title:
A Galois Correspondence for Compact Groups of Automorphisms of von
  Neumann Algebras with a Generalization to Kac Algebras

abstract:
  Let $M$ be a factor with separable predual and $G$ a compact group of
automorphisms of $M$ whose action is minimal, i.e. $M^{G^\prime}\cap M = C$,
where $M^G$ denotes the $G$-fixed point subalgebra. Then every intemediate von
Neumann algebra $M^G\subset N\subset M$ has the form $N=M^H$ for some closed
subgroup $H$ of $G$. An extension of this result to the case of actions of
compact Kac algebras on factors is also presented. No assumptions are made on
the existence of a normal conditional expectation onto $N$.


title: Compact Perturbations of Fredholm n-tuples

abstract:
  The paper gives a negative answer to the question whether one can perturb a
Fredholm pair of index 0 by compact operators to a pair with exact Koszul
complex.


title: Rings with Topologies Induced by Spaces of Functions

abstract:
  The paper contains the proof, in dimension 2, of a conjecture of R. G.
Douglas and V. Paulsen concerning the characterization of the ideals of
polynomials which are closed in the relative topology induced by the Hardy
space of the polydisk.


title: Compact Perturbations of Fredholm n-tuples II

abstract:
  The paper contains examples of Fredholm n-tuples of operators that are of
index 0 but cannot be perturbed by compact operators to n-tuples with exact
Koszul complex.


title: A Theory of Dimension

abstract:
  In which a theory of dimension related to the Jones index and based on the
notion of conjugation is developed. An elementary proof of the additivity and
multiplicativity of the dimension is given and there is an associated trace.
Applications are given to a class of endomorphisms of factors and to the theory
of subfactors. An important role is played by a notion of amenability inspired
by the work of Popa.


title: Amenability for Fell Bundles

abstract:
  Given a Fell bundle $\B$, over a discrete group $\Gamma$, we construct its
reduced cross sectional algebra $C^*_r(\B)$, in analogy with the reduced
crossed products defined for C*-dynamical systems. When the reduced and full
cross sectional algebras of $\B$ are isomorphic, we say that the bundle is
amenable. We then formulate an approximation property which we prove to be a
sufficient condition for amenability.
  A theory of $\Gamma$-graded C*-algebras possessing a conditional expectation
is developed, with an eye on the Fell bundle that one naturally associates to
the grading. We show, for instance, that all such algebras are isomorphic to
$C^*_r(\B)$, when the bundle is amenable.
  We also study induced ideals in graded C*-algebras and obtain a
generalization of results of Stratila and Voiculescu on AF-algebras, and of
Nica on quasi-lattice ordered groups. A brief comment is made on the relevance,
to our theory, of a certain open problem in the theory of exact C*-algebras.
  An application is given to the case of an $F_n$-grading of the Cuntz-Krieger
algebras $O_A$, recently discovered by Quigg and Raeburn. Specifically, we show
that the Cuntz-Krieger bundle satisfies the approximation property, and hence
is amenable, for all matrices $A$ with entries in {0,1}, even if $A$ does not
satisfy the well known property (I) studied by Cuntz and Krieger in their
paper.


title: $q$-Gaussian processes: non-commutative and classical aspects

abstract:
  We examine, for $-1<q<1$, $q$-Gaussian processes, i.e. families of operators
(non-commutative random variables) $X_t=a_t+a_t^*$ -- where the $a_t$ fulfill
the $q$-commutation relations $a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \id$ for some
covariance function $c(\cdot,\cdot)$ -- equipped with the vacuum expectation
state. We show that there is a $q$-analogue of the Gaussian functor of second
quantization behind these processes and that this structure can be used to
translate questions on $q$-Gaussian processes into corresponding (and much
simpler) questions in the underlying Hilbert space. In particular, we use this
idea to show that a large class of $q$-Gaussian processes possess a
non-commutative kind of Markov property, which ensures that there exist
classical versions of these non-commutative processes. This answers an old
question of Frisch and Bourret \cite{FB}.

categories: funct-an cond-mat hep-th math.OA math.QA q-alg


title:
On the multiplication of free $n$-tuples of non-commutative random
  variables

abstract:
  Let $a_{1},...,a_{n}, b_{1},...,b_{n}$ be random variables in some
(non-commutative) probability space, such that $\{a_{1}, ..., a_{n} \}$ is free
from $\{b_{1}, ..., b_{n} \}$. We show how the joint distribution of the
$n$-tuple $(a_{1} b_{1}, ..., a_{n} b_{n})$ can be described in terms of the
joint distributions of $(a_{1}, ..., a_{n})$ and $(b_{1}, >..., b_{n})$, by
using the combinatorics of the $n$-dimensional $R$-transform. We point out a
few applications that can be easily derived from our result, concerning the
left-and-right translation with a semicircular element (see Sections 1.6-1.10)
and the compression with a projection (see Sections 1.11-1.14) of an $n$-tuple
of non-commutative random variables. A different approach to two of these
applications is presented by Dan Voiculescu in an Appendix to the paper.


title:
$R$-diagonal pairs - a common approach to Haar unitaries and circular
  elements

abstract:
  In the free probability theory of Voiculescu two of the most frequently used
*-distributions are those of a Haar unitary and of a circular element. We
define an $R$-diagonal pair as a generalization of these distributions by the
requirement that their two-dimensional $R$-transform (or free cumulants) have a
special diagonal form. We show that the class of such $R$-diagonal pairs has an
absorption property under nested multiplication of free pairs. This implies
that in the polar decomposition of such an element the polar part and the
absolute value are free. Our calculations are based on combinatorial statements
about non-crossing partitions, in particular on a canonical bijection between
the set of intervals of NC(n) and the set of 2-divisible partitions in NC(2n).
In a forthcoming paper the theory of $R$-diagonal pairs will be used to solve
the problem of the free commutator.


title: Diagonalizing operators over continuous fields of C*-algebras

abstract:
  It is well known that in the commutative case, i.e. for $A=C(X)$ being a
commutative C*-algebra, compact selfadjoint operators acting on the Hilbert
C*-module $H_A$ (= continuous families of such operators $K(x)$, $x\in X$) can
be diagonalized if we pass to a bigger W*-algebra $L^\infty(X)={\bf A} \supset
A$ which can be obtained from $A$ by completing it with respect to the weak
topology. Unlike the "eigenvectors", which have coordinates from $\bf A$, the
"eigenvalues" are continuous, i.e. lie in the C*-algebra $A$. We discuss here
the non-commutative analog of this well-known fact. Here the "eigenvalues" are
defined not uniquely but in some cases they can also be taken from the initial
C*-algebra instead of the bigger W*-algebra. We prove here that such is the
case for some continuous fields of real rank zero C*-algebras over a
one-dimensional manifold and give an example of a C*-algebra $A$ for which the
"eigenvalues" cannot be chosen from $A$, i.e. are discontinuous. The main point
of the proof is connected with a problem on almost commuting operators. We
prove that for some C*-algebras if $h\in A$ is a selfadjoint, $u\in A$ is a
unitary and if the norm of their commutant $[u,h]$ is small enough then one can
connect $u$ with the unity by a path $u(t)$ so that the norm of $[u(t),h]$
would be also small along this path.


title:
The $:\phi^4_4:$ quantum field theory, I. Wave operator, holomorphity
  and Wick kernel

abstract:
  With the help of the complex structure and the wave operator of the nonlinear
classical Klein-Gordon equation with the interaction $u^4_4$ we define the Wick
kernel of the interacting quantum field in four-dimensional space-time and
consider its properties. In particular, the diagonal of this Wick kernel is
(real) solutions of the classical nonlinear Klein-Gordon equation with the
interaction $u^4_4.$

categories: funct-an hep-th math-ph math.FA math.MP nlin.SI quant-ph solv-int


title: Hilbert C*-modules and related subjects - a guided reference overview I

abstract:
  The overview contains 450 references of books, chapters of monographs,
papers, preprints and Ph.~D.~thesises which are concerned with the theory
and/or various applications of Hilbert C*-modules. To show a way through this
amount of literature a four pages guide is added clustering sources around
major research problems and research fields, and giving information on the
historical background. Two smaller separate parts list references treating
Hilbert modules over Hilbert*-algebras and Hilbert modules over
(non-self-adjoint) operator algebras. Any additions, corrections and
forthcoming information are welcome.


title:
Hilbert H*-modules and Hilbert modules over (non-self-adjoint) operator
  algebras -- a reference overview II

abstract:
  The two reference lists contain 54/22 references of papers and preprints
concerned with the theory and/or various applications of Hilbert modules over
Hilbert $*$-algebras and over (non-self-adjoint) operator algebras. They are
far from being complete, but they give additional information about two
research fields which are closely related to the theory of Hilbert C*-modules,
i.e. they are complements to the reference guide about this circle of problems.
Any additions, corrections and forthcoming information are welcome.


title: Two applications of free entropy

abstract:
  Using Voiculsecu's free entropy, it is shown that the free group factors
L(F_n) lack property C of Popa and that they lack finite multiplicity abelian
subalgebras. Property C is an assymptotic commutivity property.


title: Topological Hilbert Nullstellensatz for Bergman Spaces

abstract:
  The results in the paper are related to the classification problem for
invariant subspaces of multiplication operators in several variables. The main
results consist of characterizations, in the two dimensional case, of ideals of
polynomials and analytic functions which are closed in the relative topology
induced by Bergman spaces on certain domains. This provides proofs of the
Bergman space analogues of a conjecture of R. G. Douglas and V. Paulsen.


title: An Introduction to K-theory and Cyclic Cohomology

abstract:
  These lecture notes contain an exposition of basic ideas of K-theory and
cyclic cohomology. I begin with a list of examples of various situations in
which the K-functor of Grothendieck appears naturally, including the rudiments
of the topological and algebraic K-theory, K-theory of C^*-algebras, and
K-homology. I then discuss elementary properties of cyclic cohomology using the
Cuntz-Quillen version of the calculus of noncommutative differential forms on
an algebra. As an example of the relation between the two theories we describe
the Chern homomorphism and various index-theorem type statements. The remainder
of the notes contains some more detailed calculations in cyclic and reduced
cyclic cohomology. A key tool in this part is Goodwillie's theorem on the
cyclic complex of a semi-direct product algebra. The final chapter gives an
exposition of the entire cyclic cohomology of Banach algebras from the point of
view of supertraces on the Cuntz algebra. The results discussed here include
the simplicial normalization of the entire cyclic cohomology, homotopy
invariance and the action of derivations.


title:
Removal of the resolvent-like energy dependence from interactions and
  invariant subspaces of a total Hamiltonian

abstract:
  The spectral problem $(A + V(z))\psi=z\psi$ is considered where the main
Hamiltonian $A$ is a self-adjoint operator of sufficiently arbitrary nature.
The perturbation $V(z)=-B(A'-z)^{-1}B^{*}$ depends on the energy $z$ as
resolvent of another self-adjoint operator $A'$. The latter is usually
interpreted as Hamiltonian describing an internal structure of physical system.
The operator $B$ is assumed to have a finite Hilbert-Schmidt norm. The
conditions are formulated when one can replace the perturbation $V(z)$ with an
energy-independent ``potential'' $W$ such that the Hamiltonian $H=A +W$ has the
same spectrum (more exactly a part of spectrum) and the same eigenfunctions as
the initial spectral problem. The Hamiltonian $H$ is constructed as a solution
of the non-linear operator equation $H=A+V(H)$. It is established that this
equation is closely connected with the problem of searching for invariant
subspaces of the Hamiltonian
  $ {\bf H}=\left[ \begin{array}{lr} A & B B^{*} & A' \end{array}\right]. $
  The orthogonality and expansion theorems are proved for eigenfunction systems
of the Hamiltonian $ H=A + W $. Scattering theory is developed for this
Hamiltonian in the case where the operator $A$ has continuous spectrum.


title: Node Theorem for Matrix Schroedinger Operators

abstract:
  In this paper we study the ground states of a matrix Schroedinger operator,
that is an operator of the type (-Laplace) + V acting on m-component wave
functions in R^n. We prove in generalization of the classical node theorem that
the ground states of this operator have no zeros, if the potential V satisfies
the following conditions: 1) V is bounded from below. 2) V is strictly positive
at infinity with respect to the ground state energy. 3) The partial derivatives
of V do not grow too fast in the order of the derivative. Furthermore we show
that the degeneracy of the ground states is at most m. For the proof we
reformulate the problem with path integrals and perform a cluster expansion
with large/small field conditions.


title: Super-connections and non-commutative geometry

abstract:
  We show that Quillen's formalism for computing the Chern character of the
index using superconnections extends to arbitrary operators with functional
calculus. We thus remove the condition that the operators have, up to homotopy,
a gap in the spectrum. This is proved using differential graded algebras and
non-commutative differential forms. Our results also give a new proof of the
coincidence of the Chern character of a difference bundle defined using
super-connections with the classical definition.


title: Homology of pseudodifferential operators I. Manifolds with boundary

abstract:
  The Hochschild and cyclic homology groups are computed for the algebra of
`cusp' pseudodifferential operators on any compact manifold with boundary. The
index functional for this algebra is interpreted as a Hochschild 1-cocycle and
evaluated in terms of extensions of the trace functionals on the two natural
ideals, corresponding to the two filtrations by interior order and vanishing
degree at the boundary, together with the exterior derivations of the algebra.
This leads to an index formula which is a pseudodifferential extension of that
of Atiyah, Patodi and Singer for Dirac operators; together with a symbolic term
it involves the `eta' invariant on the suspended algebra over the boundary
previously introduced by the first author.


title: Covariant SPDEs and Quantum Field Structures

abstract:
  Covariant stochastic partial differential equations are studied in any
dimension. A special class of such equations is selected and it is proven that
the solutions can be analytically continued to Minkowski space-time yielding
tempered Wightman distributions which are covariant, obey the locality axiom
and a weak form of the spectral axiom. Key words: stochastic partial
differential equations, white noise, covariant Markov generalized random
fields, Euclidean QFT, Schwinger functions, Wightman distributions

categories: funct-an math-ph math.FA math.MP math.PR


title: On self-adjointness of a Schroedinger operator

abstract:
  Let $M$ be a complete Riemannian manifold and let $\Omega^*(M)$ denote the
space of differential forms on $M$. Let $d:\Omega^*(M) \to \Omega^{*+1}(M)$ be
the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian.
We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$
(where the potential $V(x):\Omega^*(M)\to \Omega^*(M)$ is a zero order
differential operator) to be self-adjoint. Our result generalizes a theorem by
Igor Oleinik about self-adjointness of a Schroedinger operator which acts on
the space of scalar valued functions.


title:
Abundance of invariant and almost invariant pure states of C*-dynamical
  systems

abstract:
  We show that invariant states of C*-dynamical systems can be approximated in
the weak*-topology by invariant pure states, or almost invariant pure states,
under various circumstances.


title: The stable rank of some free product C*-algebras

abstract:
  It is proved that the reduced group C*-algebra C*_{red}(G) has stable rank
one (i.e. its group of invertible elements is a dense subset) if G is a
discrete group arising as a free product G_1*G_2 where |G_1|>=2 and |G_2|>=3.
This follows from a more general result where it is proved that if (A,tau) is
the reduced free product of a family (A_i,tau_i), i\in I, of unital C*-algebras
A_i with normalized faithful traces tau_i, and if the family satisfies the
Avitzour condition (i.e. the traces, tau_i, are not too lumpy in a specific
sense), then A has stable rank one.


title: Operator of fractional derivative in the complex plane

abstract:
  The paper deals with a fractional derivative introduced by means of the
Fourier transform. The explicit form of the kernel of general derivative
operator acting on the functions analytic on a curve in complex plane is
deduced and the correspondence with some well known approaches is shown. In
particular, it is shown how the uniqueness of the operation depends on the
derivative order type (integer, rational, irrational, complex) and the number
of poles of considered function in the complex plane.


title: Quantized Kronecker flows and almost periodic quantum field theory

abstract:
  We define and study the properties of the infinite dimensional quantized
Kronecker flow. This $\bC^*$-dynamical system arises as a quantization of the
corresponding flow on an infinite dimensional torus. We prove an ergodic
theorem for a class of quantized Kronecker flows. We also study the closely
related almost periodic quantum field theory of bosonic, fermionic and
supersymmetric particles. We prove the existence and uniqueness of KMS and
super-KMS states for the $\bC^*$-algebras of observables arising in these
theories.


title: Projective Fourier Duality and Weyl Quantization

abstract:
  The Weyl-Wigner correspondence prescription, which makes large use of Fourier
duality, is reexamined from the point of view of Kac algebras, the most general
background for noncommutative Fourier analysis allowing for that property. It
is shown how the standard Kac structure has to be extended in order to
accommodate the physical requirements. An Abelian and a symmetric projective
Kac algebras are shown to provide, in close parallel to the standard case, a
new dual framework and a well-defined notion of projective Fourier duality for
the group of translations on the plane. The Weyl formula arises naturally as an
irreducible component of the duality mapping between these projective algebras.


title: Fourier Duality as a Quantization Principle

abstract:
  The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes
essential use of Fourier duality. The extension of this property to more
general phase spaces requires the use of Kac algebras, which provide the
necessary background for the implementation of Fourier duality on general
locally compact groups. Kac algebras -- and the duality they incorporate -- are
consequently examined as candidates for a general quantization framework
extending the usual formalism. Using as a test case the simplest non-trivial
phase space, the half-plane, it is shown how the structures present in the
complete-plane case must be modified. Traces, for example, must be replaced by
their noncommutative generalizations - weights - and the correspondence
embodied in the Weyl-Wigner formalism is no more complete. Provided the
underlying algebraic structure is suitably adapted to each case, Fourier
duality is shown to be indeed a very powerful guide to the quantization of
general physical systems.


title: Crossed Products by Dual Coactions of Groups and Homogeneous Spaces

abstract:
  Mansfield showed how to induce representations of crossed products of
C*-algebras by coactions from crossed products by quotient groups and proved an
imprimitivity theorem characterising these induced representations. We give an
alternative construction of his bimodule in the case of dual coactions, based
on the symmetric imprimitivity theorem of the third author; this provides a
more workable way of inducing representations of crossed products of
C*-algebras by dual coactions. The construction works for homogeneous spaces as
well as quotient groups, and we prove an imprimitivity theorem for these
induced representations.


title:
Hilbert $C^*$-bimodules over commutative $C^*$-algebras and an
  isomorphism condition for quantum Heisenberg manifolds

abstract:
  A study of Hilbert $C^*$-bimodules over commutative $C^*$-algebras is carried
out and used to establish a sufficient condition for two quantum Heisenberg
manifolds to be isomorphic.


title: Supermatrix Representations of Semigroup Bands

abstract:
  Various semigroups of noninvertible supermatrices of the special
(antitriangle) shape having nilpotent Berezinian which appear in supersymmetric
theories are defined and investigated. A subset of them continuously represents
left and right zero semigroups and rectangular bands. The ideal properties of
higher order rectangular band analogs and the ``wreath'' version of them are
studied in detail. We introduce the ``fine'' equivalence relations leading to
``multidimesional'' eggbox diagrams. They are full images of Green's relations
on corresponding subsemigroups.

categories: funct-an alg-geom dg-ga hep-th math.AG math.FA math.QA q-alg


title: On Hahn-Banach type theorems for Hilbert C*-modules

abstract:
  We show three Hahn-Banach type extension criteria for (sets of) bounded
C*-linear maps of Hilbert C*-modules to the underlying C*-algebras of
coefficients. One criterion establishes an alternative description of the
property of (AW*-) C*-algebras to be monotone complete or additively complete.


title:
Continuity of symplectically adjoint maps and the algebraic structure of
  Hadamard vacuum representations for quantum fields on curved spacetime

abstract:
  We derive for a pair of operators on a symplectic space which are adjoints of
each other with respect to the symplectic form (that is, they are sympletically
adjoint) that, if they are bounded for some scalar product on the symplectic
space dominating the symplectic form, then they are bounded with respect to a
one-parametric family of scalar products canonically associated with the
initially given one, among them being its ``purification''. As a typical
example we consider a scalar field on a globally hyperbolic spacetime governed
by the Klein-Gordon equation; the classical system is described by a symplectic
space and the temporal evolution by symplectomorphisms (which are
symplectically adjoint to their inverses). A natural scalar product is that
inducing the classical energy norm, and an application of the above result
yields that its ``purification'' induces on the one-particle space of the
quantized system a topology which coincides with that given by the two-point
functions of quasifree Hadamard states. These findings will be shown to lead to
new results concerning the structure of the local (von Neumann)
observable-algebras in representations of quasifree Hadamard states of the
Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as local
definiteness, local primarity and Haag-duality (and also split- and type
III_1-properties). A brief review of this circle of notions, as well as of
properties of Hadamard states, forms part of the article.


title: The Rohlin Property for Actions of $Z^{2}$ on UHF Algebras

abstract:
  We define a Rohlin property for actions of $Z^{2}$ on UHF algebras and show a
noncommutative Rohlin type theorem. Among those actions with the Rohlin
property, we classify product type actions up to outer conjugacy. In particular
we present two classes of UHF algebras. For UHF algebras in one class, which
includes the CAR algebra, there is one and only one outer conjugacy class of
product type actions and for UHF algebras in the other class, contrary to the
case of actions of $Z$, there are infinitely many outer conjugacy classesof
product type actions.


title: Normal intermediate subfactors

abstract:
  Let $N \subset M$ be an irreducible inclusion of type type II$_1$ factors
with finite Jones index. We shall introduce the notion of normality for
intermediate subfactors of the inclusion $N \subset M$. If the depth of $N
\subset M$ is 2, then an intermediate subfactor $K$ for $N \subset M$ is normal
in $ N \subset M$ if and only if the depths of $N \subset K$ and $K \subset M$
are both 2. In particular, if $M$ is the crossed product $N \rtimes G$ of a
finite group $G$, then $K = N \rtimes H$ is normal in $N \subset M$ if and only
if $H$ is a normal subgroup of $G$.


title:
$C^*$-algebras of $b$-pseudodifferential operators and an
  $\R^k$-equivariant index theorem

abstract:
  We compute $K$-theory invariants of algebras of pseudodifferential operators
on manifolds with corners and prove an equivariant index theorem for operators
invariant with respect to an action of $\R^k.$ We discuss the relation between
our results and the $\eta$-invariant.


title: Generalized Cuntz algebras associated with subfactors

abstract:
  Various generalizations of Cuntz algebras and their relations to symmetry and
duality are reviewed. New generalized Cuntz algebras are associated with a
subfactor. A characteristic Hilbert space of basic invariants (with respect to
the generalized symmetry) within these algebras is discussed.


title: Isomorphism classes for quantum Heisenberg manifolds

abstract:
  We embed the quantum Heisenberg manifold in a crossed product algebra. This
enables us to show that, in the irrational case, all tracial states on $\dc$
induce the same homomorphism on the K_0-group. We conclude that two irrational
quantum Heisenberg manifolds $\dc$ and $D^c_{\mu ' \nu '}$ are isomorphic if
and only if the parameters $(\mu,\nu)$ and $(\mu ',\nu ')$ belong to the same
orbit under the usual action of $GL_2(\ZZ)$ on the torus.


title: Topics in hidden symmetries. V

abstract:
  This note being devoted to some aspects of the inverse problem of
representation theory contains a new insight into it illustrated by two topics.
The attention is concentrated on the manner of representation of abstract
objects by the concrete ones as well as on the abstract objects themselves. The
results of researches allow to state that the actual richness and
attractiveness of the inverse problem of representation theory are based not
only on a large scope of various interesting abstract algebraic structures,
which may be concretely represented and somehow unravelled, but also on the
diversity of the manners of representation, which may produce very intriguing
unexpected and nontrivial effects under the intent look even in the simple and
almost hackneyed situations.


title:
Scaling algebras, the renormalization group and the principle of local
  stability in algebraic quantum field theory

abstract:
  We summarize the new scaling algebra approach to the structural analysis of
the short distance behaviour of quantum field theories in the operator
algebraic formulation which has recently been proposed by D. Buchholz, R. Verch
(Rev. Math. Phys. 7(1995) 1195), and further studied by D. Buchholz (Ann. Inst.
H. Poincare' 64(1996) 433, Nucl. Phys. B469(1996) 333). We also report on an
extension of this framework to quantum field theory in curved spacetime.


title: Commutators of free random variables

abstract:
  Let A be a unital $C^*$-algebra, given together with a specified state
$\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with
respect to $\phi$ (in the sense of the free probability theory of Voiculescu).
Let us denote $c:=i(ab-ba)$, where the i in front of the commutator is
introduced to make c selfadjoint. In this paper we show how the spectral
distribution of c can be calculated from the spectral distributions of a and b.
Some properties of the corresponding operation on probability measures are also
discussed. The methods we use are combinatorial, based on the description of
freeness in terms of non-crossing partitions; an important ingredient is the
notion of R-diagonal pair, introduced and studied in our previous paper
funct-an/9604012.


title:
Iterated function systems and permutation representations of the Cuntz
  algebra

abstract:
  We study a class of representations of the Cuntz algebras O_N, N=2,3,...,
acting on L^2(T) where T=R/2\pi Z. The representations arise in wavelet theory,
but are of independent interest. We find and describe the decomposition into
irreducibles, and show how the O_N-irreducibles decompose when restricted to
the subalgebra UHF_N\subset O_N of gauge-invariant elements; and we show that
the whole structure is accounted for by arithmetic and combinatorial properties
of the integers Z. We have general results on a class of representations of O_N
on Hilbert space H such that the generators S_i as operators permute the
elements in some orthonormal basis for H. We then use this to extend our
results from L^2(T) to L^2(T^d), d>1 ; even to L^2(\mathbf{T}) where \mathbf{T}
is some fractal version of the torus which carries more of the algebraic
information encoded in our representations.


title:
Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis
  of scale N

abstract:
  In this paper we show how wavelets originating from multiresolution analysis
of scale N give rise to certain representations of the Cuntz algebras O_N, and
conversely how the wavelets can be recovered from these representations. The
representations are given on the Hilbert space L^2(T) by
(S_i\xi)(z)=m_i(z)\xi(z^N). We characterize the Wold decomposition of such
operators. If the operators come from wavelets they are shifts, and this can be
used to realize the representation on a certain Hardy space over L^2(T). This
is used to compare the usual scale-2 theory of wavelets with the scale-N
theory. Also some other representations of O_N of the above form called
diagonal representations are characterized and classified up to unitary
equivalence by a homological invariant.


title:
On the infinite dimensional hidden symmetries. I. Infinite dimensional
  geometry of $q_R$-conformal symmetries

abstract:
  This paper opens the series of articles supplemental to the series
(hep-th/9405050,q-alg/9610026,q-alg/9611003,q-alg/9611019,funct-an/9611003),
which also lies in lines of general ideology exposed in the review
(mp_arc/96-477). The main purpose of the activity, which has its origin and
motivation presumably in the author's applied researches on the interactively
controlled systems (i.e.the controlled systems, in which the control is coupled
with unknown or uncompletely known feedbacks), is to explicate the essentially
infinite-dimensional aspects of the hidden symmetries, which appear in the
representation theory of the finite dimensional Lie algebras and related
algebraic structures. The series is organized as a sequence of topics, which
illustarate this basic idea on the simple and tame examples without superfluous
difficulties and details as well as in the previous series. Many of objects,
which will appear, are somehow related to ones discussed earlier. However, the
matherial will be treated more geometrically, presumably, from the points of
view of the infinite dimensional geometry, an infinite dimensional version of
the nonlinear geometric algebra and the infinite dimensional noncommutative
geometry. The intent reader will see some ideological and concrete similarities
between the subject of the paper and the infinite dimensional geometric picture
for a second quantized string.

categories: funct-an dg-ga math.DG math.OA math.QA q-alg


title: Modules with norms which take values in a C*-algebra

abstract:
  We consider modules E over a C*-algebra A which are equipped with a map into
A_+ that has the formal properties of a norm. We completely determine the
structure of these modules. In particular, we show that if A has no nonzero
commutative ideals then every such E must be a Hilbert module. The commutative
case is much less rigid: if A = C_0(X) is commutative then E is merely
isomorphic to the module of continuous sections of some bundle of Banach spaces
over X. In general E will embed in a direct sum of modules of the preceding two
types.


title:
A connection between multiresolution wavelet theory of scale N and
  representations of the Cuntz algebra O_N

abstract:
  In this paper we give a short survey of a connection between the theory of
wavelets in L^2(R) and certain representations of the Cuntz algebra on L^2(T).


title:
Generalized inverses and the maximal radius of regularity of a Fredholm
  operator

abstract:
  Operators possessing analytic generalized inverses satisfying the resolvent
identity are studied. Several characterizations and necessary conditions are
obtained. The maximal radius of regularity for a Fredholm operator T is
computed in terms of the spectral radius of a generalized inverse of T. This
provides a partial answer to a conjecture of J. Zem\'anek.


title: Some Topics in Fourier Analysis and Approximation Theory

abstract:
  This manuscript presents shortly the results obtained by participants of the
scientific seminar which is held more than twenty years under leadership of the
author at Donetsk University. In the list of references main publications are
given. These results are published in serious scientific journals and reported
at various conferences, including international ones at Moscow,ICM66;
Kaluga,1975; Kiev,1983; Haifa,1994; Z\"urich,ICM94; Moscow,1995. The area of
investigation is the Fourier analysis and the theory of approximation of
functions. Used are methods of classical analysis including special functions,
Banach spaces, etc., of harmonic analysis in finitedimensional Euclidean space,
of Diophantine analysis, of random choice, etc. The results due to the author
and active participants of the seminar, namely E. S. Belinskii, O. I.
Kuznetsova, E. R. Liflyand, Yu. L. Nosenko, V. A. Glukhov, V. P. Zastavny, Val.
V. Volchkov, V. O. Leontyev, and others, are given. Besides the participants of
the seminar and other mathematicians from Donetsk, many mathematicians from
other places were speakers at the seminar, in particular, A.A. Privalov, Z.A.
Chanturia, Yu.A. Brudnyi, N.Ya. Krugljak, V.N. Temlyakov, B.D. Kotlyar, A.N.
Podkorytov, M.A. Skopina, A.A. Ligun, A.S. Romanyuk, V.A. Martirosyan. Besides
the papers of the participants of the seminar, only monographs and survey
papers are given in the list of references. The most part of this preprint will
appear soon in extended form in the author's book "Fourier Analysis and
Approximation Theory", World Federation Publishers, Inc., P.O.Box 48654,
Atlanta, GA 30362-0654, USA.


title: On invariant measures for the group of diffeomorphisms of the circle

abstract:
  In a previous paper the author constructed biinvariant measures (possibly
having values in a line bundle) for a loop group LK (with compact simply
connected K) acting on the formal completion of its complexification LG. One
motivation for this was to find a geometric construction for the unitary
structure for the positive energy representations of LK. In this paper we
pursue an analogous construction for the group of diffeomorphisms of the
circle.


title: K-theoretic duality for shifts of finite type

abstract:
  C*-algebras generalizing Cuntz-Krieger algebras can be associated to
hyperbolic homeomorphisms of compact metric spaces. They satisfy a
non-commutative form of Spanier-Whitehead duality with respect to K-theory. We
prove this for the case of subshifts of finite type. The special feature of the
present situation is that the constructions are all done on the full Fock space
and are very explicit, while the general theorem requires much more abstract
machinery.


title: A multiplier approach to the Lance-Blecher theorem

abstract:
  A new approach to the Lance-Blecher theorem is presented resting on the
interpretation of elements of Hilbert C*-module theory in terms of multiplier
theory of operator C*-algebras: The Hilbert norm on a Hilbert C*-module allows
to recover the values of the inducing C*-valued inner product in a unique way,
and two Hilbert C*-modules {M_1, <.,.>_1}, {M_2, <.,.>_2} are isometrically
isomorphic as Banach C*-modules if and only if there exists a bijective
C*-linear map S: M_1 --> M_2 such that the identity <.,.>_1 \equiv
<S(.),S(.)>_2 is valid. In particular, the values of a C*-valued inner product
on a Hilbert C*-module are completely determined by the Hilbert norm induced
from it. In addition, we obtain that two C*-valued inner products on a Banach
C*-module inducing equivalent norms to the given one give rise to isometrically
isomorphic Hilbert C*-modules if and only if the derived C*-algebras of
''compact'' module operators are *-isomorphic. The involution and the C*-norm
of the C*-algebra of ''compact'' module operators on a Hilbert C*-module allow
to recover its original C*-valued inner product up to the following equivalence
relation: <.,.>_1 \sim <.,.>_2 if and only if there exists an invertible,
positive element $a$ of the center of the multiplier C*-algebra M(A) of A such
that the identity <.,.>_1 \equiv a \cdot <.,.>_2 holds.


title: Derivation of Local Gauge Freedom from a Measurement Principle

abstract:
  We define operator manifolds as manifolds on which a spectral measure on a
Hilbert space is given as additional structure. The spectral measure
mathematically describes space as a quantum mechanical observable.
  We show that the vectors of the Hilbert space can be represented as functions
on the manifold. The arbitrariness of this representation is interpreted as
local gauge freedom. In this way, the physical gauge principle is linked with
quantum mechanical measurements of the position variable. We derive the
restriction for the local gauge group to be U(m), where m is the number of
components of the wave functions.


title: Bell's Inequalities and Algebraic Structure

abstract:
  We provide an overview of the connections between Bell's inequalities and
algebraic structure.


title: A New Definition of Hypercomplex Analyticity

abstract:
  Complex analyticity is generalized to hypercomplex functions, quaternion or
octonion, in such a manner that it includes the standard complex definition and
does not reduce analytic functions to a trivial class. A brief comparison with
other definitions is presented.


title: Meet irreducible ideals in direct limit algebras

abstract:
  We study the meet irreducible ideals in certain direct limit algebras, namely
the strongly maximal triangular subalgebras of AF C*-algebras. These ideals
have a description in terms of the coordinates, or spectrum, that is a natural
extension of one description of meet irreducible ideals in the upper triangular
matrices. Additional information is available if the limit algebra is an
analytic subalgebra of its C*-envelope or if the analytic algebra is trivially
analytic with an injective 0-cocycle. Completely meet irreducible ideals are
considered, and a distance formula is presented.


title: A Note on Bimodules and $II_1$-Subfactors

abstract:
  A brief introduction into bimodules of $II_1$-factors is presented.
Furthermore a version of the following result due to M. Pimsner and S. Popa is
derived: Let $N=M_{-1}\subset M=M_0 \subset M_1 \subset M_2 \subset \ldots$
denote the Jones tower of a $II_1$-factor $N\subset M$ with finite index. Then
the factor obtained by the basic construction from the pair $N\subset M_{n-1}$
is equal to $M_{2n-1}$.


title: $C^*$-Tensor Categories in the Theory of $II_1$-Subfactors

abstract:
  The article contains a detailed description of the connection between finite
depth inclusions of $II_1$-subfactors and finite $C^*$-tensor categories (i.e.
$C^*$-tensor categories with dimension function for which the number of
equivalence classes of irreducible objects is finite). The $(N,N)$-bimodules
belonging to a $II_1$-subfactor $N\subset M$ with finite Jones index form a
$C^*$-tensor category with dimension function. Conversely, taking an object of
a finite $C^*$-tensor category C we construct a subfactor $A\subset R$ of the
hyperfinite $II_1$-factor R with finite index and finite depth. For this
subfactor we compute the standard invariant and show that the $C^*$-tensor
category of the corresponding $(A,A)$-bimodules is equivalent to a subcategory
of C. We illustrate the results for the $C^*$-tensor category of the unitary
finite dimensional corepresentations of a finite dimensional Hopf-*-algebra.


title:
$II_1$-Subfactors associated with the $C^*$-Tensor Category of a Finite
  Group

abstract:
  We determine the subfactors $N\subset R$ of the hyperfinite $II_1$-factor R
with finite index for which the $C^*$-tensor category of the associated
$(N,N)$-bimodules is equivalent to the $C^*$-tensor category $\C{U}_G$ of all
unitary finite dimensional representations of a given finite group G.


title:
On the infinite dimensional hidden symmetries. II. $q_R$-conformal
  modular functor

abstract:
  The article is devoted to the $q_R$-conformal modular functors, which being
``deformations'' of the conformal modular functor (the projective
representation of the category $Train(Diff_+(S^1))$, the train of the group
$Diff_+(S^1)$ of all orientation preserving diffeomorphisms of a circle) in the
class of all projective modular functors (the projective representations of the
category $Train(PSL(2,R))$, the train of the projective group $PSL(2,R)$), may
be regarded as its ``Berezin quantizations''.


title:
A cohomological characterization of approximately finite dimensional von
  Neumann algebras

abstract:
  For a von Neumann algebra M on a Hilbert space, A. Connes has constructed a
module S and a derivation of M into S, such that M is approximately finite
dimensional if and only if that derivation is inner. The paper contains a
generalization of this result to the situation with a 2-cocycle instead. The
cocycle is the obvious generalization, and the module is closely related to
Connes, but isn't a dual module.


title: On the operator algebra for the space-time uncertainty relations

abstract:
  The purpose of this note is to show that the construction of the C*-algebra
for the space-time uncertainty relations which was introduced by Doplicher,
Fredenhagen and Roberts fits comfortably into the strict deformation
quantization construction introduced earlier by the author.


title: Some Results about Frames

abstract:
  In this paper we discuss some topics related to the general theory of frames.
In particular we focus our attention to the existence of different
'reconstruction formulas' for a given vector of a certain Hilbert space and to
some refinement of the perturbative approach for the computation of the dual
frame.


title: Multi-Resolution Analysis in Arbitrary Hilbert Spaces

abstract:
  We discuss the possibility of introducing a multi-resolution in a Hilbert
space which is not necessarily a space of functions. We investigate which of
the classical properties can be translated to this more general framework and
the way in which this can be done. We comment on the procedure proposed by
means of many examples.


title: On summability of distributions and spectral geometry

abstract:
  Modulo the moment asymptotic expansion, the Cesaro and parametric behaviours
of distributions at infinity are equivalent. On the strength of this result, we
construct the asymptotic analysis for spectral densities, arising from elliptic
pseudodifferential operators. We show how Cesaro developments lead to efficient
calculations of the expansion coefficients of counting number functionals and
Green functions. The bosonic action functional proposed by Chamseddine and
Connes can more generally be validated as a Cesaro asymptotic development.


title:
On the infinite-dimensional hidden symmetries. III. $q_R$-conformal
  symmetries at $q_R\to\infty$ and Berezin-Karasev-Maslov asymptotic
  quantization of $C^\infty(S^1)$

abstract:
  The relations between the infinite dimensional geometry of $q_R$-conformal
symmetries at $q_R\to\infty$, Berezin quantization of the Lobachevskii plane
and Karasev-Maslov asymptotic quantization are explicated. Some aspects of the
``approximate'' representation theory are discussed.


title: A Riemann sum upper bound in the Riemann-Lebesque theorem

abstract:
  The Riemann-Lebesque Theorem is commonly proved in a few strokes using the
theory of Lebesque integration. Here, the upper bound $2\pi|c_k(f)|\le
S_k(f)-s_k(f)$ for the Fourier coefficients $c_k$ is proved in terms of
majoring and minoring Riemann sums $S_k(f)$ and $s_f(k)$, respectively, for
Riemann integrable functions $f(x)$. This proof has been used in a course on
methods of applied mathematics.


title: Pseudodifferential operators on differential groupoids

abstract:
  We construct an algebra of pseudodifferential operators on each groupoid in a
class that generalizes differentiable groupoids to allow manifolds with
corners. We show that this construction encompasses many examples. The
subalgebra of regularizing operators is identified with the smooth algebra of
the groupoid, in the sense of non-commutative geometry. Symbol calculus for our
algebra lies in the Poisson algebra of functions on the dual of the Lie
algebroid of the groupoid. As applications, we give a new proof of the
Poincar\'e-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization
of the Lie-Poisson structure on the dual $A^*$ of a Lie algebroid.

categories: funct-an dg-ga math.DG math.OA math.QA math.SG q-alg


title: Gamma Noise Analysis

abstract:
  We study an infinite dimensional analysis with respect to the measure on
Schwartz space of tempered distributions, corresponding to the distributional
derivative of gamma process. Laguerre polynomials being orthogonal with respect
to gamma noise measure turn out to be generalized Appell ones. This fact
enables to generalize the white noise functional approach on the stochastic
Wick-Skorokhod equations involving gamma noise. E. g. we consider Werhulst type
equation driven by gamma noise.


title: Morphisms of Multiplicative Unitaries

abstract:
  In this paper, we will give a natural definition for morphisms between
multiplicative unitaries. We will then discuss some equivalences of this
definition and some interesting properties of them. Moreover, we will define
normal sub-multiplicative unitaries for multiplicative unitaries of discrete
type and prove an imprimitivity type theorem for discrete multiplicative
unitaries.


title: On the problem of semiinfinite beam oscillation with internal damping

abstract:
  We study the Cauchy problem for the equation of the form $$ \ddot{u}(t) +
(\aa A + B)\dot{u}(t) + (A+G)u(t) = 0,\tag* $$ where $A$, $B$, and $G$ are \o s
in a Hilbert space $\Cal H$ with $A$ selfadjoint, $\sigma(A)=[0,\infty)$,
$B\ge0$ bounded, and $G$ symmetric and $A$-subordinate in a certain sense.
Spectral properties of the correspondent operator pencil
  $L(\lambda) := \lambda^2I + \lambda (\alpha A + B) + A + G$ are studied, and
existence and uniqueness of generalized and classical solutions of the Cauchy
problem are proved. Equations of the type (*) include, e.g., an abstract model
for the problem of semiinfinite beam oscillations with internal damping.


title: Dual $n_1$-Appell-like Systems in Infinite-Dimensional Analysis

abstract:
  We introduce and study dual $n_1$-Appell-like systems which are the simple
generalization of generalized dual Appell systems in Infinite-Dimensional
Analysis (IDA). We study connected with these systems objects of IDA: the
analogues of Kondratiev spaces, $S$-transform, characterization theorems etc.
The results we obtained are useful to application in the theory of probability.


title:
On Variational Inequalities with Multivalued Operators with Semi-Bounded
  Variation

abstract:
  In this paper we explore solvability of steady-state variational inequalities
with multivalued operators. Moreover, we are studying the connections between
the class of radially semi-continuous operators with semi-bounded variation and
classes of pseudo-monotone and monotone mappings, and some properties of this
operators.


title:
Stochastic equations and evolution families in the space of formal
  mappings

abstract:
  We consider stochastic equations for the class of formal mappings. Existence
and uniqueness of solution, as well as evolution property are proved.


title: Symmetrical Extensions of Dirichlet Operators

abstract:
  There is constructed and considered the extension of classical Diriclet
operator corresponding to uniformly log-concave measure in the space of
symmetric differential forms. Sufficient conditions for its essential
self-adjointness in one-dimensional case as well as for the same of its
"sypersymmetric" part in general situations are given.


title:
The asymptotical behavior of spectral function of one family of
  differential operators

abstract:
  Is considered the asymptotical behavior of spectral function $\rho(\lambda,
\epsilon),\epsilon > 0$, of one family of self adjoint differential operators
of second order, defined in space $L_2[0,+\infty)$ with potentials, depending
on $\epsilon$. Are given evaluations for the derivative of spectral function
for negative $\lambda$ and is proved the weak convergence of $\rho'(\lambda,
\epsilon)$ to $\rho'(\lambda, 0)$ with $\epsilon \to +0$.


title: An invariant for pairs of almost commuting unbounded operators

abstract:
  For a wide class of pairs of unbounded selfadjoint operators with bounded
commutator we construct a K-theoretical integer invariant which is continuous,
is equal to zero for commuting operators and is equal to one for the pair (x, i
d/dx).


title: Faithfulness of free product states

abstract:
  It is proved that the free product state, in the reduced free product of
C*-algebras, is faithful if the initial states are faithful.


title: Simplicity and the stable rank of some free product C*-algebras

abstract:
  A necessary and sufficient condition for the simplicity of the C*-algebra
reduced free product of finite dimensional abelian algebras is found, and it is
proved that the stable rank of every such free product is 1. Related results
about other reduced free products of C*-algebras are proved.


title: Projections in free product C*-algebras

abstract:
  Consider the reduced free product of C*-algebras,
(A,\phi)=(A_1,\phi_1)*(A_2,\phi_2), with respect to states \phi_1 and \phi_2
that are faithful. If \phi_1 and \phi_2 are traces, if the so-called Avitzour
conditions are satisfied, (i.e. A_1 and A_2 are not ``too small'' in a specific
sense) and if A_1 and A_2 are nuclear, then it is shown that the positive cone
of the K_0-group of A consists of those elements g in K_0(A) for which g=0 or
K_0(\phi)(g)>0. Thus, the ordered group K_0(A) is weakly unperforated.
  If, on the other hand, \phi_1 or \phi_2 is not a trace and if a certain
condition weaker than the Avitzour conditions hold, then A is properly
infinite.


title:
On invertibility preserving linear maps, simultaneous triangularization
  and Property L

abstract:
  Let F be a linear unital map of a unital matrix algebra A over the complex
numbers into the complex n by n matrices. Then F induces a linear unital map Fk
of the k by k matrices over A into the complex nk by nk matrices by the action
of F on each entry in the k by k matrices. If Fk preserves invertibility and k
is greater than dim(alg(F(A)))-dim(F(A))+2 then F is a homomorphism modulo the
Jacobson radical in alg(F(A)). The paper also contains some results related to
the property L and some sufficient conditions for simultaneous
triangularization of sets of matrices.


title:
Orbifold subfactors from Hecke algebras II --- Quantum doubles and
  braiding ---

abstract:
  A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the
system of the M_infinity-M_infinity bimodules of the asymptotic inclusion, a
subfactor analogue of the quantum double, of the Jones subfactor of type
A_2n+1. We show that this is a general phenomenon and identify some of his
orbifolds with the ones in our sense as subfactors given as simultaneous fixed
point algebras by working on the Hecke algebra subfactors of type A of Wenzl.
That is, we work on their asymptotic inclusions and show that the
M_infinity-M_infinity bimodules are described by certain orbifolds (with
ghosts) for SU(3)_3k. We actually compute several examples of the (dual)
principal graphs of the asymptotic inclusions. As a corollary of the
identification of Ocneanu's orbifolds with ours, we show that a non-degenerate
braiding exists on the even vertices of D_2n, n>2.


title: States of Toeplitz-Cuntz algebras

abstract:
  We characterize the state space of a Toeplitz-Cuntz algebra TO_n in terms of
positive operator matrices $\Omega$ on Fock space which satisfy sl(\Omega) \le
\Omega, where sl(\Omega) is the operator matrix obtained from \Omega by taking
the trace in the last variable. Essential states correspond to those matrices
\Omega which are slice-invariant. As an application we show that a pure
essential product state of the fixed-point algebra for the action of the gauge
group has precisely a circle of pure extensions to TO_n.


title: Two Approaches to Non-Commutative Geometry

abstract:
  Looking to the history of mathematics one could find out two outer approaches
to Geometry. First one (algebraic) is due to Descartes and second one
(group-theoretic)--to Klein. We will see that they are not rivalling but are
tied (by Galois). We also examine their modern life as philosophies of
non-commutative geometry. Connections between different objects (see keywords)
are discussed.
  Keywords: Heisenberg group, Weyl commutation relation, Manin plain, quantum
groups, SL(2, R), Hardy space, Bergman space, Segal-Bargmann space, Szeg"o
projection, Bergman projection, Clifford analysis, Cauchy-Riemann-Dirac
operator, Moebius transformations, functional calculus, Weyl calculus
(quantization), Berezin quantization, Wick ordering, quantum mechanics.


title: Local Hypercomplex Analyticity

abstract:
  The notion of analyticity is studied in the context of hypercomplex numbers.
A critical review of the problems arising from the conventional approach is
given. We describe a local analyticity condition which yields the desired type
of hypercomplex solutions. The result is the definition of a generalized
complex analyticity to hypercomplex space.
  02.10.Tq/Vr, 02.30.-f/Dk, 02.90.+p


title: Stability of quasi-linear hyperbolic dissipative systems

abstract:
  In this work we want to explore the relationship between certain eigenvalue
condition for the symbols of first order partial differential operators
describing evolution processes and the linear and nonlinear stability of their
stationary solutions.


title:
Classification of actions of discrete amenable groups on strongly
  amenable subfactors of type III$\lambda$

abstract:
  Using the continuous decomposition, we classify strongly free actions of
discrete amenable groups on strongly amenable subfactors of type III$_\lambda,
0 < \lambda < 1$. Winsl{\o}w's fundamental homomorphism is a complete
invariant. This removes the extra assumptions in the classification theorems of
Loi and Winsl{\o}w and gives a complete classification up to cocycle conjugacy.


title: Regular C*-valued weights

abstract:
  We introduce the notion of a C*-valued weight between two C*-algebras as a
generalization of an ordinary weight on a C*-algebra and as a C*-version of
operator valued weights on von Neumann algebras. Also, some form of lower
semi-continuity will be discussed together with an extension to the multiplier
algebra. A strong but useful condition for C*-valued weights, the so-called
regularity, is introduced. At the same time, we propose a construction
procedure for such regular C*-valued weights. This construction procedure will
be used to define the tensor product of regular C*-valued weights.


title: Umbral Calculus and Cancellative Semigroup Algebras

abstract:
  We describe some connections between three different fields: combinatorics
(umbral calculus), functional analysis (linear functionals and operators) and
harmonic analysis (convolutions on group-like structures). Systematic usage of
cancellative semigroup, their convolution algebras, and tokens between them
provides a common language for description of objects from these three fields.
  Keywords: cancellative semigroups, umbral calculus, harmonic analysis, token,
convolution algebra, integral transform


title:
A Complete Classification Of The Admissible Representations Of
  Infinite-Dimensional Classical Matrix Groups

abstract:
  This paper contains a complete description of classes of the unitary
equivalence of the admissible representations of infinite-dimensional classic
matrix groups paper.


title: Grassmannian and elliptic operators

abstract:
  The conjecture about relation between infinite-dimensional Grassmannian and
string theory is based on the fact that moduli spaces of algebraic curves are
embedded into Grassmannian via Krichever construction. We describe a
multidimensional analog of Krichever construction, that can be used in the
attempt to relate Grassmannian to membranes.


title: Examining the dual of an algebraic quantum group

abstract:
  In the first part of this paper, we implement the multiplier algebra of the
dual of an algebraic quantum group (A,Delta) as a space of linear functionals
on A. In the second part, we construct the universal corepresentation of
(A,Delta) and use it to prove a bijective correspondence between
corepresentations of (A,Delta) and homomorphisms on the dual.


title: A natural extension of a left invariant lower semi-continuous weight

abstract:
  In this paper, we describe a natural method to extend left invariant weights
on C*-algebraic quantum groups. This method is then used to improve the left
invariance property of a left invariant weight. We also prove some kind of
uniqueness result for left Haar weights on C*-algebraic quantum groups arising
from algebraic ones.


title:
Universal C*-algebraic quantum groups arising from algebraic quantum
  groups

abstract:
  In this paper, we construct a universal C*-algebraic quantum group out of an
algebraic one. We show that this universal C*-algebraic quantum group has the
same rich structure as its reduced companion. This universal C*-algebraic
quantum group also satifies an upcoming definition of Masuda, Nakagami &
Woronowicz except for the possible non-faithfulness of the left Haar weight.


title: Covariant Sectors with Infinite Dimension and Positivity of the Energy

abstract:
  We consider a Moebius covariant sector, possibly with infinite dimension, of
a local conformal net of von Neumann algebras on the circle. If the sector has
finite index, it has automatically positive energy. In the infinite index case,
we show the spectrum of the energy always to contain the positive real line,
but, as seen by an example, it may contain negative values. We then consider
nets with Haag duality on the real line, or equivalently sectors with
non-solitonic extension to the dual net; we give a criterion for irreducible
sectors to have positive energy, namely this is the case iff there exists an
unbounded Moebius covariant left inverse. As a consequence the class of sectors
with positive energy is stable under composition, conjugation and direct
integral decomposition.


title: KMS-weights on C*-algebras

abstract:
  In this paper, we build a solid framework for KMS-weights on C*-algebras. We
use another definition than the one introduced by Combes, but prove that they
are equivalent.


title: Discrete product systems and twisted crossed products by semigroups

abstract:
  A product system E over a semigroup P is a family of Hilbert spaces {E_s:s\in
P} together with multiplications E_s \times E_t\to E_{st}. We view E as a
unitary- valued cocycle on P, and consider twisted crossed products A
\times_{\beta,E} P involving E and an action \beta of P by endomorphisms of a
C*-algebra A. When P is quasi-lattice ordered in the sense of Nica, we isolate
a class of covariant representations of E, and consider a twisted crossed
product B_P \times_{\tau,E} P which is universal for covariant representations
of E when E has finite-dimensional fibres, and in general is slightly larger.
In particular, when P=N and \dim E_1=\infty, our algebra B_\NN \times_{\tau,E}
N is a new infinite analogue of the Toeplitz-Cuntz algebras TO_n. Our main
theorem is a characterisation of the faithful representations of B_P
\times_{\tau,E} P.


title:
Two 1935 questions of Mazur about polynomials in Banach spaces: a
  counter-example

abstract:
  We construct a continuous scalar-valued 2-polynomial, $W$, on the separable
Hilbert space $l_2$ and an unbounded set $R\subset l_2$ such that (i) $W$ is
bounded on an $\epsilon$-neighbourhood of $R$; (ii) $W$ is unbounded on ${1/2}
R$; (iii) consequently, $W$ does not factor through any bounded 1-polynomial on
$l_2$ sending $R$ to a bounded set. This answers in the negative two 1935
questions asked by Mazur (problems 55 and 75 in the Scottish Book). The
construction is valid both over $\R$ and $\C$. (In finite dimensions the
questions were answered in the positive by Auerbach soon after being asked.)


title: Some universal constructions in abstract topological dynamics

abstract:
  This small survey of basic universal constructions related to the actions of
topological groups on compacta is centred around a new result --- an intrinsic
description of extremely amenable topological groups (i.e., those having a
fixed point in each compactum they act upon), solving a 1967 problem by
Granirer. Another old problem whose solution (in the negative) is noted here,
is a 1957 conjecture by Teleman on irreducible representations of general
topological groups. Our exposition covers the greatest ambit and the universal
minimal flow, as well as closely related constructions and results from
representation theory. We present in a simplified fashion the known examples of
extremely amenable groups and discuss their relationship with a 1969 problem by
Ellis. Open questions, including those from bordering disciplines, are reviewed
along the way.


title: Pure E_0-semigroups and absorbing states

abstract:
  An E_0-semigroup acting on B(H) is called pure if the intersection of the
ranges $\alpha_t(B(H))$, $t>0$, is the algebra of scalars. We determine all
pure E_0-semigroups which have a weakly continuous invariant state $\omega$ and
which are minimal in an appropriate sense. In such cases, for every normal
state $\rho$ there is convergence to equilibrium in the trace norm
$\lim_{t\to\infty}\| \rho\circ\alpha_t - \omega \| = 0.$ A normal state
$\omega$ with this property is called an absorbing state. Such E_0-semigroups
mest be cocycle perturbations of CAR/CCR flows, and we develop systematic
methods for constructing those perturbations which have absorbing states with
prescribed finite eigenvalue lists.


title: The index of a quantum dynamical semigroup

abstract:
  A numerical index is introduced for semigroups of completely positive maps of
$\Cal B(H)$ which generalizes the index of E_0-semigroups. It is shown that the
index of a unital completely positive semigroup agrees with the index of its
dilation to an E_0-semigroup, provided that the dilation is minimal.


title: On the index and dilations of completely positive semigroups

abstract:
  It is known that every semigroup of normal completely positive maps $P =
{P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a
minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K
containing H. The minimal dilation of P is unique up to conjugacy. In a
previous paper a numerical index was introduced for semigroups of completely
positive maps and it was shown that the index of P agrees with the index of its
minimal dilation to an E_0-semigroup. However, no examples were discussed, and
no computations were made. In this paper we calculate the index of a unital
completely positive semigroup whose generator is a bounded operator $ L:
B(H)\to B(H) $ in terms of natrual structures associated with the generator.
This includes all unital CP semigroups acting on matrix algebras. We also show
that the minimal dilation of the semigroup $P={\exp{tL}: t\geq 0}$ to an \esg\
is is cocycle conjugate to a CAR/CCR flow.


title: Subalgebras of C*-algebras III: multivariable operator theory

abstract:
  A d-contraction is a d-tuple $(T_1,...,T_d)$ of mutually commuting operators
acting on a common Hilbert space H such that $ \|T_1\xi_1+T_2\xi_2+...
+T_d\xi_d\|^2\leq \|\xi_1\|^2+\|\xi_2\|^2+...+\|\xi_d\|^2 $ for all
$\xi_1,\xi_2,...,\xi_d\in H$. These are the higher dimensional counterparts of
contractions. We show that many of the operator-theoretic aspects of function
theory in the unit disk generalize to the unit ball B_d in complex d-space,
including von Neumann's inequality and the model theory of contractions. These
results depend on properties of the d-shift, a distinguished d-contraction
which acts on a new $H^2$ space associated with B_d, and which is the higher
dimensional counterpart of the unilateral shift. $H^2$ and the d-shift are
highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the
d-shift relative to its generated C^*-algebra we find that there is more
uniqueness in dimension $d\geq 2$ than there is in dimension one.


title: Partial Representations and Amenable Fell Bundles over Free Groups

abstract:
  We show that a Fell bundle B = {B_t}_{t \in F}, over an arbitrary free group
F, is amenable, whenever it is orthogonal (in the sense that B_x^* B_y = 0, if
x and y are distinct generators of F) and semi-saturated (in the sense that
B_{ts} coincides with the closed linear span of B_t B_s, when the
multiplication ``ts'' involves no cancelation).


title: Locally Inner Actions on $C_0(X)$-Algebras

abstract:
  We make a detailed study of locally inner actions on C*-algebras whose
primitive ideal spaces have locally compact Hausdorff complete regularizations.
We suppose that $G$ has a representation group and compactly generated
abelianization $G_{ab}$. Then if the complete regularization of $\Prim(A)$ is
$X$, we show that the collection of exterior equivalence classes of locally
inner actions of $G$ on $A$ is parameterized by the group $\E_G(X)$ of exterior
equivalence classes of $C_0(X)-actions of $G$ on $C_0(X,\K)$. Furthermore, we
exhibit a group isomorphism of $\E_G(X)$ with the direct sum $H^1(X,\sheaf
\hat{G_{ab}}) \oplus C(X,H^2(G,\T))$. As a consequence, we can compute the
equivariant Brauer group $\Br_G(X)$ for $G$ acting trivially on $X$.


title: Crossed products by $C_0(X)$-actions

abstract:
  Suppose that $G$ has a representation group $H$, that $G_{ab}:=
G/\bar{[G,G]}$ is compactly generated, and that $A$ is a \cs-algebra for which
the complete regularization of $\Prim(A)$ is a locally compact Hausdorff space
$X$. In a previous article, we showed that there is a bijection $\alpha \mapsto
(Z_\alpha,f_\alpha)$ between the collection of exterior equivalence classes of
locally inner actions $\alpha:G\to\Aut(A)$, and the collection of principal
$\hgab$-bundles $Z_\alpha$ together with continuous functions $f_\alpha:X\to
H^2(G,\T)$. In this paper, we compute the crossed products $A\rtimes_\alpha G$
in terms of the data $Z_\alpha$, $f_\alpha$, and~$\cs(H)$.


title: The Brauer Group of a Locally Compact Groupoid

abstract:
  We define the Brauer group $\Br(G)$ of a locally compact groupoid $G$ to be
the set of Morita equivalence classes of pairs $(\A,\alpha)$ consisting of an
elementary C*-bundle $\A$ over $G^{(0)}$ satisfying Fell's condition and an
action $\alpha$ of $G$ on $\A$ by $*$-isomorphisms. When $G$ is the
transformation groupoid $X\times H$, then $\Br(G)$ is the equivariant Brauer
group $\Br_H(X)$.
  In addition to proving that $\Br(G)$ is a group, we prove three isomorphism
results. First we show that if $G$ and $H$ are equivalent groupoids, then
$\Br(G)$ and $\Br(H)$ are isomorphic. This generalizes the result that if $G$
and $H$ are groups acting freely and properly on a space $X$, say $G$ on the
left and $H$ on the right then $\Br_G(X/H)$ and $\Br_H(G/ X)$ are isomorphic.
Secondly we show that the subgroup $\Br_0(G)$ of $\Br(G)$ consisting of classes
$[\A,\alpha]$ with $\A$ having trivial Dixmier-Douady invariant is isomorphic
to a quotient $\E(G)$ of the collection $\Tw(G)$ of twists over $G$. Finally we
prove that $\Br(G)$ is isomorphic to the inductive limit $\Ext(G,T)$ of the
groups $\E(G^X)$ where $X$ varies over all principal $G$ spaces $X$ and $G^X$
is the imprimitivity groupoid associated to $X$.


title: Deformation Quantization via Fell Bundles

abstract:
  A method for deforming C*-algebras is introduced, which applies to
C*-algebras that can be described as the cross-sectional C*-algebra of a Fell
bundle. Several well known examples of non-commutative algebras, usually
obtained by deforming commutative ones by various methods, are shown to fit our
unified perspective of deformation via Fell bundles. Examples are the
non-commutative spheres of Matsumoto, the non-commutative lens spaces of
Matsumoto and Tomiyama, and the quantum Heisenberg manifolds of Rieffel. In a
special case, in which the deformation arises as a result of an action of
R^{2d}, assumed to be periodic in the first d variables, we show that we get a
strict deformation quantization.


title:
The Quantum Stochastic Differential Equation Is Unitarily Equivalent to
  a Symmetric Boundary Value Problem for the Schr\"odinger Equation

abstract:
  We prove that the solution of the Hudson-Parthasarathy quantum stochastic
differential equation in the Fock space coincides with the solution of a
symmetric boundary value problem for the Schr\"odinger equation in the
interaction representation generated by the energy operator of the environment.
The boundary conditions describe the jumps in the phase and the amplitude of
the Fourier transforms of the Fock vector components as any of its arguments
changes the sign. The corresponding Markov evolution equation (the Lindblad
equation or the ``master equation'') is derived from the boundary value problem
for the Schr\"odinger equation.


title:
The functional calculus for regular operators in Hilbert C*-modules
  revisited

abstract:
  Woronowicz introduced the functional calculus for normal operators in Hilbert
C*-modules. The aim of this paper is to translate, if possible, some basic
properties of the functional calculus in Hilbert spaces to the Hilbert
C*-module framework.


title:
Infinitesimal Takesaki duality of Hamiltonian vector fields on a
  symplectic manifold

abstract:
  For an infinitesimal symplectic action of a Lie algebra ${\goth g}$ on a
symplectic manifold, we construct an infinitesimal crossed product of
Hamiltonian vector fields and Lie algebra ${\goth g}$. We obtain its second
crossed product in case ${\goth g}=R$ and show an infinitesimal version for a
theorem type of Takesaki duality.


title: Unitary Representations of Lie Groups with Reflection Symmetry

abstract:
  We consider the following class of unitary representations $\pi $ of some
(real) Lie group $G$ which has a matched pair of symmetries described as
follows: (i) Suppose $G$ has a period-2 automorphism $\tau $, and that the
Hilbert space $\mathbf{H} (\pi)$ carries a unitary operator $J$ such that $J\pi
=(\pi \circ \tau)J$ (i.e., selfsimilarity). (ii) An added symmetry is implied
if $\mathbf{H} (\pi)$ further contains a closed subspace $\mathbf{K}_0 $ having
a certain order-covariance property, and satisfying the $\mathbf{K}_0
$-restricted positivity: $< v \mid Jv > \ge 0$, $\forall v\in \mathbf{K}_0 $,
where $< \cdot \mid \cdot >$ is the inner product in $\mathbf{H} (\pi)$. From
(i)--(ii), we get an induced dual representation of an associated dual group
$G^c$. All three properties, selfsimilarity, order-covariance, and positivity,
are satisfied in a natural context when $G$ is semisimple and hermitean; but
when $G$ is the $(ax+b)$-group, or the Heisenberg group, positivity is
incompatible with the other two axioms for the infinite-dimensional irreducible
representations. We describe a class of $G$, containing the latter two, which
admits a classification of the possible spaces $\mathbf{K}_0 \subset \mathbf{H}
(\pi)$ satisfying the axioms of selfsimilarity and order-covariance.


title: Spectral asymptotics of periodic elliptic operators

abstract:
  We demonstrate that the structure of complex second-order strongly elliptic
operators $H$ on ${\bf R}^d$ with coefficients invariant under translation by
${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$,
$z\in{\bf T}^d$, of $H$ with $z$-periodic boundary conditions acting on
$L_2({\bf I}^d)$ where ${\bf I}=[0,1>$. If the semigroup $S$ generated by $H$
has a H\"older continuous integral kernel satisfying Gaussian bounds then the
semigroups $S^z$ generated by the $H_z$ have kernels with similar properties
and $z\mapsto S^z$ extends to a function on ${\bf C}^d\setminus\{0\}$ which is
analytic with respect to the trace norm. The sequence of semigroups $S^{(m),z}$
obtained by rescaling the coefficients of $H_z$ by $c(x)\to c(mx)$ converges in
trace norm to the semigroup $\hat{S}^z$ generated by the homogenization
$\hat{H}_z$ of $H_z$. These convergence properties allow asymptotic analysis of
the spectrum of $H$.


title: Light-Cone Expansion of the Dirac Sea with Light Cone Integrals

abstract:
  The Dirac sea is calculated in an expansion around the light cone. The method
is to analyze the perturbation expansion for the Dirac sea in position space.
This leads to integrals over expressions containing distributions which are
singular on the light cone. We derive asymptotic formulas for these "light cone
integrals" in terms of line integrals over the external potential and its
partial derivatives.
  The calculations are based on the perturbation expansion for the Dirac sea in
the preprint gr-qc/9606040 and yield the formulas listed in the appendix of
this preprint. The results can be obtained easier with a combination of
calculations in position and momentum space (see the corresponding preprints on
the hep-th server). Therefore the calculations are preliminary and will remain
unpublished; they are intended as reference for people who encounter similar
mathematical problems.


title: Normalization Integrals of Orthogonal Heun Functions

abstract:
  A formula for evaluating the quadratic normalization integrals of orthogonal
Heun functions over the real interval 0 <= x <= 1 is derived using a simple
limiting procedure based upon the associated differential equation. The
resulting expression gives the value of the normalization integral explicitly
in terms of the local power-series solutions about x=0 and x=1 and their
derivatives. This provides an extremely efficient alternative to numerical
integration for the development of an orthonormal basis using Heun functions,
because all of the required information is available as a by-product of the
search for the eigenvalues of the differential equation.
  02.30.Gp; 02.30.Hq; 02.70.-c; 02.60.Jh

categories: funct-an math-ph math.CA math.FA math.MP math.NA


title: Relations between asymptotic and Fredholm representations

abstract:
  We prove that for matrix algebras $M_n$ there exists a monomorphism $(\prod_n
M_n/\oplus_n M_n)\otimes C(S^1) \to {\cal Q} $ into the Calkin algebra which
induces an isomorphism of the $K_1$-groups. As a consequence we show that every
vector bundle over a classifying space $B\pi$ which can be obtained from an
asymptotic representation of a discrete group $\pi$ can be obtained also from a
representation of the group $\pi\times Z$ into the Calkin algebra. We give also
a generalization of the notion of Fredholm representation and show that
asymptotic representations can be viewed as asymptotic Fredholm
representations.


title: The C*-algebra of a Hilbert Bimodule

abstract:
  We regard a right Hilbert C*-module X over a C*-algebra A endowed with an
isometric *-homomorphism \phi: A\to L_A(X) as an object X_A of the C*-category
of right Hilbert A-modules. Following a construction by the first author and
Roberts, we associate to it a C*-algebra O_{X_A} containing X as a ``Hilbert
A-bimodule in O_{X_A}''. If X is full and finite projective O_{X_A} is the
C*-algebra C*(X), the generalization of the Cuntz-Krieger algebras introduced
by Pimsner. More generally, C*(X) is canonically embedded in O_{X_A} as the
C*-subalgebra generated by X. Conversely, if X is full, O_{X_A} is canonically
embedded in the bidual of C*(X). Moreover, regarding X as an object A_X_A of
the C*-category of Hilbert A-bimodules, we associate to it a C*-subalgebra
O_{A_X_A} of O_{X_A} commuting with A, on which X induces a canonical
endomorphism \rho. We discuss conditions under which A and O_{A_X_A} are the
relative commutant of each other and X is precisely the subspace of
intertwiners in O_{X_A} between the identity and \rho on O_{A_X_A}. We also
discuss conditions which imply the simplicity of C*(X) or of O_{X_A}; in
particular, if X is finite projective and full, C*(X) will be simple if A is
X-simple and the ``Connes spectrum'' of X is the circle.


title: Convergence of the Splitting Method for Shallow Water Equations

abstract:
  In this paper we analyze the convergence of the splitting method for shallow
water equations. In particular, we give an analytical estimation of the time
step which is necessary for the convergence and then we study the behaviour of
the motion of the shallow water in the Venice lagoon by using the splitting
method with a finite element space discretization. The numerical calculations
show that the splitting method is convergent if the time step of the first part
is sufficiently small and that it gives a good agreement with the experimental
data.


title: Pseudodifferential calculus on manifolds with corners and groupoids

abstract:
  We build a longitudinally smooth differentiable groupoid associated to any
manifold with corners. The pseudodifferential calculus on this groupoid
coincides with the pseudodifferential calculus of Melrose (also called
b-calculus). We also define an algebra of rapidly decreasing functions on this
groupoid; it contains the kernels of the smoothing operators of the (small)
b-calculus.


title: One-parameter representations on C*-algebras

abstract:
  Strongly continuous one-parameter representations on C*-algebras and their
extension to the multiplier algebra are investigated. We give also a proof of
the Stone theorem on Hilbert C*-modules and look into some related problems.


title: The analytic structure of an algebraic quantum group

abstract:
  A. Van Daele introduced and investigated so-called algebraic quantum groups.
We proved that such algebraic quantum groups give rise to C*-algebraic quantum
groups in the sense of Masuda, Nakagami & Woronowicz. We prove in this paper
that the analytic structure of these C*-algebraic quantum groups can be pulled
down to the algebraic quantum group.


title:
Operator interpretation of resonances arising in spectral problems for 2
  x 2 operator matrices

abstract:
  We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with
self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the
orthogonal sum {\cal H}={\cal H}_0\oplus{\cal H}_1 of Hilbert spaces {\cal H}_0
and {\cal H}_1. We are especially interested in the case where the spectrum of,
say, A_1 is partly or totally embedded into the continuous spectrum of A_0 and
the transfer function M_1(z)=A_1-z+V_1(z), where
V_1(z)=B_{10}(z-A_0)^{-1}B_{01}, admits analytic continuation (as an
operator-valued function) through the cuts along branches of the continuous
spectrum of the entry A_0 into the unphysical sheet(s) of the spectral
parameter plane. The values of z in the unphysical sheets where M_1^{-1}(z) and
consequently the resolvent (H-z)^{-1} have poles are usually called resonances.
A main goal of the present work is to find non-selfadjoint operators whose
spectra include the resonances as well as to study the completeness and basis
properties of the resonance eigenvectors of M_1(z) in {\cal H}_1. To this end
we first construct an operator-valued function V_1(Y) on the space of operators
in {\cal H}_1 possessing the property: V_1(Y)\psi_1=V_1(z)\psi_1 for any
eigenvector \psi_1 of Y corresponding to an eigenvalue z and then study the
equation H_1=A_1+V_1(H_1). We prove the solvability of this equation even in
the case where the spectra of A_0 and A_1 overlap. Using the fact that the root
vectors of the solutions H_1 are at the same time such vectors for M_1(z), we
prove completeness and even basis properties for the root vectors (including
those for the resonances).

categories: funct-an hep-th math-ph math.MP math.OA math.SP nucl-th


title: Green's Functions and Energy Decay on Homogeneous Spaces

abstract:
  We consider a homogeneous space X=(X,d,m) of dimension $\nu \geq 1$ 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, with local characteristic
constant c_{0}(x) and c_{1}(r), then a Green's function estimate from above and
below is obtained. A Saint-Venant-like principle is recovered in terms of the
Energy's decay.


title: Expressing the Field Operator in Terms of Currents and Densities

abstract:
  A formula is written down for the annhilation operator(bose or fermi) in
terms of the corresponding observable bilinears namely currents and densities.
The Fock space representation of these formulas is clarified. A conjecture is
written down that shows that such a correspondence is unique.


title: Abelian topological groups without irreducible Banach representations

abstract:
  We exhibit abelian topological groups admitting no nontrivial strongly
continuous irreducible representations in Banach spaces. Among them are some
abelian Banach-Lie groups and some monothetic subgroups of the unitary group of
a separable Hilbert space.


title:
Exponential Bounds on Curvature-Induced Resonances in a Two-Dimensional
  Dirichlet Tube

abstract:
  We consider curvature-induced resonances in a planar two-dimensional
Dirichlet tube of a width $ d $. It is shown that the distances of the
corresponding resonance poles from the real axis are exponentially small as $
d\to 0+ $, provided the curvature of the strip axis satisfies certain
analyticity and decay requirements.

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


title: Morita equivalences between fixed point algebras and crossed products

abstract:
  In this paper, we will prove that if $A$ is a $C^*$-algebra with an effective
coaction $\epsilon$ by a compact quantum group, then the fixed point algebra
and the reduced crossed product are Morita equivalent. As an application, we
prove an imprimitivity type theorem for crossed products of coactions by
discrete Kac $C^*$-algebras.


title:
Bound-state asymptotic estimates for window-coupled Dirichlet strips and
  layers

abstract:
  We consider the discrete spectrum of the Dirichlet Laplacian on a manifold
consisting of two adjacent parallel strips or planar layers coupled by a finite
number N of windows in the common boundary. If the windows are small enough,
there is just one isolated eigenvalue. We find upper and lower asymptotic
bounds on the gap between the eigenvalue and the essential spectrum in the
planar case, as well as for N=1 in three dimensions. Based on these results, we
formulate a conjecture on the weak-coupling asymptotic behaviour of such bound
states.


title:
Endomorphisms of B(H), extensions of pure states, and a class of
  representations of O_n

abstract:
  Let F_n be the fixed-point algebra of the gauge action of the circle on the
Cuntz algebra O_n. For every pure state \rho of F_n and every representation
\theta of C(T) we construct a representation of O_n, and we use the resulting
class of representations to parameterize the space of all states of O_n which
extend \rho. We show that the gauge group acts transitively on the pure
extensions of \rho and that the action is p-to-1 with p the period of \rho
under the usual shift. We then use the above representations of O_n to
construct endomorphisms of B(H) which we classify up to conjugacy in terms of
the parameters \rho and \theta. In particular our construction yields every
ergodic endomorphism \alpha whose tail algebra $\bigcap_k\alpha^k(B(H))$ has a
minimal projection, and our results classify these ergodic endomorphisms by an
equivalence relation on the pure states of F_n. As examples we analyze the
ergodic endomorphisms arising from periodic pure product states of F_n, for
which we are able to give a geometric complete conjugacy invariant,
generalizing results of Stacey, Laca, and Bratteli-Jorgensen-Price on the
shifts of Powers.


title: Some operator ideals in non-commutative functional analysis

abstract:
  We characterize classes of linear maps between operator spaces $E$, $F$ which
factorize through maps arising in a natural manner via the Pisier vector-valued
non-commutative $L^p$ spaces $S_p[E^*]$ based on the Schatten classes on the
separable Hilbert space $l^2$. These classes of maps can be viewed as
quasi-normed operator ideals in the category of operator spaces, that is in
non-commutative (quantized) functional analysis. The case $p=2$ provides a
Banach operator ideal and allows us to characterize the split property for
inclusions of $W^*$-algebras by the 2-factorable maps. The various
characterizations of the split property have interesting applications in
Quantum Field Theory.


title: Operator space structures and the split property II

abstract:
  A characterization of the split property for an inclusion $N\subset M$ of
$W^*$-factors with separable predual is established in terms of the canonical
non-commutative $L^2$ embedding considered in \cite{B1,B2} $$ \F_2:a\in N\to
\D_{M,\Om}^{1/4}a\Om\in L^2(M,\Om) $$ associated with an arbitrary fixed
standard vector $\Om$ for $M$. This characterization follows an analogous
characterization related to the canonical non-commutative $L^1$ embedding $$
\F_1:a\in N\to (\cdot\Om,J_{M,\Om}a\Om)\in L^1(M,\Om) $$ also considered in
\cite{B1,B2} and studied in \cite{F}. The split property for a Quantum Field
Theory is characterized by equivalent conditions relative to the
non-commutative embeddings $\F_i$, $i=1,2$, constructed by the modular
Hamiltonian of a privileged faithful state such as e.g. the vacuum state. The
above characterization would be also useful for theories on a curved space-time
where there exists no a-priori privileged state.


title: Dense analytic subspaces in fractal $L^{2}$-spaces

abstract:
  We consider self-similar measures $\mu $ with support in the interval $0\leq
x\leq 1$ which have the analytic functions $\left\{e^{i2\pi nx}:n=0,1,2,...
\right\} $ span a dense subspace in $L^{2}(\mu) $. Depending on the fractal
dimension of $\mu $, we identify subsets $P\subset \mathbb{N}_{0}=\{0,1,2,...
\} $ such that the functions $\{e_{n}:n\in P\} $ form an orthonormal basis for
$L^{2}(\mu) $. We also give a higher-dimensional affine construction leading to
self-similar measures $\mu $ with support in $\mathbb{R}^{\nu}$. It is obtained
from a given expansive $\nu $-by-$\nu $ matrix and a finite set of translation
vectors, and we show that the corresponding $L^{2}(\mu) $ has an orthonormal
basis of exponentials $e^{i2\pi \lambda \cdot x}$, indexed by vectors $\lambda
$ in $\mathbb{R}^{\nu}$, provided certain geometric conditions (involving the
Ruelle transfer operator) hold for the affine system.


title: Finite dimensional quotients of commutative operator algebras

abstract:
  The matrix normed structure of the unitization of a (non-selfadjoint)
operator algebra is determined by that of the original operator algebra. This
yields a classification up to completely isometric isomorphism of
two-dimensional unital operator algebras. This allows to define invariant
distances on the spectrum of commutative operator algebras analogous to the
Caratheodory distance for complex manifolds. Moreover, unitizations of
two-dimensional operator algebras with zero multiplication provide a rich class
of counterexamples. Especially, several badly behaved quotients of function
algebras are exhibited. Recently, Arveson has developed a model theory for
d-contractions. Quotients of the operator algebra of the d-shift are much more
well-behaved than quotients of function algebras. Completely isometric
representations of these quotients are obtained explicitly. This provides a
generalization of Nevanlinna-Pick theory. An important property of quotients of
the d-shift algebra is that their quotients of finite dimension r have
completely isometric representations by rxr-matrices. Finally, the class of
commutative operator algebras with this property is investigated.


title: Quantum Mechanics and Operator algebras on the Hilbert ball

abstract:
  Cirelli, Mani\`{a} and Pizzocchero generalized quantum mechanics by
K\"{a}hler geometry. Furthermore they proved that any unital C$^{*}$-algebra is
represented as a function algebra on the set of pure states with a
noncommutative $*$-product as an application. The ordinary quantum mechanics is
regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal
H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an
infinite dimensional K\"{a}hler manifold of positive constant holomorphic
sectional curvature. In general, such dynamical system is constructed for a
general K\"{a}hler manifold of nonzero constant holomorphic sectional curvature
$c$. The Hilbert ball $B_{{\cal H}}$ is defined by the open unit ball in ${\cal
H}$ and it is a K\"{a}hler manifold with $c<0$. We introduce the quantum
mechanics on $B_{{\cal H}}$. As an application, we show the structure of the
noncommutative function algebra on $B_{{\cal H}}$.


title:
Distributional Asymptotic Expansions of Spectral Functions and of the
  Associated Green Kernels

abstract:
  Asymptotic expansions of Green functions and spectral densities associated
with partial differential operators are widely applied in quantum field theory
and elsewhere. The mathematical properties of these expansions can be clarified
and more precisely determined by means of tools from distribution theory and
summability theory. (These are the same, insofar as recently the classic
Cesaro-Riesz theory of summability of series and integrals has been given a
distributional interpretation.) When applied to the spectral analysis of Green
functions (which are then to be expanded as series in a parameter, usually the
time), these methods show: (1) The "local" or "global" dependence of the
expansion coefficients on the background geometry, etc., is determined by the
regularity of the asymptotic expansion of the integrand at the origin (in
"frequency space"); this marks the difference between a heat kernel and a
Wightman two-point function, for instance. (2) The behavior of the integrand at
infinity determines whether the expansion of the Green function is genuinely
asymptotic in the literal, pointwise sense, or is merely valid in a
distributional (cesaro-averaged) sense; this is the difference between the heat
kernel and the Schrodinger kernel. (3) The high-frequency expansion of the
spectral density itself is local in a distributional sense (but not pointwise).
These observations make rigorous sense out of calculations in the physics
literature that are sometimes dismissed as merely formal.

categories: funct-an gr-qc hep-th math-ph math.FA math.MP


title:
On the AF embeddability of crossed products of AF algebras by the
  integers

abstract:
  It is shown that if A is an AF algebra then a crossed product of A by the
integers can be embedded into an AF algebra if and only if the crossed product
is stably finite. This equivalence follows from a simple K-theoretic
characterization of AF embeddability. It is then shown that if a crossed
product of an AF algebra by the integers is AF embeddable then the AF embedding
can be chosen in such a way as to induce a rationally injective map on K_0 of
the crossed product.


title:
The Laplace Transform Method for Linear Differential Equations of the
  Fractional Order

abstract:
  The Laplace transform method for solving of a wide class of initial value
problems for fractional differential equations is introduced. The method is
based on the Laplace transform of the Mittag-Leffler function in two
parameters. To extend the proposed method for the case of so-called
"sequential" fractional differential equations, the Laplace transform for the
''sequential'' fractional derivative is also obtained. Besides that, tools
necessary for testing candidate solutions by direct substitution in
corresponding equations are introduced: fractional derivatives of the
Mittag-Leffler function and the rule for the fractional differentiation of
integrals depending on a parameter. Definition of the fractional Green's
function is given and some of its properties, necessary for constructing
solutions of initial-value problems for fractional linear differential
equations, are presented. Explicit expressions for the fractional Green's
function for the special cases of one-, two-, three- and four-term equations
are given, as well as the explicit expression for an arbitrary n-term
fractional linear ordinary differential equation with constant coefficients.
Several examples of solution of various types of fractional differential
equations, including fractional diffusion-wave equation. The bibliography (54
items) covers many field of possible application of fractional derivatives.


title:
On the von Neumann Inequality for Linear Matrix Functions of Several
  Variables

abstract:
  The theorem on the existence of three commuting contractions on a Hilbert
space and of a linear homogeneous matrix function of three independent
variables for which the generalized von Neumann inequality fails is proved.


title: On a class of stochastic differential equations used in quantum optics

abstract:
  Stochastic differential equations for processes with values in Hilbert spaces
are now largely used in the quantum theory of open systems. In this work we
present a class of such equations and discuss their main properties; moreover,
we explain how they are derived from purely quantum mechanical models, where
the dynamics is represented by a unitary evolution in a Hilbert space, and how
they are related to the theory of continual measurements. An essential tool is
an isomorphism between the bosonic Fock space and the Wiener space, which allow
to connect certain quantum objects with probabilistic ones.


title: An Addendum to Krein's Formula

abstract:
  We provide additional results in connection with Krein's formula, which
describes the resolvent difference of two self-adjoint extensions A_1 and A_2
of a densely defined closed symmetric linear operator A with (possibly
infinite) equal deficiency indices. In particular, we explicitly derive the
linear fractional transformation relating the operator-valued Weyl-Titchmarsh
M-functions M_1(z) and M_2(z) corresponding to A_1 and A_2.


title: Pure states on O_d

abstract:
  We study representations of the Cuntz algebras O_d and their associated
decompositions. In the case that these representations are irreducible, their
restrictions to the gauge-invariant subalgebra UHF_d have an interesting cyclic
structure. If S_i, 1 \leq i \leq d, are representatives of the Cuntz relations
on a Hilbert space H, special attention is given to the subspaces which are
invariant under S_i^*. The applications include wavelet multiresolutions
corresponding to wavelets of compact support (to appear in the later paper
\cite{BEJ97}), and finitely correlated states on one-dimensional quantum spin
chains.


title:
Boundary values of holomorphic functions and some spectral problems for
  unitary representations

abstract:
  We consider hilbert spaces of holomorphic functions in Cartan domains (in
particular in ball and polydisk) and operator of restriction of holomorphic
function to a submanifold in Shilov boundary. We discuss conditions when this
operator exists.
  Using such 'trace theorems' it is possible to construct discrete increments
to spectra of some unitary representation and to catch singular unitary
representations in the spectra. We also discuss spectral problems related to
Berezin type kernels on riemann symmetric spaces.


title:
Sufficient Conditions for Conservativity of Minimal Quantum Dynamical
  Semigroups

abstract:
  The conservativity of a minimal quantum dynamical semigroup is proved
whenever there exists a ``reference'' subharmonic operator bounded from below
by the dissipative part of the infinitesimal generator. We discuss applications
of this criteria in mathematical physics and quantum probability.


title: Cyclic Cohomology of Etale Groupoids; The General Case

abstract:
  We give a general method for computing the cyclic cohomology of crossed
products by etale groupoids, extending the Feigin-Tsygan-Nistor spectral
sequences. In particular we extend the computations performed by Brylinski,
Burghelea,Connes and Nistor for the convolution algebra -of compactly supported
smooth functions- of an etale groupoid, removing the Hausdorffness assumption
and including the computation of the hyperbolic components. Examples like group
actions on manifolds and foliations are considered.