorem gives the spectral invariance of G\rtimes A if A is the set
of C^{\infty}-vectors for the action of G on B, or if B= C_{0}(M), and A is a
set of G-differentiable Schwartz functions S(M) on M. This gives many examples
of spectral invariant dense subalgebras for the C*-algebras associated with
dynamical systems. We also obtain relevant results about exact sequences,
subalgebras, tensoring by smooth compact operators, and strong spectral
invariance in L_{1}(G, B).


title:
Dense m-convex Frechet Subalgebras of Operator Algebra Crossed Products
  by Lie Groups

abstract:
  Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require
Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily con-
nected, which acts on both $A$ and B by *-automorphisms, and let \s be a sub-
multiplicative function from G to the nonnegative real numbers. If \s and the
action of G on A satisfy certain simple properties, we define a dense Frechet
*-subalgebra G\rtimes^{\s} A of the crossed product L^{1}(G, B). Our algebra
consists of differentiable A-valued functions on G, rapidly vanishing in \s.
  We give conditions on the action of G on A which imply the m-convexity of the
dense subalgebra G\rtimes^{\s}A. A locally convex algebra is said to be m-con-
vex if there is a family of submultiplicative seminorms for the topology of the
algebra. The property of m-convexity is important for a Frechet algebra, and is
useful in modern operator theory.
  If G acts as a transformation group on a manifold M, we develop a class of
dense subalgebras for the crossed product L^{1}(G, C_{0}(M)), where C_{0}(M)
denotes the continuous functions on M vanishing at infinity with the sup norm
topology.We define Schwartz functions S(M) on M, which are differentiable with
respect to some group action on M, and are rapidly vanishing with respect to
some scale on M. We then form a dense m-convex Frechet *-subalgebra G\rtimes^
{\s} S(M) of rapidly vanishing, G-differentiable functions from G to S(M).
  If the reciprocal of \s is in L^{p}(G) for some p, we prove that our group
algebras S^{\s}(G) are nuclear Frechet spaces, and that G\rtimes^{\s}A is the
projective completion S^{\s}(G) \otimes A.


title: On certain free product factors via an extended matrix model

abstract:
  Voiculescu's random matrix model for freeness is extended to the non-Gaussian
case and also the case of constant block diagonal matrices. Thus we are able to
investigate free products of free group factors with matrix algebras and with
the hyperfinite II$_1$ factor, showing that $$ L(F_n) * R = L(F_(n+1)) $$ for
$n \ge 1$, (where $L(F_1)=L(Z)$).


title: Interpolated free group factors

abstract:
  The interpolated free group factors L(F_r), 1 < r <= \infty, are defined and
proofs of their properties with respect to compression by projections and
taking free products are proved. Hence it follows that all the free group
factor are isomorphic to each other or none of them are. These factors were
defined and these properties were proved independently by F. Radulescu, and
those given in this paper are equivalent, but use different techniques.
Specifically, we develop algebraic techniques that allow us to show that R*R =
L(F_2), where R is the hyperfinite II_1 factor.


title: Free products of hyperfinite von Neumann algebras and free dimension

abstract:
  The free product of an arbitrary pair of finite hyperfinite von Neumann
algebras is examined, and the result is determined to be the direct sum of a
finite dimensional algebra and an interpolated free group factor $L(\freeF_r)$.
The finite dimensional part depends on the minimal projections of the original
algebras and the "dimension", r, of the free group factor part is found using
the notion of free dimension. For discrete amenable groups $G$ and $H$ this
implies that the group von Neumann algebra $L(G*H)$ is an interpolated free
group factor and depends only on the orders of $G$ and $H$.


title: Non-commutative spheres and numerical quantum mechanics

abstract:
  We discuss some basic issues that arise when one attempts to model quantum
mechanical systems on a computer, and we describe the mathematical structure of
the resulting discretized cannonical commutation relations.


title: Discretized CCR algebras

abstract:
  We discuss how the canonical commutation relations must be modified in order
to make appropriate numerical models of quantum systems. The C*-algebras
associated with the discretized CCRs are the non-commutative spheres of
Bratteli, Elliott, Evans and Kishimoto.


title:
The free abelian topological group and the free locally convex space on
  the unit interval

abstract:
  We give a complete description of the topological spaces $X$ such that the
free abelian topological group $A(X)$ embeds into the free abelian topological
group $A(I)$ of the closed unit interval. In particular, the free abelian
topological group $A(X)$ of any finite-dimensional compact metrizable space $X$
embeds into $A(I)$. The situation turns out to be somewhat different for free
locally convex spaces. Some results for the spaces of continuous functions with
the pointwise topology are also obtained. Proofs are based on the classical
Kolmogorov's Superposition Theorem.


title: On von Neumann algebras which are complemented subspaces of B(H)

abstract:
  If there exists a completely bounded projection of B(H) onto a von Neumann
algebra M on H, then M is injective. If there exists a bounded projection and M
is properly infinite, the same conclusion holds.


title: Compression Limit Algebras

abstract:
  This paper studies direct limits of full upper triangular matrix algebras
with embeddings which are not *-extendible. A representation of the limit
algebra is found so that the generated C*-algebra is the C*-envelope. Some
examples are described.


title: The Soft Torus II: A Variational Analysis of Commutator Norms

abstract:
  The field of C*-algebras over the interval [0,2] for which the fibers are the
Soft Tori is shown to be continuous. This result is applied to show that any
pair of non-commuting unitary operators can be perturbed (in a weak sense) in
such a way to decrease the commutator norm. Perturbations in norm are also
considered and a characterization is given for pairs of unitary operators which
are local minimum points for the commutator norm in the finite dimensional
case.


title: A Fredholm Operator Approach To Morita Equivalence

abstract:
  Given C*-algebras A and B and an imprimitivity A-B-bimodule X, we construct
an explicit isomorphism X_* : K_i(A) --> K_i(B) where K_i denote the complex
K-theory functors for i=0, 1. Our techniques do not require separability nor
existence of countable approximate identities. We thus extend, to general
C*-algebras, the result of Brown, Green and Rieffel according to which strongly
Morita equivalent C*-algebras have isomorphic K-groups. The method employed
includes a study of Fredholm operators on Hilbert modules.


title: Lie-algebras and linear operators with invariant subspaces

abstract:
  A general classification of linear differential and finite-difference
operators possessing a finite-dimensional invariant subspace with a polynomial
basis (the generalized Bochner problem) is given. The main result is that any
operator with the above property must have a representation as a polynomial
element of the universal enveloping algebra of some algebra of differential
(difference) operators in finite-dimensional representation plus an operator
annihilating the finite-dimensional invariant subspace. In low dimensions a
classification is given by algebras $sl_2({\bold R})$ (for differential
operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference
operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann
variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$),
$sl_3({\bold R})$, $sl_2({\bold R}) \oplus sl_2({\bold R})$ and $gl_2 ({\bold
R}) \ltimes {\bold R}^{r+1}\ , r$ a natural number (operators in ${\bold
R^2}$). A classification of linear operators possessing infinitely many
finite-dimensional invariant subspaces with a basis in polynomials is
presented. A connection to the recently-discovered quasi-exactly-solvable
spectral problems is discussed.


title: Summary of Spectral Invariance Results

abstract:
  The author's recent results on spectral invariant dense subalgebras of
C*-algebras associated with dynamical systems are summarized. If G is a
compactly generated polynomial growth Type R Lie group, and the action of G on
S(M) (Schwartz functions on a locally compact G-space M) is tempered in a
certain sense, then there is a natural smooth crossed product S(G X M) which is
dense and spectral invariant in the C*-crossed product C*(G X M).


title: A Factorization Theorem for Smooth Crossed Products

abstract:
  We show that if E is a Frechet G\rtimes S(M)-module, for which the canonical
map from the projective completion G\rtimes S(M) {\widehat \otimes} E to E is
surjective, then every element of E can be written as a finite sum of elements
of the form ae where e\in E and a is an element of the smooth crossed product
G\rtimes S(M). We require that the Schwartz functions S(M) vanish rapidly with
repsect to a continuous, proper map \s : M ---> [0, \infty).


title: "Vector bundles" over quantum Heisenberg manifolds

abstract:
  By means of techniques from the Morita equivalence theory, we get finitely
generated and projective modules over the quantum Heisenberg manifolds. This
enables us to get some information about the range of the trace of these
algebras, at the level of the K_{0}-group, in terms of the Poisson bracket in
whose direction the manifolds are deformed.


title: Generalized fixed-point algebras of certain actions on crossed products

abstract:
  Let G and H be two locally compact groups acting on a C*-algebra A by
commuting actions. We construct an action on the crossed product AXG out of a
unitary 2-cocycle u and the action of H on A. For A commutative, and free and
proper actions of G and H, we show that if the roles of these two actions are
reversed, and u is replaced by u*, then the corresponding generalized
fixed-point algebras, in the sense of Rieffel, are strong-Morita equivalent. We
apply this result to the computation of the K-theory of quantum Heisenberg
manifolds.


title: The Bunce-Deddens Algebras as Crossed Products by Partial Automorphisms

abstract:
  We describe both the Bunce-Deddens C*-algebras and their Toeplitz versions,
as crossed products of commutative C*-algebras by partial automorphisms. In the
latter case, the commutative algebra has, as its spectrum, the union of the
Cantor set and a copy the set of natural numbers N, fitted together in such a
way that N is an open dense subset. The partial automorphism is induced by a
map that acts like the odometer map on the Cantor set while being the
translation by one on N. From this we deduce, by taking quotients, that the
Bunce-Deddens C*-algebras are isomorphic to the (classical) crossed product of
the algebra of continuous functions on the Cantor set by the odometer map.


title: Algebraic orders on $K_0$ and approximately finite operator algebras

abstract:
  This is a revised and corrected version of a preprint circulated in 1990 in
which various non-self-adjoint limit algebras are classified. The principal
invariant is the scaled $K_0$ group together with the algebraic order on the
scale induced by partial isometries in the algebra.


title:
On the outer automorphism groups of triangular alternation limit
  algebras

abstract:
  Let $A$ denote the alternation limit algebra, studied by Hopenwasser and
Power, and by Poon, which is the closed direct limit of upper triangular matrix
algebras determined by refinement embeddings of multiplicity $r_k$ and standard
embeddings of multiplicity $s_k$. It is shown that the quotient of the
isometric automorphism group by the approximately inner automorphisms is the
abelian group $ \ZZ ^d$ where $d$ is the number of primes that are divisors of
infinitely many terms of each of the sequences $(r_k)$ and $(s_k)$. This group
is also the group of automorphisms of the fundamental relation of $A$.


title: Time-Frequency Localization and the Gabor Transform

abstract:
  The time-frequency content of a signal can be measured by the Gabor transform
or windowed Fourier transform. This is a function defined on phase space that
is computed by taking the Fourier transform of the product of the signal
against a translate of a fixed window. The problem of finding signals with
Gabor transform that are maximally concentrated within a given region of phase
space is discussed. It is well known that such problems give rise to an
eigenvalue problem for an associated self-adjoint, positive {\em concentration}
operator that has its spectrum contained in the unit interval. In this paper,
the asymptotic behavior of these eigenvalues as the concentration region gets
large is studied. The smoothness of the eigenfunctions is also examined.


title:
Factoriality and the Connes invariant T(M) for free products of von
  Neumann algebras

abstract:
  This revised version corrects some substancial errors in the proofs. The old
proofs were only valid for (finite dimensional) * (finite dimensional). The
scope of the results remains the same, however for type I_\infty algebras, the
results of the original version were incorrect.


title: On the Fock representation of the q-commutation relations

abstract:
  The q-commutation relations in the title are those that have recently
received much attention, and that for -1<q<1 provide an interpolation between
Bosonic and Fermionic statistics, passing through free statistics at q=0. We
look at the C*-algebra R^q generated by the representation of these relations
on the twisted Fock space of Bozejko-Speicher, and we find a canonical unitary
U_q from the twisted Fock space to usual full Fock space, such that U_q R^q
(U_q)^* contains the Cuntz algebra R^0, and such that we have equality for
|q|<0.44.


title: Operator spaces and residually finite-dimensional $C^\ast$-algebras

abstract:
  For every operator space $X$ the $C^\ast$-algebra containing it in a
universal way is residually finite-dimensional (that is, has a separating
family of finite-dimensional representations). In particular, the free
$C^\ast$-algebra on any normed space so is. This is an extension of an earlier
result by Goodearl and Menal, and our short proof is based on a criterion due
to Exel and Loring.


title: Modular Structure and Duality in Conformal Quantum Field Theory

abstract:
  Making use of a recent result of Borchers, an algebraic version of the
Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e.
the Tomita-Takesaki modular group associated with the von Neumann algebra of a
wedge region and the vacuum vector concides with the evolution given by the
rescaled pure Lorentz transformations preserving the wedge. A similar geometric
description is valid for the algebras associated with double cones. Moreover
essential duality holds on the Minkowski space $M$, and Haag duality for double
cones holds provided the net of local algebras is extended to a pre-cosheaf on
the superworld $\tilde M$, i.e. the universal covering of the Dirac-Weyl
compactification of $M$. As a consequence a PCT symmetry exists for any
algebraic conformal field theory in even space-time dimension. Analogous
results hold for a Poincar\'e covariant theory provided the modular groups
corresponding to wedge algebras have the expected geometrical meaning and the
split property is satisfied. In particular the Poincar\'e representation is
unique in this case.


title: Spectral and polar decomposition in AW*-algebras

abstract:
  We show the possibility and the uniqueness of polar decomposition of elements
of arbitrary AW*-algebras inside them. We prove that spectral decomposition of
normal elements of normal AW*-algebras is possible and unique inside them. The
possibility of spectral decomposition of normal elements does not depend on the
normality of the AW*-algebra under consideration.


title: Coherent States of the q--Canonical Commutation Relations

abstract:
  For the $q$-deformed canonical commutation relations $a(f)a^\dagger(g) =
(1-q)\,\langle f,g\rangle{\bf1}+q\,a^\dagger(g)a(f)$ for $f,g$ in some Hilbert
space ${\cal H}$ we consider representations generated from a vector $\Omega$
satisfying $a(f)\Omega=\langle f,\phi\rangle\Omega$, where $\phi\in{\cal H}$.
We show that such a representation exists if and only if $\Vert\phi\Vert\leq1$.
Moreover, for $\Vert\phi\Vert<1$ these representations are unitarily equivalent
to the Fock representation (obtained for $\phi=0$). On the other hand
representations obtained for different unit vectors $\phi$ are disjoint. We
show that the universal C*-algebra for the relations has a largest proper,
closed, two-sided ideal. The quotient by this ideal is a natural $q$-analogue
of the Cuntz algebra (obtained for $q=0$). We discuss the Conjecture that, for
$d<\infty$, this analogue should, in fact, be equal to the Cuntz algebra
itself. In the limiting cases $q=\pm1$ we determine all irreducible
representations of the relations, and characterize those which can be obtained
via coherent states.


title: A Wiener estimate for relaxed Dirichlet problems in dimension $N\geq 2$

abstract:
  We prove a Wiener energy estimate for relaxed Dirichlet problems $Lu + \mu u
=\nu$ in $\Omega$, with $L$ an uniformly elliptic operator with bounded
coefficients, $\mu$ a measure of ${\cal M}_0(\Omega)$, $\nu$ a Kato measure and
$\Omega$ a bounded open set of ${\bf R}^N$, $N \geq 2$. Choosing a particular
$\mu$, we obtain an energy estimate also for classical variational Dirichlet
problems.


title:
Approximation of Relaxed Dirichlet Problems by Boundary Value problems
  in perforated domains

abstract:
  Given an elliptic operator~$L$ on a bounded domain~$\Omega \subseteq {\bf
R}^n$, and a positive Radon measure~$\mu$ on~$\Omega$, not charging polar sets,
we discuss an explicit approximation procedure which leads to a sequence of
domains~$\Omega_h \subseteq \Omega$ with the following property: for
every~$f\in H^{-1}(\Omega)$ the sequence~$u_h$ of the solutions of the
Dirichlet problems~$L\, u_h=f$ in~$\Omega_h$, $u_h=0$ on~$\partial \Omega_h$,
extended to 0 in~$\Omega \setminus \Omega_h$, converges to the solution of the
\lq\lq relaxed Dirichlet problem\rq\rq\ $L\,u+\mu u=f$ in~$\Omega$, $u=0$
on~$\partial \Omega$.


title: On a groupoid construction for actions of certain inverse semigroups

abstract:
  We consider a version of the notion of F-inverse semigroup (studied in the
algebraic theory of inverse semigroups). We point out that an action of such an
inverse semigroup on a locally compact space has associated a natural groupoid
construction, very similar to the one of a transformation group. We discuss
examples related to Toeplitz algebras on subsemigroups of discrete groups,to
Cuntz-Krieger algebras, and to crossed-products by partial automorphisms in the
sense of Exel.


title:
Connection between the harmonic analysis on the sphere and the harmonic
  analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint

abstract:
  In a previous paper $[$B,V-1$]$, an algebra of holomorphic ``perikernels'' on
a complexified hyperboloid $ X^{(c)}_{d-1} $ (in $\Bbb C^d)$ has been
introduced; each perikernel $ {\cal K} $ can be seen as the analytic
continuation of a kernel $ {\bf K} $ on the unit sphere $ \Bbb S^{d-1} $ in an
appropriate ``cut-domain'' , while the jump of $ {\cal K} $ across the
corresponding ``cut'' defines a Volterra kernel $ K $ (in the sense of J.
Faraut $\lbrack$Fa-1$\rbrack$) on the one-sheeted hyperboloid $ X_{d-1} $ (in $
\Bbb R^d). $ \par In the present paper, we obtain results of harmonic analysis
for classes of perikernels which are invariant under the group $ {\rm
SO}(d,\Bbb C) $ and of moderate growth at infinity. For each perikernel $ {\cal
K} $ in such a class, the Fourier-Legendre coefficients of the corresponding
kernel $ {\bf K} $ on $ \Bbb S^{d-1} $ admit a carlsonian analytic
interpolation $ \tilde F(\lambda) $ in a half-plane, which is the ``spherical
Laplace transform''\ of the associated Volterra kernel $K$ on $ X_{d-1}. $
Moreover, the composition law $ {\cal K }= {\cal K}_1\ast^{( c)}{\cal K}_2 $
for perikernels (interpreted in terms of convolutions for the


title: q-Supersymmetric Generalization of von Neumann's Theorem

abstract:
  Assuming that there exist operators which form an irreducible representation
of the q-superoscillator algebra, it is proved that any two such
representations are equivalent, related by a uniquely determined superunitary
transformation. This provides with a q-supersymmetric generalization of the
well-known uniqueness theorem of von Neumann for any finite number of degrees
of freedom.


title: On some operator identities and representations of algebras

abstract:
  Certain infinite families of operator identities related to powers of
positive root generators of (super) Lie algebras of first-order differential
operators and $q$-deformed algebras of first-order finite-difference operators
are presented.


title: A calculation with a bi-orthogonal wavelet transformation

abstract:
  We explore the use of bi-orthogonal basis for continuous wavelet
transformations, thus relaxing the so-called admissibility condition on the
analyzing wavelet. As an application, we determine the eigenvalues and
corresponding radial eigenfunctions of the Hamiltonian of relativistic
Hydrogen-like atoms.


title: Commuting difference operators with polynomial eigenfunctions

abstract:
  We present explicit generators of an algebra of commuting difference
operators with trigonometric coefficients. The operators are simultaneously
diagonalized by recently discovered q-polynomials (viz. Koornwinder's
multivariable generalization of the Askey-Wilson polynomials). From the
viewpoint of physics the algebra can be interpreted as consisting of the
quantum integrals of a novel difference-type integrable sytem. This system
generalizes the Calogero-Moser systems associated with non-exceptional root
systems.

categories: funct-an hep-th math.OA nlin.SI solv-int


title: The role of C*-algebras in infinite dimensional numerical linear algebra

abstract:
  This is a survey of four recent papers which deal with the relationship of
simple C*-algebras to the problem of computing the spectra of self-adjoint
operators in the general case, especially when the spectrum is not discrete. It
is an expanded version of a talk presented at the 50 year C*-algebra
celebration, held at the annual meeting of the AMS in San Antonio during
January, 1993.


title:
Classification of linear differential operators with an invariant
  subspace of monomials

abstract:
  A complete classification of linear differential operators possessing
finite-dimensional invariant subspace with a basis of monomials is presented.


title: Singular Traces and Compact Operators. I

abstract:
  We give a necessary and sufficient condition on a positive compact operator
$T$ for the existence of a singular trace (i.e. a trace vanishing on the finite
rank operators) which takes a finite non-zero value on $T$. This generalizes
previous results by Dixmier and Varga. We also give an explicit description of
these traces and associated ergodic states on $\ell^\infty(\Bbb N)$ using tools
of non standard analysis in an essential way.


title: Non Symmetric Dirichlet Forms on Semifinite von Neumann Algebras

abstract:
  The theory of non symmetric Dirichlet forms is generalized to the non abelian
setting, also establishing the natural correspondences among Dirichlet forms,
sub-Markovian semigroups and sub-Markovian resolvents within this context.
Examples of non symmetric Dirichlet forms given by derivations on Hilbert
algebras are studied.


title:
Metric and Generalized Projection Operators in Banach Spaces: Properties
  and Applications

abstract:
  Metric projection operators can be defined in similar wayin Hilbert and
Banach spaces. At the same time, they differ signifitiantly in their
properties. Metric projection operator in Hilbert space is a monotone and
nonexpansive operator. It provides an absolutely best approximation for
arbitrary elements from Hilbert space by the elements of convex closed sets .
This leads to a variety of applications of this operator for investigating
theoretical questions in analysis and for approximation methods. Metric
projection operators in Banach space do not have properties mentioned above and
their applications are not straightforward. Two of the most important
applications of the method of metric projection operators are as follows:
  1. Solve a variational inequality by the iterative-projection method,
  2. Find common point of convex sets by the iterative-projection method.
  In Banach space these problems can not be solved in the framework of metric
projection operators. Therefore, in the present paper we introduce new
generalized projection operators in Banach space as a natural generalization of
metric projection operators in Hilbert space. In Sections 2 and 3 we introduce
notations and recall some results from the theory of variational inequalities
and theory of approximation. Then in Sections 4 and 5 we describe the
properties of metric projection operators $P_\Omega$ in Hilbert and Banach
spaces and also formulate equivalence theorems between variational inequalities
and direct projection equations with these


abstract:
  This paper is an account (without proofs) of the results of our work "Metric
and Generalized Projection Operators in Banach Spaces: Properties and
Applications", funct-an/9311001. The Section 9 establishing a connection
between variational inequalities and Wienner-Hopf equations in Banach spaces by
means of metric and generalized projection operators, is added.


title: On Some Estimates for Projection Operator in Banach Space

abstract:
  Previously unknown estimates of uniform continuity of projection operators in
Banach space have been obtained. They can be used in the investigations of
approximation methods, in particular, the method of quasisolutions, methods of
regularization and penalty functions, for solving nonlinear problems on exact
and peturbed sets.


title: Exponential Stability of Linear Delay Impulsive Differential Equations

abstract:
  For ordinary differential equations and functional differential equations the
following result is well known. Suppose any solution is bounded on the
half-line for each bounded on the half-line right-hand side. Then under certain
conditions the equations is exponentially stable.
  We prove the same result for a delay differential equation
  $$ \dot{x}(t) + \sum_{i=1}^k A_i (t)x[h_i(t)] = f(t), $$
  with impulses
  $$ x(\tau_i + 0) = B_i x(\tau_i - 0) $$
  at fixed moments $\tau_i$. The proof is based on a solution representation
formula obtained here.


title:
Boundedness and Stability of Impulsively Perturbed Systems in a Banach
  Space

abstract:
  Consider a linear impulsive equation in a Banach space
  $$\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0,$$
  $$x(\tau_i +0)= B_i x(\tau_i -0) + \alpha_i,$$
  with $\lim_{i \rightarrow \infty} \tau_i = \infty $. Suppose each solution of
the corresponding semi-homogeneous equation
  $$\dot{x}(t)+A(t)x(t) = 0,$$
  (2) is bounded for any bounded sequence $\{ \alpha_i \}$.
  The conditions are determined ensuring
  (a) the solution of the corresponding homogeneous equation has an exponential
estimate;
  (b) each solution of (1),(2) is bounded on the half-line for any bounded $f$
and bounded sequence $\{ \alpha_i \}$ ;
  (c) $\lim_{t \rightarrow \infty}x(t)=0$ for any $f, \alpha_i$ tending to
zero;
  (d) exponential estimate of $f$ implies a similar estimate for $x$.


title: Direct integrals and Hilbert W*-Modules

abstract:
  Investigating the direct integral decomposition of von Neumann algebras of
bounded module operators on self-dual Hilbert W*-moduli an equivalence
principle is obtained which connects the theory of direct disintegration of von
Neumann algebras on separable Hilbert spaces and the theory of von Neumann
representations on self-dual Hilbert {\bf A}-moduli with countably generated
{\bf A}-pre-dual Hilbert {\bf A}-module over commutative separable W*-algebras
{\bf A}. Examples show posibilities and bounds to find more general relations
between these two theories, (cf. R. Schaflitzel's results). As an application
we prove a Weyl--Berg--Murphy type theorem: For each given commutative
W*-algebra {\bf A} with a special approximation property (*) every normal
bounded {\bf A}-linear operator on a self-dual Hilbert {\bf A}-module with
countably generated {\bf A}-pre-dual Hilbert {\bf A}-module is decomposable
into the sum of a diagonalizable normal and of a ''compact'' bounded {\bf
A}-linear operator on that module.


title:
Global Version of Bjornestal's Estimate for Metric Projection Operator
  in Banach Space

abstract:
  In 1979, B.Bjornestal obtained local estimate for a modulus of uniform
continuity of metric projection operator on closed subspace in uniformly convex
and uniformly smooth Banach space $B$. In the present paper we give the global
version of this result for the projection operator on an arbitrary closed
convex set in $B$.


title:
Positive representations of general commutation relations allowing Wick
  ordering

abstract:
  We consider the problem of representing in Hilbert space commutation
relations of the form $$ a_ia_j^*=\delta_{ij}{\bold1} +
\sum_{k\ell}T_{ij}^{k\ell} a_\ell^*a_k \quad,$$ where the $T_{ij}^{k\ell}$ are
essentially arbitrary scalar coefficients. Examples comprise the $q$-canonical
commutation relations introduced by Greenberg, Bozejko, and Speicher, and the
twisted canonical (anti-)commutation relations studied by Pusz and Woronowicz,
as well as the quantum group S$_\nu$U$(2)$. Using these relations, any
polynomial in the generators $a_i$ and their adjoints can uniquely be written
in ``Wick ordered form'' in which all starred generators are to the left of all
unstarred ones. In this general framework we define the Fock representation, as
well as coherent representations. We develop criteria for the natural scalar
product in the associated representation spaces to be positive definite, and
for the relations to have representations by bounded operators in a Hilbert
space. We characterize the relations between the generators $a_i$ (not
involving $a_i^*$) which are compatible with the basic relations. The relations
may also be interpreted as defining a non-commutative differential calculus.
For generic coefficients $T_{ij}^{k\ell}$, however, all differential forms of
degree 2 and higher vanish. We exhibit conditions for this not to be the case,
and relate them to the ideal structure of the Wick algebra, and conditions of
positivity. We show that the differential calculus is compatible with the
involution iff the coefficients $T$ define a representation of the braid group.
This condition is also shown to imply improved bounds for the positivity of the
Fock representation. Finally, we study the KMS states of the group of gauge
transformations defined by $a_j\mapsto\exp(it)a_j$.


title:
The Penalty Method for Variational Inequalities with Nonsmooth Unbounded
  Operators in Banach Space

abstract:
  The existence of a solution, convergence and stability of the penalty method
for variational inequalities with nonsmooth unbounded uniformly and properly
monotone operators in Banach spase $B$ are investigated. All the objects of the
inequality - the operator A, "the right-hand part" $f$ and the set of
constrains $\Omega $ - are to be perturbed. The stability theorems are
formulated in terms of geometric characteristics of the spaces $B$ and $B^*$.
The results of this paper are continuity and generalization of the Lions' ones,
published earlier in \cite{l}. They are new even in Hilbert spaces.


title:
A New Approach to Investigation of Evolution Differential Equations in
  Banach Spaces

abstract:
  Known investigations of nonlinear evolution equations
  $${dx\over dt} + A(t)x(t) = f(t)\ ,\quad x(t_{0}) = x^{0},\ \quad t_{0} \le t
< \infty\ , \eqno(0.1)$$
  with monotone operators $A(t)$ acting from reflexive Banach space $B$ to dual
space $B^*$, usually assume that along with $B$ and $B^*$ there is a Hilbert
space $H$ and continuous imbedding $B \hookrightarrow H$ in the triplet
  $$B \hookrightarrow H \hookrightarrow B^*\ ; \eqno(0.2)$$
  and that $B$ is dense in $H$. The stabilization of solutions of evolution
equations has been proven either in the sense of weak convergence in $B$ or in
the norm of $H$ space, and only asymptotic estimates of stabilization rate have
been obtained [15].
  In the present paper we consider equations of type (0.1) without conditions
(0.2) and establish stabilization with both


title:
Boundedness and Stability of Impulsively Perturbed Delay Differential
  Equations

abstract:
  Suppose any solution of a linear impulsive delay differential equation
  $$ \dot{x} (t) + \sum_{i=1}^m A_i (t) x[h_i (t)] = 0,~t \geq 0, x(s) = 0, s <
0, $$
  $$ x(\tau_j +0) = B_j x(\tau_j -0) + \alpha_j, ~j=1,2, ... ,$$
  is bounded for any bounded sequence $\{ \alpha_i \}$. The conditions ensuring
exponential stability of this equation are presented. The behavior of solutions
of the non-homogeneous equation is analyzed.


title: Group Cohomology, Modular Theory and Space-time Symmetries

abstract:
  The Bisognano-Wichmann property on the geometric behavior of the modular
group of the von Neumann algebras of local observables associated to wedge
regions in Quantum Field Theory is shown to provide an intrinsic sufficient
criterion for the existence of a covariant action of the (universal covering
of) the Poincar\'e group. In particular this gives, together with our previous
results, an intrinsic characterization of positive-energy conformal
pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore
theory of central extensions of locally compact groups by polish groups,
selecting and making an analysis of a wider class of extensions with natural
measurable properties and showing henceforth that the universal covering of the
Poincar\'e group has only trivial central extensions (vanishing of the first
and second order cohomology) within our class.


title:
Infinite-Dimensional Geometry of the Universal Deformation of the
  Complex Disk

abstract:
  The universal deformation of the complex disk is studied from the viewpoint
of infinite-dimensional geometry. The structure of a subsymmetric space on the
universal deformation is described. The foliation of the universal deformation
by subsymmetry mirrors is shown to determine a real polarization.
  The subject of the paper maybe of interest to specialists in algebraic
geometry and representation theory as well as to researchers dealing with
mathematical problems of modern quantum field theory.
  Contents.
  I. The infinite-dimensional geometry of the flag manifold of the
Virasoro-Bott group (the base of the universal deformation of the complex
disk).
  II. The infinite-dimensional geometry of the skeleton of the flag manifold of
the Virasoro-Bott group.
  III. The infinite-dimensional geometry of the universal deformation of the
complex disk.

categories: funct-an alg-geom hep-th math.AG math.OA


title: On Integrable Solutions of Impulsive Delay Differential Equations

abstract:
  The connection of function properties of solutions with exponential stability
of linear impulsive differential equation
  $$\dot{x} (t) - \sum_{k=1}^m {A_k (t) x[h_k(t)]} = r(t),~ t \geq 0, x(\xi ) =
\varphi (\xi),~ \xi < 0,$$
  $$x(\tau_j) = B_j x(\tau_j - 0) , ~j=1,2, \dots .$$
  The explicit stability results and sufficient conditions for existence of
integrable solutions are presented.


title: Order Preservation in Limit Algebras

abstract:
  The matrix units of a digraph algebra, A, induce a relation, known as the
diagonal order, on the projections in a masa in the algebra. Normalizing
partial isometries in A act on these projections by conjugation; they are said
to be order preserving when they respect the diagonal order. Order preserving
embeddings, in turn, are those embeddings which carry order preserving
normalizers to order preserving normalizers. This paper studies operator
algebras which are direct limits of finite dimensional algebras with order
preserving embeddings. We give a complete classification of direct limits of
full triangular matrix algebras with order preserving embeddings. We also
investigate the problem of characterizing algebras with order preserving
embeddings.


title: On Polynomial Relations in the Heisenberg Algebra

abstract:
  Polynomial relations between the generators of the classical and quantum
Heisenberg algebras are presented. Some of those relations can have a meaning
of the formulas of the normal ordering for the creation/annihilation operators
occurred in the method of the second quantization.


title:
Crossed product decompositions of a purely infinite von Neumann algebra
  with faithful, almost periodic weight

abstract:
  For $\MvN$ a separable, purely infinite von Neumann algebra with almost
periodic weight $\phi$, a decomposition of $\MvN$ as a crossed product of a
semifinite von Neumann algebra by a trace--scaling action of a countable
abelian group is given. Then Takasaki's continuous decomposition of the same
algebra is related to the above discrete decomposition via Takesaki's notion of
induced action, but here one induces up from a dense subgroup. The above
results are used to give a model for the one--parameter trace--scaling action
of $\Real_+$ on the injective II$_\infty$ factor. Finally, another model of the
same action, due to work of Aubert and explained by Jones, is described.


title: Regular Lie groups and a theorem of Lie-Palais

abstract:
  In 1984 Milnor had shown how to deduce the Lie-Palais theorem on integration
of infinitesimal actions of finite-dimensional Lie algebras on compact
manifolds from general theory of regular Lie groups modelled on locally convex
spaces. We show how, in the case of effective action, one can eliminate from
Milnor's argument the abstract Lie-Cartan theorem, making the deduction rather
elementary. A machinery employed in the proof provides a partial solution to a
problem examined in 1972 by van Est and \'Swierczkowski.


title: Classification of direct limits of even Cuntz-circle algebras

abstract:
  We prove a classification theorem for purely infinte simple C*-algebras that
is strong enough to show that the tensor products of two different irrational
rotation algebras with the same even Cuntz algebra are isomorphic. In more
detail, let C be be the class of simple C*-algebras A which are direct limits A
= lim A_k, in which each A_k is a finite direct sum of algebras of the form
C(X) \otimes M_n \otimes O_m, where m is even, O_m is the Cuntz algebra, X is
either a point, a compact interval, or the circle S^1, and each map A_k ---> A
is approximately absorbing. ("Approximately absorbing" is defined in Section 1
of the paper.) We show that two unital C*-algebras A and B in the class C are
isomorphic if and only if (K_0 (A), [1_A], K_1 (A)) is isomorphic to (K_0 (B),
[1_B], K_1 (B)). This class is large enough to exhaust all possible K-groups:
if G_0 and G_1 are countable odd torsion groups and g is in G_0, then there is
a C*-algebra A in C with (K_0 (A), [1_A], K_1 (A)) isomorphic to (G_0, g, G_1).
The class C contains the tensor products of irrational rotation algebras with
even Cuntz algebras. It is also closed under several natural operations.


title: Singular Perturbations, Regularization and Extension Theory

abstract:
  We present an abstract result on removing regularization for singular
potentials which are not semibounded from below. The relation between ``right''
regularizations and ``right'' self-adjoint extensions of the perturbed
Schr\"odinger operators is examined.


title:
Twisted Partial Actions, A Classification of Stable C*-Algebraic Bundles
  (Preliminary Version)

abstract:
  We introduce the notion of continuous twisted partial actions of a locally
compact group on a C*-algebra. With such, we construct an associated
C*-algebraic bundle called the semidirect product bundle. Our main theorem
shows that, given any C*-algebraic bundle which is second countable and whose
unit fiber algebra is stable, there is a continuous twisted partial action of
the base group on the unit fiber algebra, whose associated semidirect product
bundle is isomorphic to the given one.


title: No More Than Mechanics. I

abstract:
  One can introduce so-called {\em Plain Mechanics} having an {\bf operator
realization}. Then the set of one-dimension representations of this operator
realization may be identified with the Classical Mechanics. Different
irreducible infinite-dimension representations may be recognized as Quantum
Mechanics for different $\hbar$ (the Planck constant). It can be done in the
such manner that the following diagram will be commutative.
  Plain Mechanics
  / \
  / \
  / \
  Quantum Mechanics --> Classical Mechanics h->0
  Here the horizontal arrow is well known correspondence between Quantum and
Classical Mechanics if Planck constant tensing to zero. A {\em realization} of
this scheme for a particle in $n$-dimensional space by two-sided convolutions
on the Heisenberg group is constructed. We also introduce the {\em motion
equations} for observables in this realization. The left arrow of the given
diagram carries this equation to the Heisenberg one and the right arrow maps it
to the Hamilton equation.


title: Epimorphisms of C*-algebras are surjective

abstract: We answer a question raised by V.G.Pestov in the affirmative.


title:
Homology for operator algebras I: Spectral homology for reflexive
  algebras

abstract:
  A stable homology theory is defined for completely distributive CSL algebras
in terms of the point-neighbourhood homology of the partially ordered set of
meet-irreducible elements of the invariant projection lattice. This specialises
to the simplicial homology of the underlying simplicial complex in the case of
a digraph algebra. These groups are computable and useful. In particular it is
shown that if the first spectral homology group is trivial then Schur
automorphisms are automatically quasispatial. This motivates the introduction
of essential Hochschild cohomology which we define by using the point weak star
closure of coboundaries in place of the usual coboundaries.


title:
Homology for operator algebras II : Stable homology for non-self-adjoint
  algebras

abstract:
  A new homology is defined for a non-self-adjoint operator algebra and
distinguished masa which is based upon cycles and boundaries associated with
complexes of partial isometries in the stable algebra. Under natural hypotheses
the zeroth order group coincides with the $\ K_0$\ group of the generated
C*-algebra. Several identifications and applications are given, and in
particular it is shown how stable homology is significant for the
classification of regular subalgebras and regular limit algebras.


title: Infinite Lexicographic Products of Triangular Algebras

abstract:
  Some new connections are given between linear orderings and triangular
operator algebras. A lexicograhic product is defined for triangular operator
algebras and the Jacobson radical of an infinite lexicographic product of upper
triangular matrix algebras is determined.


title: Lexicographic semigroupoids

abstract:
  The natural lexicographic semigroupoids associated with Cantor product spaces
indexed by countable linear orders are classified. Applications are given to
the classification of triangular operator algebras which are direct limits of
upper triangular matrix algebras.


title:
On the Banach space isomorphism type of AF C*-algebras and their
  triangular subalgebras

abstract:
  It is shown that all the approximately finite dimensional C*-algebras which
are not of Type I are isomorphic as Banach spaces. This generalises the matroid
case given previously by Arazy. Analogous results are obtained for various
families of triangular subalgebras of AF C*-algebras. In addition the
classification of various continua of Type I AF C*-algebras is discussed.


title: Douady's conjecture on Banach analytic spaces

abstract:
  We show that, as conjectured by Adrien Douady back in 1972, every complete
metric space is homeomorphic (moreover, isometric) to the locus of zeros of an
analytic map between two Banach spaces. As a corollary, a paracompact
topological space admits the structure of a Banach analytic space if and only
if it is metrizable with a complete metric.


title: An Algebraic Spin and Statistics Theorem

abstract:
  A spin-statistics theorem and a PCT theorem are obtained in the context of
the superselection sectors in Quantum Field Theory on a 4-dimensional
space-time. Our main assumption is the requirement that the modular groups of
the von Neumann algebras of local observables associated with wedge regions act
geometrically as pure Lorentz transformations. Such a property, satisfied by
the local algebras generated by Wightman fields because of the
Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.


title: Endomorphisms of B(H)

abstract:
  The unital endomorphisms of B(H) of (Powers) index n are classified by
certain U(n)-orbits in the set of non-degenerate representations of the Cuntz
algebra O_n on H. Using this, the corre- sponding conjugacy classes are
identified, and a set of labels is given. This set is given as P modulo a
certain non-smooth equivalence, where P is a set of pure state on the UHF
algebra of Glimm type n^infinity. Several subsets of P, giving concrete
examples of non- conjugate shifts, are worked out in detail, including sets of
product states, and a set of nearest neighbor states.


title:
Completely Positive Maps on Coxeter Groups, Deformed Commutation
  Relations, and Operator Spaces

abstract:
  In this article we prove that quasi-multiplicative (with respect to the usual
length function) mappings on the permutation group $\SSn$ (or, more generally,
on arbitrary amenable Coxeter groups), determined by self-adjoint contractions
fulfilling the braid or Yang-Baxter relations, are completely positive. We
point out the connection of this result with the construction of a Fock
representation of the deformed commutation relations $d_id_j^*-\sum_{r,s}
t_{js}^{ir} d_r^*d_s=\delta_{ij}\id$, where the matrix $t_{js}^{ir}$ is given
by a self-adjoint contraction fulfilling the braid relation. Such deformed
commutation relations give examples for operator spaces as considered by
Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by
$G_i=d_i+d_i^*$, are typically not injective.


title:
Interpolations between Bosonic and Fermionic Relations given by
  Generalized Brownian Motions (revised version)

abstract:
  We present an interpolation between the bosonic and fermionic relations. This
interpolation is given by an object which we call `generalized Brownian motion'
and which is characterized by a generalization of the pairing rule for the
calculation of the moments of bosonic and fermionic fields. We develop some
basic theory for such generalized Brownian motions and consider more closely
one example, which turns out to be intimately connected with Voiculescu's
concept of `free product'.


title: Hilbert C*-modules over monotone complete C*-algebras

abstract:
  The aim of the present paper is to describe self-duality and C*- reflexivity
of Hilbert {\bf A}-modules $\cal M$ over monotone complete C*-algebras {\bf A}
by the completeness of the unit ball of $\cal M$ with respect to two types of
convergence being defined, and by a structural criterion. The derived results
generalize earlier results of {\sc H.~Widom} [Duke Math.~J.~23, 309-324, MR 17
\# 1228] and {\sc W.~L.~Paschke} [Trans. Amer.~Math.~Soc.~182, 443-468, MR 50
\# 8087, Canadian J.~Math.~26, 1272-1280, MR 57 \# 10433]. For Hilbert
C*-modules over commutative AW*-algebras the equivalence of the self-duality
property and of the Kaplansky-Hilbert property is reproved, (cf. {\sc M.~Ozawa}
[J.~Math.~Soc.~Japan 36, 589-609, MR 85m:46068] ). Especially, one derives that
for a C*-algebra {\bf A} the {\bf A}-valued inner pro\-duct of every Hilbert
{\bf A}-module $\cal M$ can be continued to an {\bf A}-valued inner product on
it's {\bf A}-dual Banach {\bf A}-module $\cal M$' turning $\cal M$' to a
self-dual Hilbert {\bf A}-module if and only if {\bf A} is monotone complete
(or, equivalently, additively complete) generalizing a result of {\sc
M.~Hamana} [Internat.~J.~Math.~3(1992), 185-204]. A classification of countably
generated self-dual Hilbert {\bf A}-modules over monotone complete C*-algebras
{\bf A} is established. The set of all bounded module operators ${\bf End}_A
(\cal M)$ on self-dual Hilbert {\bf A}-modules $\cal M$ over monotone complete
C*-algebras {\bf A} is proved again to be a monotone complete


title: Approximate unitary equivalence of homomorphisms from O_{\infty}

abstract:
  We prove that if two homomorphisms from O_{\infty} to a purely infinite
simple C*-algebra have the same class in KK-theory, and if either both are
unital or both are nonunital, then they are approximately unitarily equivalent.
It follows that O_{\infty} is classifiable in the sense of Rordam. In
particular, Rordam's classification theorem for direct limits of matrix
algebras over even Cuntz algebras extends to direct limits involving both
matrix algebras over even Cuntz algebras and corners of O_{\infty}, for which
the K_0-group can be an arbitrary countable abelian group with no even torsion.


title:
On an Application of Hypoellipticity to Solutions of Functional
  Equations

abstract:
  We study the regularity of solutions of functional equations of a generalized
mean value type. In this paper we give sufficient conditions for the regularity
by using hypoellipticity which is a concept of the theory of partial
differential equations. Further we also give an affirmative answer to a
conjecture of H.\'Swiatak. A part of the results was announced in the
comprehensive paper [8] on series of our joint works. To prove the regularity
of solutions of functional equation is in general one of central problem in the
theory of functional equations.


title:
Classical and quantum dynamics of noncanonically coupled oscillators and
  Lie superalgebras

abstract:
  The classical and quantum dynamics of noncanonically coupled os- cillators is
investigated in its relation to Lie superalgebras. It is shown that the quantum
dynamics admits a hidden (super)hamiltonian formulation and, hence, preserves
the initial operator relations.

categories: funct-an chao-dyn math.OA nlin.CD nlin.SI solv-int


title:
The equivariant Brauer groups of commuting free and proper actions are
  isomorphic

abstract:
  If $X$ is a locally compact space which admits commuting free and proper
actions of locally compact groups $G$ and $H$, then the Brauer groups
$\Br_H(G/X)$ and $\Br_G(X/H)$ are naturally isomorphic.


title: Relative Convolutions. I. Properties and Applications

abstract:
  To study operator algebras with symmetries in a wide sense we introduce a
notion of {\em relative convolution operators} induced by a Lie algebra.
Relative convolutions recover many important classes of operators, which have
been already studied (operators of multiplication, usual group convolutions,
two-sided convolution etc.) and their different combinations. Basic properties
of relative convolutions are given and a connection with usual convolutions is
established. Presented examples show that relative convolutions provide us with
a base for systematical applications of harmonic analysis to PDO theory,
complex and hypercomplex analysis, coherent states, wavelet transform and
quantum theory.
  KEYWORDS: Lie groups and algebras, convolution operator, representation
theory, Heisenberg group, integral representations, Hardy space, Szeg\"o
projector, Toeplitz operators, Fock space, Segal--Bargmann space, Bargmann
projector, Dirac equation, Clifford analysis, coherent states, wavelet
transform, quantization.


title: $L^p$-Spaces as Quasi *-Algebras

abstract:
  The Banach space $L^p(X,\mu)$, for $X$ a compact Hausdorff measure space, is
considered as a special kind of quasi *-algebra (called CQ*-algebra) over the
C*-algebra $C(X)$ of continuous functions on $X$. It is shown that, for $p \geq
2$, $(L^p(X,\mu), C(X))$ is *-semisimple (in a generalized sense). Some
consequences of this fact are derived.


title:
Crossed products whose primitive ideal spaces are generalized trivial
  $\widehat G$-bundles

abstract:
  We characterize when the primitive ideal space of a crossed product $\acg$ of
a \cs-algebra $A$ by a locally compact abelian group $G$ is a $\sigma$-trivial
$\ghat G$-space for the dual $\ghat G$-action. Specifically, we show that
$\Prim(\acg)$ is $\sigma$-trivial if and only if the quasi-orbit space is
Hausdorff, the map which assigns to each quasi-orbit $\w$ a certain subgroup
$\ttg(\alpha^\w)$ of the Connes spectrum of the system $(A_\w,G,\alpha^\w)$ is
continuous, and there is a generalized Green twisting map for $(A,G,\alpha)$.
Our proof requires a substantial generalization of a theorem of Olesen and
Pedersen in which we show that there is a generalized Green twisting map for
$(A,G,\alpha)$ if and only if $\acg$ is isomorphic to a generalized induced
algebra.


title: Convolution and Limit Theorems for Conditionally Free Random Variables

abstract:
  We introduce the notion of a conditionally free product and conditionally
free convolution. We describe this convolution both from a combinatorial point
of view, by showing its connection with the lattice of non-crossing partitions,
and from an analytic point of view, by presenting the basic formula for its
$R$-transform. We calculate explicitly the distributions of the conditionally
free Gaussian and conditionally free Poisson distribution.


title:
Landau Hamiltonians with Random Potentials: Localization and the Density
  of States

abstract:
  We prove the existence of localized states at the edges of the bands for the
two-dimensional Landau Hamiltonian with a random potential, of arbitrary
disorder, provided that the magnetic field is sufficiently large. The
corresponding eigenfunctions decay exponentially with the magnetic field and
distance. We also prove that the integrated density of states is Lipschitz
continuous away from the Landau energies. The proof relies on a Wegner estimate
for the finite-area magnetic Hamiltonians with random potentials and
exponential decay estimates for the finite-area Green's functions. The proof of
the decay estimates for the Green's functions uses fundamental results from
two-dimensional bond percolation theory.


title: An Adiabatic Theorem for Singularly Perturbed Hamiltonians

abstract:
  The adiabatic approximation in quantum mechanics is considered in the case
where the self-adjoint hamiltonian $H_0(t)$, satisfying the usual spectral gap
assumption in this context, is perturbed by a term of the form $\epsilon
H_1(t)$. Here $\epsilon \to 0$ is the adiabaticity parameter and $H_1(t)$ is a
self-adjoint operator defined on a smaller domain than the domain of $H_0(t)$.
Thus the total hamiltonian $H_0(t)+\epsilon H_1(t)$ does not necessarily
satisfy the gap assumption, $\forall \epsilon >0$. It is shown that an
adiabatic theorem can be proven in this situation under reasonnable hypotheses.
The problem considered can also be viewed as the study of a time-dependent
system coupled to a time-dependent perturbation, in the limit of large coupling
constant.


title: Existence results for non-coercive variational problems

abstract:
  The aim of this paper is to give an existence result for a class of
one-dimensional, non-convex, non-coercive problems in the Calculus of
Variations. The main tools for the proof are an existence theorem in the convex
case and the closure of the convex hull of the epigraph of functions strictly
convex at infinity.


title: An existence result for non-coercive non-convex problems

abstract:
  We consider a class of one--dimensional non--convex non--coercive problems in
the Calculus of Variations. We prove an existence result for this class of
problems using a Liapunov type theorem on the range of non--atomic measures.


title: Time-Optimal Control Problem for the Swing and the Ski

abstract:
  This paper is concerned with a class of time-optimal control problems for the
swing and the ski. We first consider the motion of a man standing on a swing.
For simplicity, we neglect friction and air resistance and assume that the mass
of the swinger is concentrated in his baricenter $B$. For the ski, we use the
model considered by Aldo Bressan (1991), with a special approximation of the
skier. Our main concern is the existence and the structure of time-optimal
controls. As first step, we introduce an auxiliary control system with control
entering linearly and show the relationships between this and the original
system. We then perform a detailed study of the auxiliary system, using
geometric techniques. In turn this provides accurate information on the
time-optimal controls for the original system.


title:
Determinants of elliptic boundary problems for Dirac operators I. Local
  boundary conditions

abstract:
  We study functional determinants for Dirac operators on manifolds with
boundary and discuss the ellipticity of boundary problems by using the
Calder\'on projector. We give, for local boundary conditions, an explicit
formula relating these determinants to the corresponding Green functions. We
finally apply this result to the case of a bidimensional disk under bag-like
conditions.


title: Path spaces, continuous tensor products, and E_0 semigroups

abstract:
  We classify all continuous tensor product systems of Hilbert spaces which are
``infinitely divisible" in the sense that they have an associated logarithmic
structure. These results are applied to the theory of E_0 semigroups to deduce
that every E_0 semigroup which possesses sufficiently many ``decomposable"
operators must be cocycle conjugate to a CCR flow. A *path space* is an
abstraction of the set of paths in a topological space, on which there is given
an associative rule of concatenation. A metric path space is a pair (P,g)
consisting of a path space P and a function g:P^2 --> complex numbers which
behaves as if it were the logarithm of a multiplicative inner product. The
logarithmic structures associated with infinitely divisible product systems are
such objects. The preceding results are based on a classification of metric
path spaces.


title:
Characteristics of pairs of operators, Lie hybrids, Poisson brackets and
  nonlinear geometric algebra

abstract:
  Some algebraic, geometric and geometroalgebraic characteristics of pairs of
operators are discussed.


title: Diagonalizing ''compact'' operators on Hilbert W*-modules

abstract:
  For W*-algebras A and self-dual Hilbert A-modules M we show that every
self-adjoint, ''compact'' module operator on M is diagonalizable. Some specific
properties of the eigenvalues and of the eigenvectors are described.


title: Berezin's quantization on flag manifolds and spherical modules

abstract:
  It is shown that the theory of spherical Harish-Chandra modules naturally
provides the algebras of covariant, contravariant and mixed symbols on
generalized flag manifolds. The general proof of the correspondence principle
for all these symbol algebras is given.


title: An Equivariant Brauer Group and Actions of Groups on C*-algebras

abstract:
  Suppose that $(G,T)$ is a second countable locally compact transformation
group given by a homomorphism $\ell:G\to\Homeo(T)$, and that $A$ is a separable
continuous-trace \cs-algebra with spectrum $T$. An action $\alpha:G\to\Aut(A)$
is said to cover $\ell$ if the induced action of $G$ on $T$ coincides with the
original one. We prove that the set $\brgt$ of Morita equivalence classes of
such systems forms a group with multiplication given by the balanced tensor
product: $[A,\alpha][B,\beta] = [A\Ttensor B,\alpha\tensor\beta]$, and we refer
to $\brgt$ as the Equivariant Brauer Group. We give a detailed analysis of the
structure of $\brgt$ in terms of the Moore cohomology of the group $G$ and the
integral cohomology of the space $T$. Using this, we can characterize the
stable continuous-trace \cs-algebras with spectrum $T$ which admit actions
covering $\ell$. In particular, we prove that if $G=\R$, then every stable
continuous-trace \cs-algebra admits an (essentially unique) action
covering~$\ell$, thereby substantially improving results of Raeburn and
Rosenberg. Versions of this paper in *.dvi and *.ps form are available via
World wide web servers at http://coos.dartmouth.edu/~dana/dana.html


title: On dense subspaces in a class of Fr\'echet function spaces on R^n

abstract:
  When dealing with concrete problems in a function space on R^n, it is
sometimes helpful to have a dense subspace consisting of functions of a
particular type, adapted to the problem under consideration. We give a theorem
that allows one to write down many of such subspaces in commonly occurring
Fr\'echet function spaces. These subspaces are all of the form $\{pf_0 |
p\in{\cal P}\}$ where $f_0$ is a fixed function and ${\cal P}$ is an algebra of
functions. Classical results like the Stone-Weierstrass theorem for polynomials
and the completeness of the Hermite functions are related by this theorem.


title:
Diagonalization of compact operators in Hilbert modules over finite
  W*-algebras

abstract:
  It is known that a continuous family of compact operators can be diagonalized
pointwise. One can consider this fact as a possibility of diagonalization of
the compact operators in Hilbert modules over a commutative W*-algebra. The aim
of the present paper is to generalize this fact for a finite W*-algebra $A$ not
necessarily commutative. We prove that for a compact operator $K$ acting in the
right Hilbert $A$-module $H^*_A$ dual to $H_A$ under slight restrictions one
can find a set of "eigenvectors" $x_i\in H^*_A$ and a non-increasing sequence
of "eigenvalues" $\lambda_i\in A$ such that $K\,x_i = x_i\,\lambda_i$ and the
autodual Hilbert $A$-module generated by these "eigenvectors" is the whole
$H_A^*$. As an application we consider the Schr\"odinger operator in magnetic
field with irrational magnetic flow as an operator acting in a Hilbert module
over the irrational rotation algebra $A_{\theta}$ and discuss the possibility
of its diagonalization.


title: Lie-algebraic discretization of differential equations

abstract:
  A certain representation for the Heisenberg algebra in finite-difference
operators is established. The Lie-algebraic procedure of discretization of
differential equations with isospectral property is proposed. Using
$sl_2$-algebra based approach, (quasi)-exactly-solvable finite-difference
equations are described. It is shown that the operators having the Hahn,
Charlier and Meixner polynomials as the eigenfunctions are reproduced in
present approach as some particular cases. A discrete version of the classical
orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is
introduced.


title: Connection between Different Function Theories in Clifford Analysis

abstract:
  We describe an explicit connection between solutions to equations $Df=0$ (the
Generalized Cauchy-Riemann equation) and $(D+M)f=0$, where operators $D$ and
$M$ commute. The described connection allows to construct a ``function theory''
(the Cauchy theorem, the Cauchy integral, the Taylor and Laurent series etc.)
for solutions of the second equation from the known function theory for
solution of the first (generalized Cauchy-Riemann) equation. As well known,
many physical equations related to the orthogonal group of rotations or the
Lorentz group (the Dirac equation, the Maxwell equation etc.) can be naturally
formulated in terms of the Clifford algebra. For them our approach gives an
explicit connection between solutions with zero and non-zero mass (or external
fields) and provides with a family of formulas for calculations.
\keywords{Dirac equation with mass, Clifford analysis.} \AMSMSC{30G35}{34L40,
81Q05}


title: Estimates of Lebesgue constants via Fourier transforms. Many dimensions

abstract:
  This is an attempt of a comprehensive survey of the results in which
estimates of the norms of linear means of multiple Fourier series, the Lebesgue
constants, are obtained by means of estimating the Fourier transform of a
function generating such a method. Only few proofs are given in order to
illustrate a general idea of techniques applied. Among the results are well
known elsewhere as well as less known or published in an unacceptable journals
and several new unpublished results.


title: Lusin's C-property is not valid for functional Hilbert modules

abstract:
  We show that elements of Hilbert $A$-module obtained by completion of the
space of square-integrable functions on a space with measure $X$ taking values
in a $C^*$-algebra $A$ cannot be viewed as $A$-valued functions on $X$ defined
almost everywhere


title: Capacity theory for monotone operators

abstract:
  If $Au=-div(a(x,Du))$ is a monotone operator defined on the Sobolev space
$W^{1,p}(R^n)$, $1<p<+\infty$, with $a(x,0)=0$ for a.e. $x\in R^n$, the
capacity $C_A(E,F)$ relative to $A$ can be defined for every pair $(E,F)$ of
bounded sets in $R^n$ with $E\subset F$. We prove that $C_A(E,F)$ is increasing
and countably subadditive with respect to $E$ and decreasing with respect to
$F$. Moreover we investigate the continuity properties of $C_A(E,F)$ with
respect to $E$ and $F$.


title: Toward a general theory of transmutation

abstract:
  A general construction of transmutation operators is developed for
selfadjoint operators in Gelfand triples. Theorems regarding analyticity of
generalized eigenfunctions and Paley-Wiener properties are proved.


title: Lefschetz Numbers and Geometry of Operators in W*-modules

abstract:
  The main goal of the present paper is to generalize the results
of~\cite{TroLNM,TroBoch} in the following way: To be able to define
$K_0(A)\o\C$-valued Lefschetz numbers of the first type of an endomorphism $V$
on a C*-elliptic complex one usually assumes that $V=T_g$ for some
representation $T_g$ of a compact group $G$ on the C*-elliptic complex. We try
to refuse this restriction in the present paper. The price to pay for this is
twofold:
  (i) $ $ We have to define Lefschetz numbers valued in some larger group as
$K_0(A)\o\C$.
  (ii) We have to deal with W*-algebras instead of general unital C*-algebras.
  To obtain these results we have got a number of by-product facts on the
theory of Hilbert W*- and C*-modules and on bounded module operators on them
which are of independent interest.


title:
Diagonalization of compact operators in Hilbert modules over C*-algebras
  of real rank zero

abstract:
  It is known that the classical Hilbert--Schmidt theorem can be generalized to
the case of compact operators in Hilbert $A$-modules $H_A^*$ over a
$W^*$-algebra of finite type, i.e. compact operators in $H_A^*$ under slight
restrictions can be diagonalized over $A$. We show that if $B$ is a weakly
dense $C^*$-subalgebra of real rank zero in $A$ with some additional property
then the natural extension of a compact operator from $H_B$ to $H_A^*\supset
H_B$ can be diagonalized with diagonal entries being from the $C^*$-algebra
$B$.


title:
On the $\Gamma$-equivariant form of the Berezin's quantization of the
  upper half plane

abstract:
  Let $\Gamma$ be a fuchsian subgroup of \pslr. In this paper we consider the
$\Gamma$-equivariant form of the Berezin's quantization of the upper half plane
which will correspond to a deformation quantization of the (singular) space
$\Bbb H/\Gamma$. Our main result is that the von Neumann algebra associated to
the $\Gamma-$ equivariant form of the quantization is stable isomorphic with
the von Neumann algebra associated to $\Gamma$.


title: Oscillation of a Linear Delay Impulsive Differential Equation

abstract:
  The main result of the paper is that the oscillation (non-oscillation) of the
impulsive delay differential equation $\dot {x}(t)+\sum_{k=1}^m
A_k(t)x[h_k(t)]=0,~~t\geq 0$, $x(\tau_j)=B_jx(\tau_j-0), \lim \tau_j = \infty$
is equivalent to the oscillation (non-oscillation) of the equation without
impulses $\dot {x}(t)=\sum_{k=1}^m A_k(t) \prod_{h_k(t)<\tau_j\leq t}
B_j^{-1}x[h_k(t)]=0, t \geq 0$. Explicit oscillation results are presented.


title: Uniqueness theorem for unbounded domain

abstract:
  We prove in this paper the uniqueness theorem for a certain class of harmonic
functions defined in unbounded domain lying in a band.


title: On the discrete spectrum of non-analytic matrix-valued Friedrichs model

abstract:
  We have found the sufficient conditions for the spectrum of matrix-valued
Friedrichs model to be finite.


title: Impulsive Stabilization of Linear Delay Differential Equations

abstract:
  The paper is concerned with stabilization of a scalar delay differention
equation $$ {\dot x}(t) - \sum_{k=1}^m A_k(t)x[h_k(t)] = 0,~t\geq 0,~
x(\xi)=\varphi (\xi), \xi <0, $$ by introducing impulses in certain moments of
time $$ x(\tau_j) = B_j x(\tau_j -0), ~j=1,2, \dots ~. $$ Explicit stability
results are presented both for the equation with positive coefficients and for
the equation with $A_k$ being of arbitrary sign.


title:
The Riesz-Clifford Functional Calculus for Non-Commuting Operators and
  Quantum Field Theory

abstract:
  We present a Riesz-like hyperholomorphic functional calculus for a set of
non-commuting operators based on the Clifford analysis. Applications to the
quantum field theory are described. Keywords: Functional calculus, Weyl
calculus, Riesz calculus, Clifford analysis, quantization, quantum field
theory. AMSMSC Primary:47A60, Secondary: 81T10


title: Some Conditions for Existence and Integrability of the Fourier Transform

abstract:
  The Fourier transform is naturally defined for integrable functrions.
Otherwise, it should be stipulated in which sense the Fourier transform is
understood. We consider some class of radial and, generally saying,
nonintegrable functions. The Fourier transform is calculated as an improper
integral and the limit coincides with the Fourier transform in the
distributional sense. The inverse Fourier formula is proved as well. Given are
some applications of the result obtained.


title: Unconditional Integrability for Dual Actions

abstract:
  The dual action of a locally compact abelian group, in the context of
C*-algebraic bundles, is shown to satisfy an integrability property, similar to
Rieffel's proper actions. The tools developed include a generalization of
Bochner's integral as well as a Fourier inversion formula for operator valued
maps.


title:
Isomorphisms of Hilbert C*-Modules and $*$-Isomorphisms of Related
  Operator C*-Algebras

abstract:
  Let $\cal M$ be a Banach C*-module over a C*-algebra $A$ carrying two
$A$-valued inner products $< .,. >_1$, $<.,. >_2$ which induce equivalent to
the given one norms on $\cal M$. Then the appropriate unital C*-algebras of
adjointable bounded $A$-linear operators on the Hilbert $A$-modules $\{ {\cal
M}, < .,. >_1 \}$ and $\{ {\cal M}, < .,. >_2 \}$ are shown to be
$*$-isomorphic if and only if there exists a bounded $A$-linear isomorphism $S$
of these two Hilbert $A$-modules satisfying the identity $< .,. >_2 \equiv <
S(.),S(.) >_1$. This result extends other equivalent descriptions due to
L.~G.~Brown, H.~Lin and E.~C.~Lance. An example of two non-isomorphic Hilbert
C*-modules with $*$-isomorphic C*-algebras of ''compact''/adjointable bounded
module operators is indicated.


title: Endomorphism Semigroups and Lightlike Translations

abstract:
  Borchers and Wiesbrock have demonstrated certain results concerning the
one-parameter semigroups of endomorphisms of von Neumann algebras that appear
as lightlike translations in the theory of algebras of local observables. These
results are abstracted and analyzed as essentially operator-theoretic. Criteria
are then demonstrated for a spatial derivation of a von Neumann algebra to
generate a one-parameter semigroup of endomorphisms. All this is combined to
establish a von Neumann-algebraic converse to the Borchers-Wiesbrock results.
This analysis is then applied to questions of isotony and covariance for local
algebras.


title: On the C*-envelope of approximately finite-dimensional operator algebras

abstract:
  The C*-envelope of the limit algebra (or limit space) of a contractive
regular system of digraph algebras (or digraph spaces) is shown to be an
approximately finite C*-algebra and the direct system for the C*-envelope is
determined explicitly.


title: A remark on trace properties of K-cycles

abstract:
  In this paper we discuss trace properties of $d^+$-summable $K$-cycles
considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a
trace theorem on the algebra $\A$ of a $K$--cycle stated in [\rfr(Conn4)],
namely we show that a natural functional on $\A$ is a trace functional. Then we
discuss whether this functional gives a trace on the whole universal graded
differential algebra $\Q(\A)$. On the one hand we prove that the regularity
conditions on $K$-cycles considered in [\rfr(Conn4)] imply the trace property
on $\Q(\A)$. On the other hand, by constructing an explicit counterexample, we
remark that the sole $K$-cycle assumption is not sufficient for such a property
to hold.


title: Continuous Fell Bundles Associated to Measurable Twisted Actions

abstract:
  Given a measurable twisted action of a second-countable, locally compact
group G on a separable C*-algebra A, we prove the existence of a topology on
AxG making it a continuous bundle, whose cross sectional C*-algebra is
isomorphic to the Busby--Smith--Packer--Raeburn crossed product.


title: A Note on the Wodzicki Residue

abstract:
  In this note we explain the relationship of the Wodzicki residue of (certain
powers of) an elliptic differential operator $P$\ acting on sections of a
complex vector bundle $E$\ over a closed compact manifold $M$\ and the
asymptotic expansion of the trace of the corresponding heat operator $e^{-tP}$.
In the special case of a generalized laplacian $\triangle$\ and ${{\rm dim}\;M
> 2}$, we thereby obtain a simple proof of the fact already shown in [KW], that
the Wodzicki residue $res(\triangle^{-{n\over 2}+1} )$\ is the integral of the
second coefficient of the heat kernel expansion of $\triangle$\ up to a
proportional factor.


title: C*-Algebras with the Approximate Positive Factorization Property

abstract:
  We say that a unital C*-algrebra A has the approximate positive factorization
property (APFP) if every element of A is a norm limit of products of positive
elements of A. (There is also a definition for the nonunital case.) T. Quinn
has recently shown that a unital AF algebra has the APFP if and only if it has
no finite dimensional quotients. This paper is a more systematic investigation
of C*-algebras with the APFP. We prove various properties of such algebras. For
example: They have connected invertible group, trivial K_1, and stable rank 1.
In the unital case, the K_0 group separates the tracial states. The APFP passes
to matrix algebras. and if I is an ideal in A such that I and A/I have the
APFP, then so does A. We also give some new examples of C*-algebras with the
APFP, including type II_1 factors and infinite-dimensional simple unital direct
limits with slow dimension growth, real rank zero, and trivial K_1 group. An
infinite- dimensional simple unital direct limit with slow dimension growth and
with the APFP must have real rank zero, but we also give examples of unital
algebras with the APFP which do not have real rank zero.
  Our analysis also leads to the introduction of a new concept of rank for a
C*-algebra that may be of interest in the future.


title: Approximate Unitary Equivalence of Homomorphisms from O_infinity

abstract:
  We prove that if two nonzero homomorphisms from the Cuntz algebra O_infinity
to a purely infinite simple C*-algebra have the same class in KK-theory, and if
either both are unital or both are nonunital, then they are approximately
unitarily equivalent. It follows that O_infinity is classifiable in the sense
of Rordam. In particular, Rordam's classification theorem for direct limits of
matrix algebras over even Cuntz algebras extends to direct limits involving
both matrix algebras over even Cuntz algebras and corners of O_infinity for
which the K_0 group can be an arbitrary countable abelian group with no even
torsion.


title: A space of multipliers on L

abstract:
  Conditions for a function (number sequence) to be a multiplier on the space
of integrable functions on $\Bbb R$ ($\Bbb T$) are given. This generalizes
recent results of Giang and Moricz.


title: A Classification Theorem for Nuclear Purely Infinite Simple C*-Algebras

abstract:
  Starting from Kirchberg's theorems announced in 1994, namely O_2 tensor A is
isomorphic to O_2 for separable unital nuclear simple A and O_infinity tensor A
is isomorphic to A if in addition A is purely infinite, we prove that
KK-equivalence implies isomorphism for nonunital separable nuclear purely
infinite simple C*-algebras. It follows that if A and B are unital separable
nuclear purely infinite simple C*-algebras which satisfy the Universal
Coefficient Theorem, and if there is a graded isomorphism from K_* (A) to K_*
(B) which preserves the class of the identity, then A is isomorphic to B.
  Our main technical results are, we believe, of independent interest. We say
that two asymptotic morphisms t ---> \phi_t and t ---> \psi_t from A to B are
asymptotically unitarily equivalent if there exists a continuous unitary path t
---> u_t in the unitization B^+ such that || u_t \phi_t (a) u_t^* - \psi_t (a)
|| ---> 0 for all a in A. Let A be separable, nuclear, unital, and simple, and
let D be unital. We prove that any asymptotic morphism from A to K tensor
O_infinity tensor D is asymptotically unitarily equivalent to a homomorphism,
and two homotopic homomorphisms from A to K tensor O_infinity tensor D are
asymptotically unitarily equivalent.


title:
Dynamical inverse problem of representation theory and noncommutative
  geometry

abstract:
  Dynamical inverse problem of representation theory, which has its origin in a
classical paper of E.P.Wigner on a determination of commutation relations of
quantum mechanical quantities by the quantum dynamical equations, is
illustrated on the simplest examples.


title:
The classical limit of dynamics for spaces quantized by an action of
  $R^d$

abstract:
  We have previously shown how to construct a deformation quantization of any
locally compact space on which a vector group acts. Within this framework we
show here that, for a natural class of Hamiltonians, the quantum evolutions
will have the classical evolution as their classical limit.


title: The General Analytic Solution of a Functional Equation of Addition Type

abstract:
  The general analytic solution to the functional equation $$ \phi_1(x+y)= { {
\biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over {
\biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is
characterised. Up to the action of the symmetry group, this is described in
terms of Weierstrass elliptic functions. We illustrate our theory by applying
it to the classical addition theorems of the Jacobi elliptic functions and the
functional equations $$ \phi_1(x+y)=\phi_4(x)\phi_5(y)+\phi_4(y)\phi_5(x) $$
and \[ \Psi _1(x+y)=\Psi _2(x+y) \phi_2(x)\phi_3(y) +\Psi_3(x+y)
\phi_4(x)\phi_5(y). \]


title: Contact interactions on graph superlattices

abstract:
  We consider a quantum mechanical particle living on a graph and discuss the
behaviour of its wavefunction at graph vertices. In addition to the standard
(or delta type) boundary conditions with continuous wavefunctions, we
investigate two types of a singular coupling which are analogous to the delta'
interaction and its symmetrized version for particle on a line. We show that
these couplings can be used to model graph superlattices in which point
junctions are replaced by complicated geometric scatterers. We also discuss the
band spectra for rectangular lattices with the mentioned couplings. We show
that they roughly correspond to their Kronig-Penney analogues: the delta'
lattices have bands whose widths are asymptotically bounded and do not approach
zero, while the delta lattice gap widths are bounded. However, if the
lattice-spacing ratio is an irrational number badly approximable by rationals,
and the delta coupling constant is small enough, the delta lattice has no ggaps
above the threshold of the spectrum. On the other hand, infinitely many gaps
emerge above a critical value of the coupling constant; for almost all ratios
this value is zero.


title:
A duality between Schroedinger operators on graphs and certain Jacobi
  matrices

abstract:
  The known correspondence between the Kronig-Penney model and certain Jacobi
matrices is extended to a wide class of Schroedinger operators on graphs.
Examples include rectangular lattices with and without a magnetic field, or
comb-shaped graphs leading to a Maryland-type model.


title:
Representations for the Three-Body T-Matrix, Scattering Matrices and
  Resolvent in Unphysical Energy Sheets

abstract:
  Explicit representations for the Faddeev components of the three-body
T-matrix continued analytically into unphysical sheets of the energy Riemann
surface are formulated. According to the representations, the T-matrix in
unphysical sheets is explicitly expressed in terms of its components taken in
the physical sheet only. The representations for the T-matrix are then used to
construct similar representations for the analytic continuation of the
three-body scattering matrices and the resolvent. Domains on unphysical sheets
are described where the representations obtained can be applied. A method for
finding three-body resonances based on the Faddeev differential equations is
proposed.


title: Virtual Extensions

abstract:
  A process of extending sets which can be used as foundation for an
alternative organization for Differential and Integral Calculus is presented.


title: Virtual Calculus - Part I

abstract:
  An alternative organization for Differential and Integral Calculus, based on
an extension of real numbers that include infinitesimal and infinite
quantities, is presented. Only Elementary Set Theory is used, without reference
to methods or results from Mathematical Logic.


title: Virtual Calculus - Part II

abstract:
  A simultaneous extension of real numbers set and the class of real functions
is discussed.


title: On Dirac's Delta Calculus

abstract:
  It is shown that theories already presented as rigorous mathematical
formalizations of widespread manipulations of Dirac's delta function are all
unsatisfactory, and a new alternative is proposed.


title: Invariant measures for unitary forms of Kac-Moody groups, Parts I-III

abstract:
  The purpose of this paper is to describe some conjectures and results on the
existence and uniqueness of invariant measures on formal completions of
Kac-Moody groups and associated homogeneous spaces. Existence is rigorously
established in all affine type cases.


title: Some directional microlocal classes defined using wavelet transforms

abstract:
  In this short paper we discuss how the position - scale half-space of wavelet
analysis may be cut into different regions. We discuss conditions under which
they are independent in the sense that the T\"oplitz operators associated with
their characteristic functions commute modulo smoothing operators. This shall
be used to define microlocal classes of distributions having a well defined
behavior along lines in wavelet space. This allows us the description of
singular and regular directions in distributions. As an application we discuss
elliptic regularity for these microlocal classes for domains with cusp-like
singularities.


title:
Some estimates for the Banach space norms in the von Neumann algebras
  associated with the Berezin's quantization of compact Riemann

abstract:
  Let $\G$ be any cocompact, discrete subgroup of $\pslr$. In this paper we
find estimates for the predual and the uniform Banach space norms in the von
Neumann algebras associated with the Berezin' s quantization of a compact
Riemann surface $\Bbb D/\G$. As a corollary, for large values of the
deformation parameter $1/h$, these von Neumann algebras are isomorphic. Using
the results in [AS], [AC], [GHJ] on the von Neumann dimension of the Hilbert
spaces in the discrete series of unitary representations of $PSL(2,\Bbb R)$, as
left modules over $\Gamma$ we deduce that the fundamental group ([MvN]) of the
von Neumann $\Cal L(\Gamma)$ contains the positive rational numbers.
Equivalently, this proves that the algebras $\Cal L(\Gamma)\otimes M_n(\Bbb
C)$, are isomorphic for all $n$.


title: An obstruction for q-deformation of the convolution product

abstract:
  We consider two independent q-Gaussian random variables X and Y and a
function f chosen in such a way that f(X) and X have the same distribution. For
0 < q < 1 we find that at least the fourth moments of X + Y and f(X) + Y are
different. We conclude that no q-deformed convolution product can exist for
functions of independent q-Gaussian random variables.


title: Partial actions of groups and actions of inverse semigroups

abstract:
  Given a group G, we construct, in a canonical way, an inverse semigroup S(G)
associated to G. The actions of S(G) are shown to be in one-to-one
correspondence with the partial actions of G, both in the case of actions on a
set, and that of actions as operators on a Hilbert space. In other words, G and
S(G) have the same representation theory.
  We show that S(G) governs the subsemigroup of all closed linear subspaces of
a G-graded C*-algebra, generated by the grading subspaces. In the special case
of finite groups, the maximum number of such subspaces is computed.
  A ``partial'' version of the group C*-algebra of a discrete group is
introduced. While the usual group C*-algebra of finite commutative groups
forgets everything but the order of the group, we show that the partial group
C*-algebra of the two commutative groups of order four, namely Z/4 and Z/2+Z/2,
are not isomorphic.


title: Weakly coupled states on branching graphs

abstract:
  We consider a Schr\"odinger particle on a graph consisting of $\,N\,$ links
joined at a single point. Each link supports a real locally integrable
potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type
boundary condition at the vertex. If all the links are semiinfinite and ideally
coupled, the potential decays as $\,x^{-1-\epsilon}$ along each of them, is
non--repulsive in the mean and weak enough, the corresponding Schr\"odinger
operator has a single negative eigenvalue; we find its asymptotic behavior. We
also derive a bound on the number of bound states and explain how the
$\,\delta\,$ coupling constant may be interpreted in terms of a family of
squeezed potentials.

categories: funct-an math-ph math.FA math.MP math.QA quant-ph


title: Hidden $sl_2$-algebra of finite-difference equations

abstract:
  The connection between polynomial solutions of finite-difference equations
and finite-dimensional representations of the $sl_2$-algebra is established
(the talk given at the Wigner Symposium, Guadalajara, Mexico, August 1995, to
be published in the Proceedings)

categories: funct-an hep-th math.OA math.QA nlin.SI q-alg solv-int


title: Noncommutative flows I: dynamical invariants

abstract:
  We show that a noncommutative dynamical system of the type that occurs in
quantum theory can often be associated with a dynamical principle; that is, an
infinitesimal structure that completely determines the dynamics. The nature of
these dynamical principles is similar to that of the second order differential
equations of classical mechanics, in that one can locate a space of momentum
operators, a ``Riemannian metric", and a potential. These structures are
classified in terms of geometric objects which, in the simplest cases, occur in
finite dimensional matrix algebras. As a consequence, we obtain a new
classification of E_0-semigroups acting on type I factors.


title: Minimal E_0-semigroups

abstract:
  It is known that every semigroup of normal completely positive maps of a von
Neumann can be ``dilated" in a particular way to an E_0-semigroup acting on a
larger von Neumann algebra. The E_0-semigroup is not uniquely determined by the
completely positive semigroup; however, it is unique (up to conjugacy) provided
that certain conditions of {\it minimality} are met. Minimality is a subtle
property, and it is often not obvious if it is satisfied for specific examples
even in the simplest case where the von Neumann algebra is $\Cal B(H)$. In this
paper we clarify these issues by giving a new characterization of minimality in
terms projective cocycles and their limits. Our results are valid for
semigroups of endomorphisms acting on arbitrary von Neumann algebras with
separable predual.


title: The topology of ideals in some triangular AF algebras

abstract:
  A set of meet-irreducible ideals is described for a class of maximal
triangular almost finite algebras. This set forms a topological space under the
hull-kernel closure, and there is a one-to-one correspondence between closed
sets in this space and ideals in the AF algebra. Some connections with nest
representations and nest-primitive ideals are also described.


title: Bound states in a locally deformed waveguide: the critical case

abstract:
  We consider the Dirichlet Laplacian for a strip in $\,\R^2$ with one straight
boundary and a width $\,a(1+\lambda f(x))\,$, where $\,f\,$ is a smooth
function of a compact support with a length $\,2b\,$. We show that in the
critical case, $\,\int_{-b}^b f(x)\, dx=0\,$, the operator has no bound states
for small $\,|\lambda|\,$ if $\,b<(\sqrt{3}/4)a\,$. On the other hand, a weakly
bound state exists provided $\,\|f'\|< 1.56 a^{-1}\|f\|\,$; in that case there
are positive $\,c_1, c_2\,$ such that the corresponding eigenvalue satisfies
$\,-c_1\lambda^4\le \epsilon(\lambda)- (\pi/a)^2 \le -c_2\lambda^4\,$ for all
$\,|\lambda|\,$ sufficiently small.

categories: funct-an cond-mat math-ph math.FA math.MP quant-ph


title:
Free products of finite dimensional and other von Neumann algebras with
  respect to non-tracial states

abstract:
  The von Neumann algebra free product of arbitary finite dimensional von
Neumann algebras with respect to arbitrary faithful states, at least one of
which is not a trace, is found to be a type~III factor possibly direct sum a
finite dimensional algebra. The free product state on the type~III factor is
what we call an extremal almost periodic state, and has centralizer isomorphic
to $L(\freeF_\infty)$. This allows further classification the type~III factor
and provides another construction of full type~III$_1$ factors having arbitrary
$\Sd$~invariant of Connes. The free products considered in this paper are not
limited to free products of finite dimensional algebras, but can be of a quite
general form.


title:
Purely infinite simple $C^*$-algebras arising from free product
  constructions

abstract:
  Examples of simple, separable, unital, purely infinite $C^*$--algebras are
constructed, including:
  (1) some that are not approximately divisible;
  (2) those that arise as crossed products of any of a certain class of
$C^*$--algebras by any of a certain class of non--unital endomorphisms;
  (3) those that arise as reduced free products of pairs of $C^*$--algebras
with respect to any from a certain class of states.


title:
Asymptotic estimates for bound states in quantum waveguides coupled
  laterally through a narrow window

abstract:
  Consider the Laplacian in a straight planar strip of width $\,d\,$, with the
Neumann boundary condition at a segment of length $\,2a\,$ of one of the
boundaries, and Dirichlet otherwise. For small enough $\,a\,$ this operator has
a single eigenvalue $\,\epsilon(a)\,$; we show that there are positive
$\,c_1,c_2\,$ such that $\,-c_1 a^4 \le \epsilon(a)- \left(\pi/ d\right)^2 \le
-c_2 a^4\,$. An analogous conclusion holds for a pair of Dirichlet strips, of
generally different widths, with a window of length $\,2a\,$ in the common
boundary.


title:
Complex structure and solutions of classical nonlinear equation with the
  interaction $u^4_4$

abstract:
  We consider the (real) nonlinear wave equation $$\Box u + m^2 u + \lambda u^3
= 0, \quad m > 0, \quad \lambda > 0, $$ on four-\-dimensional Minkowski space.
We introduce the complex structure and show that the (nonlinear) operator of
dynamics, the wave and scattering operators define complex analytic maps on the
space of initial Cauchy data with finite energy. In other words, let
$R(\varphi, \pi) = \varphi + i\mu^{-1}\pi$ be the map of initial data on the
positive frequency part of the solution of the free Klein-\-Gordon equation
with these initial data. The operators $RU(t)R^{-1},$ $RWR^{-1},$ and
$RSR^{-1}$ are defined correctly and are complex analytic on the complex
Hilbert space $H^1({\R}^3,\C).$


title: Imprimitivity for $C^*$-Coactions of Non-Amenable Groups

abstract:
  We give a condition on a full coaction $(A,G,\delta)$ of a (possibly)
nonamenable group $G$ and a closed normal subgroup $N$ of $G$ which ensures
that Mansfield imprimitivity works; i.e. that $A\times_{\delta{\vert}} G/N$ is
Morita equivalent to $A\times_\delta G\times_{\deltahat,r} N$. This condition
obtains if $N$ is amenable or $\delta$ is normal. It is preserved under Morita
equivalence, inflation of coactions, the stabilization trick of Echterhoff and
Raeburn, and on passing to twisted coactions.


title: Duality of Restriction and Induction for $C^*$-Coactions

abstract:
  Consider a coaction $\delta$ of a locally compact group $G$ on a \cstar
algebra $A$, and a closed normal subgroup $N$ of $G$. We prove, following
results of Echterhoff for abelian $G$, that Mansfield's imprimitivity between
$A\times_{\delta|}G/N$ and $A\times_\delta G\times_{\deltahat,r}N$ implements
equivalences between Mansfield induction of representations from $A\times G/N$
to $A\times G$ and restriction of representations from $A\times G\times_r N$ to
$A\times G$, and between restriction of representations from $A\times G$ to
$A\times G/N$ and Green induction of representations from $A\times G$ to
$A\times G\times_r N$. This allows us to deduce properties of Mansfield
induction from the known theory of ordinary crossed products.


title: A new perspective on Functional Integration

abstract:
  The core of this article is a general theorem with a large number of
specializations. Given a manifold $N$ and a finite number of one-parameter
groups of point transformations on $N$ with generators $Y, X_{(1)}, \cdots,
X_{(d)} $, we obtain, via functional integration over spaces of pointed paths
on $N$ (paths with one fixed point), a one-parameter group of functional
operators acting on tensor or spinor fields on $N$. The generator of this group
is a quadratic form in the Lie derivatives $\La_{X_{(\a)}}$ in the
$X_{(\a)}$-direction plus a term linear in $\La_Y$.
  The basic functional integral is over $L^{2,1}$ paths $x: {\bf T} \ra N$
(continuous paths with square integrable first derivative). Although the
integrator is invariant under time translation, the integral is powerful enough
to be used for systems which are not time translation invariant. We give seven
non trivial applications of the basic formula, and we compute its semiclassical
expansion.
  The methods of proof are rigorous and combine Albeverio H\oegh-Krohn
oscillatory integrals with Elworthy's parametrization of paths in a curved
space. Unlike other approaches we solve Schr\"odinger type equations directly,
rather than solving first diffusion equations and then using analytic
continuation.


title: C*-crossed products by partial actions and actions of inverse semigroups

abstract:
  The recently developed theory of partial actions of discrete groups on
$C^*$-algebras is extended. A related concept of actions of inverse semigroups
on $C^*$-algebras is defined, including covariant representations and crossed
products. The main result is that every partial crossed product is a crossed
product by a semigroup action.


title: A Paley-Wiener Theorem for Nilpotent Lie Groups

abstract:
  A version of Paley-Wiener like theorem for connected, simply connected
nilpotent Lie groups is proven.


title: Trace-scaling automorphisms of certain stable AF algebras

abstract:
  Trace scaling automorphisms of stable AF algebras with dimension group
totally ordered are outer conjugate if the scaling factors are the same (not
equal to one). This is an adaptation of a similar result for the AFD type
II_infty factor by Connes and extends the previous result for stable UHF
algebras.


title: Characterisations of crossed products by partial actions

abstract:
  Partial actions of discrete groups on $C^*$-algebras and the associated
crossed products have been studied by Exel and McClanahan. We characterise
these crossed products in terms of the spectral subspaces of the dual coaction,
generalising and simplifying a theorem of Exel for single partial
automorphisms. We then use this characterisation to identify the Cuntz algebras
and the Toeplitz algebras of Nica as crossed products by partial actions.


title: Crossed product duality for partial $C^*$-automorphisms

abstract:
  For partial automorphisms of $C^*$-algebras, Takai-Takesaki crossed product
duality tends to fail, in proportion to the extent to which the partial
automorphism is not an automorphism.


title: Harnack Inequality on Homogeneous Spaces

abstract:
  We consider a homogeneous space $X=(X,d,m) $ of dimension $\nu\geq1$ and a
local regular Dirichlet form in $L^{2}(X,m) .$ We prove that if a Poincar\'{e}
inequality holds on every pseudo-ball $B(x,R) $ of $X$, then an Harnack's
inequality can be proved on the same ball with local characteristic constant
$c_{0}$ and $c_{1}$