Holomorphic Lagrangian fibrations on hypercomplex manifolds

Andrey Soldatenkov111Andrey Soldatenkov is partially supported by AG
Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023, Misha
Verbitsky222Misha Verbitsky is partially supported by RFBR grant
10-01-93113-NCNIL-a, RFBR grant 09-01-00242-a, Science Foundation of the SU-
HSE award No. 10-09-0015 and AG Laboratory NRU-HSE, RF government grant, ag.
11.G34.31.0023.

Abstract

A hypercomplex manifold is a manifold equipped with a triple of complex
structures satisfying the quaternionic relations. A holomorphic Lagrangian
variety on a hypercomplex manifold with trivial canonical bundle is a
holomorphic subvariety which is calibrated by a form associated with the
holomorphic volume form; this notion is a generalization of the usual
holomorphic Lagrangian subvarieties known in hyperkähler geometry. An HKT
(hyperkähler with torsion) metric on a hypercomplex manifold is a metric
determined by a local potential, in a similar way to the Kähler metric. We
prove that a base of a holomorphic Lagrangian fibration is always Kähler, if
its total space is HKT. This is used to construct new examples of hypercomplex
manifolds which do not admit an HKT structure.

###### Contents

  1. 1 Introduction
    1. 1.1 HKT metrics and $SL(n,{\mathbb{H}})$-structures
    2. 1.2 Calibrations on manifolds and Lagrangian fibrations
  2. 2 Preliminaries
    1. 2.1 Hypercomplex manifolds and quaternionic Dolbeault complex
    2. 2.2 $SL(n,{\mathbb{H}})$-manifolds and holomorphic Lagrangian calibration
  3. 3 Holomorphic Lagrangian fibrations on $SL(n,{\mathbb{H}})$-manifolds
    1. 3.1 HKT structures and Lagrangian fibrations
    2. 3.2 Examples of Lagrangian fibrations

##  1 Introduction

###  1.1 HKT metrics and $SL(n,{\mathbb{H}})$-structures

Let $I,J,K$ be complex structures on a manifold $M$ satisfying the
quaternionic relation $I\circ J=-J\circ I=K$. Then $(M,I,J,K)$ is called a
hypercomplex manifold. Hypercomplex manifolds are quaternionic analogues of
complex manifolds, in the same way as hyperkähler manifolds are quaternionic
analogues of Kähler manifolds. However, the theory of hypercomplex manifolds
is much richer in examples and (unlike that of hyperkähler manifolds) still
not well understood.

For more details on hypercomplex and hyperkähler structures, please see
Subsection 2.1.

The term hypercomplex is due to C. P. Boyer, [Bo], who classified compact
hypercomplex manifolds in $\dim_{\mathbb{H}}=1$. These are: K3 surfaces,
compact 2-dimensional complex tori, and Hopf surfaces. However, in dimension
$>1$ there is no classification apparent.

The notion of hypercomplex structure was considered as early as in 1955 by
[Ob], who proved existence and uniqueness of a torsion-free connection
preserving a hypercomplex structure. However, the first non-trivial examples
were obtained only in 1988 by physicists Ph. Spindel, A. Sevrin, W. Troost, A.
Van Proeyen ([SSTV]), and the intensive study of hypercomplex structures began
with the paper [J] by D. Joyce, who classified hypercomplex structures on
homogeneous spaces, and constructed one on each compact Lie group multiplied
by a compact torus of appropriate dimension.

The main geometric tool allowing one to work with the hypercomplex manifolds
is a notion of an HKT (“hyperkähler with torsion”) metric, due to physicists
Howe and Papadopoulos ([HP]). For a definition and a discussion of HKT
structure, please see Subsection 2.1. HKT structures were much used in physics
since mid-1990-ies ([GP1], [GP2], [GPS]), and then mathematicians started
using HKT structures to study the hypercomplex manifolds.

In [V1], the second named author developed Hodge theory for HKT manifolds, and
it turned out to be quite useful. To illustrate the usability of HKT metrics,
let us quote the following result, obtained in [V4]. Let $(M,I,J,K)$ be a
hypercomplex manifold with $(M,I)$ a complex manifold of Kähler type. Then
$(M,I)$ is in fact hyperkähler (that it, admits a hyperkähler structure). The
idea of the proof is that any Kähler metric on $(M,I)$ gives an HKT metric on
$(M,I,J,K)$, and this metric can be used to apply Hodge theory. The interplay
between Kähler geometry and HKT geometry is quite extensive, and can be used
to study the geometry of hypercomplex manifolds extensively.

In the present paper, we obtain a result in a similar vein, constructing a
Kähler metric on a base of holomorphic Lagrangian fibration on an HKT
manifold.

Of course, the notion of “holomorphic Lagrangian fibration” is itself highly
non-trivial, because an HKT manifold is not necessarily holomorphically
symplectic. It was developed in [GV], using the theory of calibrations and the
holonomy of the Obata connection.

The Obata connection is an immensely powerful tool, known since 1950-ies
([Ob]), but even the most primitive invariants of Obata connection, such as
its holonomy, are hard to compute, and still unknown in most cases. When
$(M,I,J,K)$ admits a hyperkähler metric, Obata connection is equal to the
Levi-Civita connection of the hyperkähler metric and its holonomy is
$\operatorname{Sp}(n)$. The converse is also true: whenever holonomy of a
torsion-free connection is $Sp(n)$, it is a Levi-Civita connection of a
hyperkähler manifold.

In a general situation, it’s hard to say anything definite about the holonomy
$\operatorname{Hol}(\nabla)$ of the Obata connection. It is clear that
$\operatorname{Hol}(\nabla)\subset GL({\mathbb{H}},n)$, but the equality is
rarely realized. In fact, before the publication of the paper [Sol] in 2011,
there was no single example of a hypercomplex manifold with
$\operatorname{Hol}(\nabla)=GL({\mathbb{H}},n)$. In [Sol], the first named
author provided an example of such a manifold, by proving that
$\operatorname{Hol}(\nabla)=GL({\mathbb{H}},2)$ for the homogeneous
hypercomplex structure on $SU(3)$, constructed by D. Joyce in [J].

However, there are many examples of hypercomplex manifolds with
$\operatorname{Hol}(\nabla)$ $\subsetneq GL({\mathbb{H}},n)$. Two biggest
known families of hypercomplex manifolds are the Joyce’s homogeneous manifolds
from [J], and hypercomplex nilmanifolds ([BD]). As shown in [BDV], for all
nilmanifolds, the holonomy of the Obata connection lies in

$SL({\mathbb{H}},n)=[GL({\mathbb{H}},n),GL({\mathbb{H}},n)]\subset
GL({\mathbb{H}},n).$

A hypercomplex manifold with holonomy in $SL(n,{\mathbb{H}})$ is called an
$SL(n,{\mathbb{H}})$-manifold. Such manifold might have holonomy group which
is strictly less than $SL(n,{\mathbb{H}})$; in fact, no example of a manifold
with $\operatorname{Hol}(\nabla)=SL({\mathbb{H}},n)$ is known so far.

For any $SL(n,{\mathbb{H}})$-manifold, the Obata connection on the canonical
bundle $K(M,I)=\Lambda^{2n,0}(M,I)$ of $(M,I)$ preserves a non-trivial
section. This implies that $K(M,I)$ is holomorphically trivial (see [V3]). In
the presence of an HKT metric, the converse is also true: any compact
hypercomplex manifold with holomorphically trivial canonical bundle $K(M,I)$
and an HKT metric satisfies $\operatorname{Hol}(\nabla)\subset
SL({\mathbb{H}},n)$. This result is obtained in [V3] using the Hodge theory
for HKT manifolds developed in [V1].

The Hodge-theoretic constructions of [V1] work for
$SL(n,{\mathbb{H}})$-manifolds with HKT-structure especially well. For such
manifolds, one obtains Hodge-type decomposition on the holomorphic cohomology
bundle $H^{*}({\cal O}_{(M,I)})$.

###  1.2 Calibrations on manifolds and Lagrangian fibrations

Let $M$ be a Riemannian manifold. A calibration on $M$ is a closed $k$-form
$\eta$, such that $\eta(x_{1},...,x_{k})\leqslant 1$ for each orthonormal
$k$-tuple $x_{1},...,x_{k}\in TM$. A $k$-dimensional oriented subspace
$V\subset T_{x}M$ is called calibrated if the Riemannian volume of $V$ is
equal to $\eta{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$, and a subvariety
$Z\subset M$ of dimension $k$ is called calibrated if its singularities are of
Hausdorff codimension $\geqslant 1$, and all smooth tangent planes
$T_{z}Z\subset T_{z}M$ are calibrated.

The theory of calibrations has a long and distinguished history, starting from
the work of Harvey and Lawson [HL]. In [GV], several families of calibrations
were constructed on hyperkähler manifolds, using the quaternionic linear
algebra. Let $(M,I,J,K)$ be a hyperkähler manifold, and
$\omega_{I},\omega_{J},\omega_{K}$ its Kähler forms. These forms generate an
interesting commutative subalgebra in $\Lambda^{*}(M)$ (see [GV], and [V0] for
structure results about this algebra). In [GV], Grantcharov and Verbitsky
discovered several new calibrations which are expressed as polynomials of
$\omega_{I},\omega_{J},\omega_{K}$. One of these calibrations, denoted as
$\Psi$ in the sequel (see (2.2)), is called holomorphic Lagrangian
calibration. On a hyperkähler manifold, it calibrates holomorphic Lagrangian
subvarieties in $(M,I)$.

It is surprising (and quite wonderful) that this form remains closed, even
when the metric is not hyperkähler, provided that the Obata connection of
$(M,I,J,K)$ belongs to $SL(n,{\mathbb{H}})$. The hyperkähler condition can be
weakened drastically: it suffices to assume that the metric $g$ is
quaternionic Hermitian, and the Obata-parallel section of $K(M,I)$ has
constant length with respect to $g$. In this case $\Psi$ is closed. If
rescaled properly, this form is a calibration, calibrating a new class of
complex subvarieties of $(M,I)$ called holomorphic Lagrangian (see [GV] and
3.1 for more details).

In [SV] it was shown that holomorphic Lagrangian subvarieties of
$SL(n,{\mathbb{H}})$-manifolds exist only in a countable number of complex
structure of the form $L=aI+bJ+cK$ (such complex structures are called induced
by a hypercomplex structure, see Subsection 2.1). In the present paper, we
study holomorphic Lagrangian fibrations on hypercomplex manifolds. Such
fibrations are often present in examples ([GV]; see also Subsection 3.2).

The holomorphic Lagrangian fibrations are of significant interest for
mathematicians and physicists; for an early survey of the subject, please see
the paper [Saw], and for applications, see e.g. [KRS]. The non-Kähler version
of this geometry defined above should be even more useful, because the number
of examples is much greater.

In the present paper, we consider HKT metrics on
$SL(n,{\mathbb{H}})$-manifolds admitting holomorphic Lagrangian fibrations. We
prove that a base of such fibration is always Kähler, if the fibration is
smooth (3.1). This allows us to construct new examples of hypercomplex
manifolds admitting no HKT metrics.

Originally, it was conjectured that any compact hypercomplex manifold admits
an HKT metric. Examples of hypercomplex nilmanifolds not admitting HKT metrics
were constructed by Fino and Grantcharov ([FG]). Since then, hypercomplex
nilmanifolds admitting HKT metrics were classified completely in [BDV]. It was
found that in fact most of hypercomplex nilmanifolds are not HKT. However, no
other hypercomplex manifolds not admitting an HKT metrics were known before
now.

Acknowledgements. The work on this paper began during the visit to the
University of Science and Technology of China. The authors would like to
express their gratitude to USTC for the hospitality an to Prof. Xiuxiong Chen
for the invitation.

##  2 Preliminaries

###  2.1 Hypercomplex manifolds
and quaternionic Dolbeault complex

A $C^{\infty}$-manifold $M$ is called hypercomplex if $M$ is equipped with
complex structures $I$, $J$, $K$, that satisfy the quaternionic relations

$IJ=-JI=K.$

It was shown in [Ob] that any hypercomplex manifold admits a unique torsion-
free connection that preserves $I,J$ and $K$. This connection is called the
Obata connection. Note that the holonomy of the Obata connection is a subgroup
of $GL(n,{\mathbb{H}})$.

Any almost-complex structure which is preserved by a torsion-free connection
is integrable (this follows from Newlander-Nirenberg theorem). Therefore, for
any $a,b,c\in{\mathbb{R}}$ with $a^{2}+b^{2}+c^{2}=1$, the almost-complex
structure $L=aI+bJ+cK$ is integrable. We denote by $(M,L)$ the corresponding
complex manifold. Such complex structures are called induced by quaternions.

Let $(M,I,J,K)$ be a hypercomplex manifold of real dimension $4n$. The
hypercomplex structure induces the action of $SU(2)$ on all tensor bundles
over $M$. Recall that any irreducible complex representation of $SU(2)$ is of
the form $S^{k}U$, where $U$ is the standard two-dimensional representation
and $S^{k}$ denotes the symmetric power. We will refer to any representation
of the form $\left(S^{k}(U)\right)^{\oplus m}$ (for arbitrary $m$) as a weight
k representation of $SU(2)$.

We make the following observation about the weight decomposition of the
exterior algebra
$\Lambda_{\mathbb{C}}^{*}M=\Lambda^{*}M\otimes_{\mathbb{R}}{\mathbb{C}}$.
First, note that $\Lambda^{1}_{\mathbb{C}}M$ is an $SU(2)$ representation of
weight one. It follows from the Clebsch-Gordan formula that
$\Lambda^{k}_{\mathbb{C}}M$ is a sum of representations of weight $\leq k$. By
duality, the same is true about $\Lambda^{4n-k}_{\mathbb{C}}M$. Denote by
$\Lambda^{k}_{+}M$ the maximal subrepresentation of weight $k$ for $k\leq 2n$
and of weight $4n-k$ for $k>2n$ inside $\Lambda^{k}_{\mathbb{C}}M$. Denote by
$\Pi_{+}\colon\Lambda^{*}M\to\Lambda^{*}_{+}M$ the equivariant projection onto
the component of maximal weight.

Note that the Hodge decomposition with respect to an arbitrary complex
structure (we can pick $I$ without loss of generality) is compatible with the
weight decomposition:

$\Lambda^{k}_{+}M=\bigoplus_{p+q=k}\Lambda^{p,q}_{I,+}M,$

where $\Lambda^{p,q}_{I,+}M=\Lambda^{p,q}_{I}M\cap\Lambda^{k}_{+}M$. This is
actually a weight decomposition of an $SU(2)$-representation for a special
choice of Cartan subalgebra corresponding to $I$. Therefore,
$\Lambda^{k}_{+}M$ is generated by $\Lambda^{k,0}_{I,+}M=\Lambda^{k,0}_{I}M$
as an $SU(2)$-module. It follows that the summands $\Lambda^{p,q}_{I,+}M$ are
of the same dimension and actually isomorphic to $\Lambda^{p+q,0}_{I}M$. We
will denote these isomorphisms by

$\mathcal{R}_{p,q}\colon\Lambda^{p+q,0}_{I}M\tilde{\to}\Lambda^{p,q}_{I,+}M,$

see [V2] and formula (2.3) below.

Given a hyperhermitian metric $g$ on $M$, we can consider the corresponding
Kähler forms $\omega_{I}$, $\omega_{J}$ and $\omega_{K}$. It is easy to check
that the form

$\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$

lies in $\Lambda^{2,0}_{I}M$. If $d\Omega_{I}=0$, then the manifold $M$ is
called hyperkähler. This condition implies that $g$ is Kähler with respect to
any induced complex structure. If $\Omega_{I}$ satisfies a strictly weaker
condition $\partial\Omega_{I}=0$, then $M$ is called an HKT manifold, and $g$
is called an HKT metric (HKT stands for HyperKähler with Torsion; see [GP] for
a general introduction to HKT geometry).

We can reformulate the HKT condition as follows. Denote by
$d_{+}\colon\Lambda^{k}_{+}M\to\Lambda^{k+1}_{+}M$ the composition
$\Pi_{+}\circ d$ of the de Rham differential and the projection onto the
component of maximal weight. Note that $\omega_{I}\in\Lambda^{1,1}_{I,+}M$.
Then by Theorem 5.7 in [V1], the condition $\partial\Omega_{I}=0$ is
equivalent to

$d_{+}\omega_{I}=0.$ (2.1)

That is, for an HKT metric $g$ the exterior differential of its Kähler form
$d\omega_{I}$ has weight one. This observation will be important for the proof
of the main theorem.

###  2.2 $SL(n,{\mathbb{H}})$-manifolds
and holomorphic Lagrangian calibration

As it was mentioned above, the holonomy of the Obata connection on a
hypercomplex manifold is a subgroup of $GL(n,{\mathbb{H}})$. Recall that this
group can be defined as follows. Consider a vector space $V$ of real dimension
$4n$ with a quaternionic structure $I,J,K$. Then $GL(n,{\mathbb{H}})$ consists
of those linear transformations of $V$ that commute with $I$, $J$ and $K$.
Consider the Hodge decomposition
$V_{\mathbb{C}}=V\otimes{\mathbb{C}}=V^{1,0}_{I}\oplus V^{0,1}_{I}$. The
action of $GL(n,{\mathbb{H}})$ preserves this decomposition, hence it induces
an action on all exterior powers of $V^{1,0}_{I}$. Denote by
$SL(n,{\mathbb{H}})$ the subgroup of $GL(n,{\mathbb{H}})$ consisting of those
elements that act identically on $V^{2n,0}_{I}$.

Consider a hypercomplex manifold $(M,I,J,K)$. If the holonomy of the Obata
connection is contained in $SL(n,{\mathbb{H}})$, then we call $M$ an
$SL(n,{\mathbb{H}})$-manifold. It follows almost immediately from the
definition (see [V3], Claim 1.1), that on an $SL(n,{\mathbb{H}})$-manifold the
canonical bundle $\Lambda^{2n,0}_{I}M$ is flat with respect to the Obata
connection. It also follows ([V3], Claim 1.2), that any parallel section of
$\Lambda^{2n,0}_{I}M$ is holomorphic. Moreover, if the manifold $(M,I,J,K)$ is
HKT and compact, the condition $\operatorname{Hol}(M)\subset
SL(n,{\mathbb{H}})$ is equivalent to holomorphic triviality of the canonical
bundle ([V3], Theorem 2.3).

Denote by $\Phi_{I}\in\Lambda^{2n,0}_{I}M$ a parallel section which we call a
holomorphic volume form. Note that the operator $J$ defines a real structure
(that is, a complex-antilinear involution) on $\Lambda^{2n,0}_{I}M$ by
$\eta\mapsto\overline{J\eta}$. We can always assume that $\Phi_{I}$ is real
with respect to this structure.

Given an $SL(n,{\mathbb{H}})$-manifold $M$ with a hyperhermitian metric $g$,
denote by $\Phi_{I}$ the holomorphic volume form and by
$\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$ the $(2,0)$-form associated with
the metric. In [GV], a number of calibrations (in the sense of [HL]) on
$SL(n,{\mathbb{H}})$-manifolds were constructed. In particular, it was shown
that the form

$\Psi=(-\sqrt{-1})^{n}\mathcal{R}_{n,n}(\Phi_{I})\in\Lambda^{n,n}_{I,+}M$
(2.2)

is a calibration with respect to a properly rescaled metric. It was shown that
this form calibrates $\Omega_{I}$-Lagrangian subvarieties of $M$. Recall, that
a complex (with respect to the complex structure $I$) subvariety $N\subset M$
of complex dimension $n$ is called $\Omega_{I}$-Lagrangian if the restriction
of $\Omega_{I}$ to the smooth part of $N$ vanishes. This condition is
equivalent to the following: $T_{x}N$ is orthogonal to $J(T_{x}N)$ with
respect to the metric $g$ at any smooth point $x\in N$. To see this, observe
that for any vector fields $X,Y\in T^{1,0}_{I}M$ we have
$\Omega_{I}(X,Y)=2g(JX,Y)$.

We will not need the construction from [GV] in its full generality, but only
some basic properties of the form $\Psi$. For the convenience of the reader we
will formulate these properties in the following lemma and give a short proof,
independent from [GV].

Lemma 2.1:  Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold of real
dimension $4n$, $\Phi_{I}\in\Lambda^{2n,0}_{I}M$ a holomorphic volume form and
$\Psi=(-\sqrt{-1})^{n}\mathcal{R}_{n,n}(\Phi_{I})\in\Lambda^{n,n}_{I,+}M$. Let
$N\subset M$ be an $I$-complex subvariety, $\dim_{\mathbb{C}}N=n$, with
$T_{x}N\cap J(T_{x}N)=0$ at all smooth points $x\in N$. Then:

  1. 1.

$d\Psi=0$,

  2. 2.

$\Psi|_{N}$ is a strictly positive volume form on the smooth part of $N$.

Proof: 1\. As it was mentioned above (see also formula (2.3) below), the
operator $\mathcal{R}_{n,n}$ is induced by the $SU(2)$-action on the exterior
algebra of $M$. Since the Obata connection $\nabla$ preserves the hypercomplex
structure, it commutes with the $SU(2)$-action and with the operator
$\mathcal{R}_{n,n}$. It was explained above that $\nabla\Phi_{I}=0$, thus
$\nabla\Psi=0$ also. The Obata connection is torsion-free, hence
$\nabla\Psi=0$ implies $d\Psi=0$.

2\. Let $x\in N$ be a smooth point. By the assumptions of the lemma, we can
choose a basis $e_{1},\ldots,e_{n}$ of $T^{1,0}_{I,x}N$, such that
$e_{1},\ldots,e_{n},J\overline{e}_{1},\ldots,J\overline{e}_{n}$ will form a
basis of $T^{1,0}_{I,x}M$. To show that $\Psi|_{N}$ is strictly positive, we
have to evaluate the form $\Psi$ on the polyvector
$\xi=(-\sqrt{-1})^{n}e_{1}\wedge\overline{e}_{1}\wedge\cdots\wedge
e_{n}\wedge\overline{e}_{n}\in\Lambda^{n,n}_{I}(T_{x}M)$.

We will use the following explicit description of the operator
$\mathcal{R}_{n,n}$. Consider the operators: $\mathcal{H}=-\sqrt{-1}I$,
$\mathcal{X}=\frac{1}{2}(\sqrt{-1}K-J)$ and
$\mathcal{Y}=\frac{1}{2}(\sqrt{-1}K+J)$ acting on $\Lambda^{1}_{\mathbb{C}}M$.
It is straightforward to check that these operators satisfy the standard
$\mathfrak{sl}_{2}({\mathbb{C}})$ commutator relations:
$[\mathcal{X},\mathcal{Y}]=\mathcal{H}$,
$[\mathcal{H},\mathcal{X}]=2\mathcal{X}$,
$[\mathcal{H},\mathcal{Y}]=-2\mathcal{Y}$. Also observe that

$\mathcal{H}|_{\Lambda^{1,0}_{I}M}=1,\hskip
10.0pt\mathcal{X}|_{\Lambda^{1,0}_{I}M}=0,\hskip
10.0pt\mathcal{Y}|_{\Lambda^{1,0}_{I}M}=J,$
$\mathcal{H}|_{\Lambda^{0,1}_{I}M}=-1,\hskip
10.0pt\mathcal{X}|_{\Lambda^{0,1}_{I}M}=-J,\hskip
10.0pt\mathcal{Y}|_{\Lambda^{0,1}_{I}M}=0.$

We can extend $\mathcal{H}$, $\mathcal{X}$, $\mathcal{Y}$ as derivations to
the whole exterior algebra. Then we have

$\mathcal{R}_{n,n}=\mathcal{Y}^{n},$ (2.3)

and

$\langle\Psi,\xi\rangle=(-\sqrt{-1})^{n}\langle\mathcal{Y}^{n}\Phi_{I},\xi\rangle=(\sqrt{-1})^{n}\langle\Phi_{I},\mathcal{Y}^{n}\xi\rangle.$

Observe, that $\mathcal{Y}|_{T^{1,0}_{I}M}=0$,
$\mathcal{Y}|_{T^{0,1}_{I}M}=J$, and since $\mathcal{Y}$ acts as a derivation,
we have

$\mathcal{Y}^{n}\xi=n!(-\sqrt{-1})^{n}e_{1}\wedge
J\overline{e}_{1}\wedge\cdots\wedge e_{n}\wedge J\overline{e}_{n}.$

The volume form is given by $\Phi_{I}=a\,e^{*}_{1}\wedge
J\overline{e^{*}_{1}}\wedge\cdots\wedge e^{*}_{n}\wedge J\overline{e^{*}_{n}}$
for some positive real number $a$, and we see that $\langle\Psi,\xi\rangle>0$.

##  3 Holomorphic Lagrangian fibrations
on $SL(n,{\mathbb{H}})$-manifolds

###  3.1 HKT structures and Lagrangian fibrations

Definition 3.1:  A holomorphic Lagrangian subvariety of an
$SL(n,{\mathbb{H}})$-manifold is a subvariety calibrated by the form $\Psi$ of
2.2.

Definition 3.2: Let $M$ be an $SL(n,{\mathbb{H}})$-manifold, and
$\varphi\colon(M,I)\to X$ a smooth holomorphic fibration. It is called a
holomorphic Lagrangian fibration if all its fibers are holomorphic Lagrangian
subvarieties. We will say that the fibration $\varphi\colon(M,I)\to X$ is
smooth if $\varphi$ is a submersion (hence all the fibers are smooth).

Remark 3.3: Suppose that we are given an arbitrary holomorphic fibration
$\varphi\colon(M,I)\to X$ with $M$ compact and $\dim_{\mathbb{C}}X={1\over
2}\dim_{\mathbb{C}}M$. Denote by $F_{x}$ the fiber of $\varphi$ over $x\in X$.
Consider the set $U=\\{x\in X\colon J(TF_{x})\cap TF_{x}=0\\}$. It is clear
that $U$ is open and all the fibers over $U$ can be made Lagrangian with
respect to some hyperhermitian metric. Namely, the condition $J(TF_{x})\cap
TF_{x}=0$ implies that each fiber of the vector bundle $J(TF_{x})$ projects
onto $T_{x}X$ by $\varphi$, so that we can lift any Hermitian metric from $U$
to $J(TF_{x})$. Then we can uniquely extend it to a hyperhermitian metric on
$\varphi^{-1}(U)$ with $J(TF_{x})$ orthogonal to $TF_{x}$. This means that the
set of Lagrangian fibers of $\varphi$ is open in $M$.

The main result of this paper is the following theorem.

Theorem 3.4:  Let $M$ be a compact $SL(n,{\mathbb{H}})$-manifold, and
$\varphi\colon(M,I)\to X$ a smooth holomorphic Lagrangian fibration. Assume
that $M$ admits an HKT-structure. Then $X$ is Kähler.

Proof: Let $\omega_{I}\in\Lambda^{1,1}_{I}M$ be a Kähler form, and $\Psi$ the
holomorphic Lagrangian calibration defined by (2.2). Consider the form

$\Theta=\Psi\wedge\omega_{I}\in\Lambda^{n+1,n+1}_{I}M.$

By 2.2, $\Psi$ is closed, so we have $d\Theta=\Psi\wedge
d\omega_{I}\in\Lambda^{2n+3}M$. It follows from (2.1) that $d\omega_{I}$ has
weight one, hence Clebsch-Gordan formula implies that $d\Theta$ is a sum of
weight $2n+1$ and weight $2n-1$ components. But the weight decomposition of
$\Lambda^{2n+3}M$ has components only up to weight $2n-3$, so $d\Theta$ has to
vanish.

We will obtain a Kähler metric on $X$ as a push-forward $\pi_{*}\Theta$, which
is a $(1,1)$-current on $X$. We have to show that this current is strictly
positive and smooth.

Consider any Hermitian metric on $X$ and denote by $\eta$ the corresponding
$(1,1)$-form. It is always possible to rescale $\eta$ so that it satisfies the
condition $\pi^{*}\eta\leq\omega_{I}$. In this case we have
$\Theta=\Psi\wedge\omega_{I}\geq\Psi\wedge\pi^{*}\eta$. The inequality is
preserved under taking push-forwards, so we have
$\pi_{*}\Theta\geq\pi_{*}(\Psi\wedge\pi^{*}\eta)$.

For any test-form $\alpha\in\Lambda^{n-1,n-1}_{I}X$ we have by definition

$\displaystyle\int_{X}\pi_{*}(\Psi\wedge\pi^{*}\eta)\wedge\alpha=\int_{M}\Psi\wedge\pi^{*}\eta\wedge\pi^{*}\alpha$
$\displaystyle=\int_{M}\Psi\wedge\pi^{*}(\eta\wedge\alpha)=\int_{X}\pi_{*}\Psi\wedge\eta\wedge\alpha.$

Thus, we have the inequality $\pi_{*}\Theta\geq\pi_{*}(\Psi)\wedge\eta$ where
$\pi_{*}(\Psi)$ is a closed $0$-current on $X$, that is a constant. To see
that this constant is positive, take $\alpha=\eta^{n-1}$ and observe that
$\Psi\wedge\pi^{*}(\eta^{n})$ is a smooth positive form of top degree on $M$.
It suffices to check that this form is non-zero at some point of $M$. Pick any
non-critical point $x$ of $\pi$ and decompose the tangent space
$T_{x}M=V_{0}\oplus V_{1}$, where $V_{0}$ is tangent to the fiber and $V_{1}$
is any complementary subspace. Since $\pi^{*}\eta^{n}|_{V_{0}}=0$ and
$\pi^{*}\eta^{n}|_{V_{1}}$ is a volume form on $V_{1}$, we should check that
$\Psi|_{V_{0}}$ is non-zero. But this follows from the second statement in
2.2, since the fiber is Lagrangian.

It remains to check that $\pi_{*}\Theta$ is smooth. This follows from
smoothness of the fibration $\varphi\colon(M,I)\to X$. Since $\varphi$ is a
submersion, we can choose a splitting $TM=\mathcal{V}\oplus\mathcal{H}$, where
$\mathcal{V}$ is a subbundle, tangent to the fibers of $\varphi$. Using this
splitting we can lift vector field from $X$ to $M$. Denote by $F_{x}$ the
fiber $\varphi^{-1}(x)$ over a point $x\in X$. Consider a $(1,1)$-form
$\theta$ on $X$, which is given by

$\theta(\xi,\overline{\eta})_{x}=\int_{F_{x}}(\pi^{*}\xi)\lrcorner\,(\pi^{*}\overline{\eta})\lrcorner\,\Theta,$

where $\xi,\eta\in T^{1,0}X$ and $\pi^{*}\xi$, $\pi^{*}\eta$ denote the the
lifts of these vector fields to $\mathcal{H}$. Note, that $\theta$ does not
actually depend on the choice of $\mathcal{H}$, since convolution of $\Theta$
with a vertical vector field from $\mathcal{V}$ gives a differential form that
restricts trivially to the fiber. The form $\theta$ coincides with
$\pi_{*}\Theta$ as a current, thanks to Fubini theorem. But it is clear from
the definition that $\theta$ is smooth. This completes the proof.

Remark 3.5: Note that for a non-smooth holomorphic Lagrangian fibration
$\varphi\colon(M,I)\to X$ the above proof shows that
$\pi_{*}(\Theta\wedge\omega_{I})$ is a Kähler current. Therefore, $X$ is in
Fujiki class $\mathcal{C}$, whenever $M$ admits an HKT-structure (see [DP]).

###  3.2 Examples of Lagrangian fibrations

In this subsection we construct a class of hypercomplex
$SL(n,{\mathbb{H}})$-manifolds that do not admit an HKT metric. The idea,
which was also used in [KS], is to consider a torus fibration over an affine
base.

Let $(X,I)$ be a complex manifold, $\dim_{\mathbb{C}}X=n$. We call $X$ an
affine complex manifold if there exists a torsion-free connection $D\colon
TX\to TX\otimes\Lambda^{1}X$ which is flat and preserves the complex
structure: $DI=0$.

Fix a point $x\in X$ and consider the holonomy group
$\mathrm{Hol}_{x}(D)\subset GL(T_{x}X)$. We call $X$ an affine manifold with
integer monodromy if there exists a lattice $\Lambda_{x}\subset T_{x}X$ which
is preserved by holonomy: $\mathrm{Hol}_{x}(D)\Lambda_{x}=\Lambda_{x}$. In
this case, we can construct a subset $\Lambda\subset TX$ in the total space of
the tangent bundle, obtained as parallel translation of $\Lambda_{x}$. For any
point $y\in X$ the intersection $\Lambda\cap T_{y}X$ is a lattice in $T_{y}X$.

Let $M=TX/\Lambda$, that is, a manifold obtained as a fiberwise quotient of
$TX$. We will introduce a pair of anticommuting complex structures on $TX$
that will descend to $M$. The connection $D$ defines a splitting

$T(TX)=\mathcal{V}\oplus\mathcal{H}$ (3.1)

into a direct sum of the vertical subbundle $\mathcal{V}$ which is tangent to
the fibers of the projection $TX\to X$, and a horizontal complement
$\mathcal{H}$. Note that for any point $(x,v)\in TX$ with $v\in T_{x}X$ we
have natural isomorphisms $\mathcal{H}_{(x,v)}\simeq
T_{x}X\simeq\mathcal{V}_{(x,v)}$, thus we can identify the components of the
splitting (3.1). This also shows that the complex structure $I$ acts naturally
on both $\mathcal{H}$ and $\mathcal{V}$. With these observations in mind,
define a pair of operators from $\operatorname{End}(T(TM))$:

$\mathcal{I}=\begin{pmatrix}-I&0\\\
0&I\end{pmatrix},\quad\mathcal{J}=\begin{pmatrix}0&\operatorname{Id}\\\
-\operatorname{Id}&0\end{pmatrix},$

where the block-matrix form corresponds to the decomposition (3.1). It is
clear that these operators define a pair of anticommuting almost-complex
structures. We need to check that these structures are integrable. This can be
done locally.

Using the fact that the connection $D$ is flat and $I$ is parallel, we can
choose a parallel local frame in $TX$ of the form $e_{1},Ie_{1},\ldots
e_{n},Ie_{n}$. Since $D$ is torsion-free, these vector fields commute, hence
they define a local coordinate system on $X$. Since the sections $e_{i}$ and
$Ie_{i}$ are flat, they are tangent to the subbundle $\mathcal{H}$. Therefore,
they also define a local coordinate system in the total space of $TX$ in which
the complex structures $\mathcal{I}$ and $\mathcal{J}$ act as standard complex
structures of ${\mathbb{H}}^{n}$. Thus, they are integrable.

This construction gives a hypercomplex structure on the total space of $TX$.
Observe that this structure is invariant with respect to translations along
the fiber. This implies that the hypercomplex structure descends to $M$. We
call the manifold $M$ constructed this way quaternionic double of an affine
complex manifold $X$.

The hypercomplex structure on the quaternionic double $M$ is locally modeled
on ${\mathbb{H}}^{n}$ by construction. Thus, the Obata connection on this
manifold is flat (and it is actually induced from the flat connection $D$ on
$X$). Since the connection $D$ preserves the lattice in the tangent bundle, it
also preserves the holomorphic volume form. To see this, we can again choose a
local frame $e_{1},Ie_{1},\ldots e_{n},Ie_{n}$ as above. We can assume that
the elements of this frame form a local basis for the lattice $\Lambda$
preserved by $D$. Then the volume form
$(e_{1}^{*}-\sqrt{-1}Ie_{1}^{*})\wedge\cdots\wedge(e_{n}^{*}-\sqrt{-1}Ie_{n}^{*})$
is well-defined globally and parallel with respect to $D$, hence holomorphic.
It follows that the Obata connection on $M$ also preserves an $I$-holomorphic
volume form. We conclude that the manifold $M$ is an
$SL(n,{\mathbb{H}})$-manifold.

Theorem 3.6: Let $M$ be a quaternionic double of an affine complex manifold
$X$. If $X$ is not Kähler, then $M$ does not admit an HKT metric.

Proof: We can choose a Hermitian metric $g$ on $X$. Then we can define a
hyperhermitian metric $h=g\oplus g$ on $M$, using the decomposition (3.1).
With respect to this metric the fibers of the projection $M\to X$ are
Lagrangian, hence we can apply 3.1.

We conclude this section by some examples of affine complex manifolds with
integer monodromy. We remark that the manifolds in the examples will be non-
Kähler (actually any complex affine manifold admitting a Kähler metric is a
quotient of a torus, as follows from the Calabi-Yau theorem and the
Bieberbach’s solution of Hilbert’s 18-th problem; see e.g. [Bes]). Therefore,
the corresponding quaternionic doubles possess no HKT metrics.

Example 3.7: Let $N$ be a three-dimensional real Heisenberg group, that is the
group of upper-triangular unipotent $3\times 3$ matrices with real elements.
Consider the nilpotent Lie group $G=N\times{\mathbb{R}}$. Then $G$ is
diffeomorphic to ${\mathbb{C}}^{2}$ and one can check (see [Ha], Example 2)
that the structure of the Lie group on ${\mathbb{C}}^{2}$ can be explicitly
given by

$(w_{1},w_{2})\cdot(z_{1},z_{2})=(w_{1}+z_{1},w_{2}-\sqrt{-1}\overline{w_{1}}z_{1}+z_{2}).$

It is clear from this formula, that left translations are holomorphic affine
transformations of $C^{2}$, so $G$ has a left-invariant complex structure
which is preserved by a flat torsion-free connection. We can choose a lattice
in ${\mathbb{C}}^{2}$, say $\Lambda={\mathbb{Z}}[\sqrt{-1}]^{2}$. Then the
manifold $X=\Lambda\backslash G$ (which is called a primary Kodaira surface)
will be an affine complex manifold with integer monodromy.

Example 3.8: Let $G$ be a three-dimensional complex Heisenberg group. It can
be described (see [Ha]) as ${\mathbb{C}}^{3}$ with multiplication given by

$(w_{1},w_{2},w_{3})\cdot(z_{1},z_{2},z_{3})=(w_{1}+z_{1},w_{2}+z_{2},w_{3}+z_{3}+w_{1}z_{2}).$

We can choose a lattice in $G$, for example
$\Lambda={\mathbb{Z}}[\sqrt{-1}]^{3}$ and take a quotient $X=\Lambda\backslash
G$. This is an affine complex manifold with integer monodromy (which is called
an Iwasawa manifold).

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Misha Verbitsky
Laboratory of Algebraic Geometry,
National Research University HSE,
7 Vavilova Str. Moscow, Russia, 117312
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Andrey Soldatenkov
Laboratory of Algebraic Geometry,
National Research University HSE,
7 Vavilova Str. Moscow, Russia, 117312
<EMAIL_ADDRESS>


# Quasi-Hermitian pair and co-amenability

Bat-Od Battseren<EMAIL_ADDRESS>
###### Abstract.

We adapt the notion of quasi-Hermition group to the pairs $(G,H)$ of discrete
group $G$ and its subgroup $H$. We show that a quasi-Hermitian pair is
amenable in the sense of Eymard.

##  1\. Introduction

Hermitian Banach $*$-algebras make an important class of $*$-algebras that
extends the class of $C^{*}$-algebras. It has being an effectively elaborated
in approximation theory, time-frequency analysis and signal process, and non-
commutative geometry. In this paper, we are mainly focus on the
(quasi-)Hermitian property of Banach $*$-algebras derived from a group.

Spectrum of a function in group algebra plays a crucial role in study of
analytic group theory. For example, amenability is characterized by the fact
that the spectrum of a normalized characteristic function has 1 as a spectral
point in the reduced group $C^{*}$-algebra. The property of being Hermitian is
a variant of such spectral study. A locally compact group $G$ is called
Hermitian (resp. quasi-Hermitian) if the spectrum
$\operatorname{Sp}(f,L^{1}(G))$ is in the real line for all self-adjoint
functions $f=f^{*}$ in $L^{1}(G)$ (resp. in $C_{c}(G)$). It was a long-
standing conjecture that Hermitian groups are amenable until it was
affirmatively proven in [SW20] even for quasi-Hermitian groups. In this paper,
we extend the notion of quasi-Hermition group to pairs $(G,H)$ of discrete
group $G$ and its subgroup $H$ and adapt the Samei-Wiersma’s results for
quasi-Hermition pairs.

As amenability is characterized in many ways, it has many versions such as
weak amenability, a-T-menability (also known as Haagerup property), Haagerup-
Kraus’s approximation property, co-amenability, etc. We are more interested in
co-amenability in the sense of Eymard. Recall the original definition of
amenability given by John von Neumann: A discrete group $G$ is amenable if
there is a $G$-invariant state on $\ell^{\infty}(G)$. Similarly, a subgroup
$H$ of $G$ is co-amenable in $G$ if there is a $G$-invariant state on
$\ell^{\infty}(G/H)$. Our main result is as follows.

######  Theorem 1.

If $(G,H)$ is quasi-Hermtian, then $H$ is co-amenable in $G$.

The notion of quasi-Hermitian pair extends the usual notion of quasi-Hermitian
group. In particular, if we take $H$ to be the trivial subgroup, we recover
Samei-Wiersma’s result for discrete groups. Indeed, $G$ is quasi-Hermitian if
and only if $(G,\\{e\\})$ is quasi-Hermitian, and $G$ is amenable if and only
if $\\{e\\}$ is co-amenable in $G$. This extension allows us to target a
richer class.

About the proof of Theorem 1, we essentially follow the steps of [SW20]. The
first step is that we construct Banach $*$-algebras $PF^{*}_{p}(G:H)$
associated to the pair $(G,H)$ and $1\leq p\leq 2$. It generalizes the usual
Banach $*$-algebras $PF^{*}_{p}(G)$. When $p=2$, we have
$PF^{*}_{p}(G:H)=C^{*}_{\lambda_{G/H}}(G)$. The second step is to see that the
triple

$\displaystyle(PF^{*}_{p_{1}}(G:H),PF^{*}_{p_{2}}(G:H),PF^{*}_{p_{3}}(G:H))$

is a spectral interpolation of $*$-semisimple Banach $*$-algebras relative to
$PF^{*}_{1}(G:H)$ for any $1\leq p_{1}<p_{2}<p_{3}\leq 2$. Then in Theorem 10,
we will prove that $(G,H)$ is quasi-Hermitian if and only if the spectrum of
$[f]\in[\operatorname{\mathbb{C}}G]$ in $PF^{*}_{p}(G:H)$ does not depend on
$1\leq p\leq 2$. This latter was proven in [Bar90] for amenable Hermitian
groups, during which time it was unknown that Hermitian groups are amenable.
The third step is to see that if the universal $C^{*}$-enveloping algebra of
$PF^{*}_{1}(G,H)$ is canonically isomorphic to $C^{*}_{\lambda_{G/H}}(G)$,
then $H$ is co-amenable in $G$. Then Proposition 2.6 of [SW20] concludes our
proof.

##  2\. Preliminaries

We recall some definitions and useful results.

###  2.1. Banach $*$-algebras

Let $A$ be a Banach $*$-algebra and let $S$ be a $*$-subalgebra of $A$. For
$a\in A$, we denote by $\operatorname{Sp}(a,A)$ the spectrum of $a$ in $A$,
and by $r(a,A)$ the spectral radius of $a$ in $A$. Denote by $S_{h}$ the self-
adjoint elements of $S$. We say that $S$ is quasi-Hermitian in $A$ if
$\operatorname{Sp}(a,A)\subseteq\operatorname{\mathbb{R}}$ for all $a\in
S_{h}$. Similarly, we say that $S$ is quasi-symmetric in $A$ if
$\operatorname{Sp}(a^{*}a,A)\subseteq\operatorname{\mathbb{R}}_{+}$ for all
$a\in S$. Banach $*$-algebra $A$ is called Hermitian (resp. symmetric) if it
is quasi-Hermitian (resp. quasi-symmetric) in itself. Shirali-Ford’s theorem
states that a Banach $*$-algebra is Hermitian if and only if it is symmetric
(see for example [BD73, Theorem 41.5]).

By a $*$-representation of $A$, we understand a pair
$(\pi,\operatorname{\mathcal{H}})$ where $\operatorname{\mathcal{H}}$ is a
Hilbert space and
$\pi:A\rightarrow\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$ is a
$*$-homomophism into bounded operators on $\operatorname{\mathcal{H}}$. It is
a well known fact that any $*$-representation is norm decreasing. We say that
$A$ is $*$-semisimple if for any non-zero element $a\in A$, there is a
$*$-representation that sends $a$ into a non-zero element. In the sequel, $A$
is always assumed to be $*$-semisimple. For a $*$-representation
$(\pi,\operatorname{\mathcal{H}})$ of $A$, we denote by $C^{*}_{\pi}(A)$ the
completion of $\pi(A)$ in
$\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$. Let
$(\pi,\operatorname{\mathcal{H}})$ and $(\rho,\operatorname{\mathcal{K}})$ be
$*$-representations of $A$. We say that $\pi$ weakly contains $\rho$ and write
$\rho\prec\pi$ if $\|\rho(a)\|\leq\|\pi(a)\|$ for all $a\in A$. We say that
$\pi$ and $\rho$ are weakly equivalent and write $\pi\sim\rho$ if they weakly
contains each other. We say that $\pi$ and $\rho$ are isomorphic and write
$\pi\cong\rho$ if there is a unitary operator
$U\in\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}},\operatorname{\mathcal{K}})$
such that $\pi(a)=U^{*}\rho(a)U$ for all $a\in A$. Note that for any family
$\\{(\pi_{i},\operatorname{\mathcal{H}})\\}$ of $*$-representations, their
$\ell^{2}$-direct-sum gives a $*$-representation
$(\pi,\operatorname{\mathcal{H}})$ such that $\pi_{i}\prec\pi$ for all $a\in
A$. This yields the existence of the universal $*$-representation
$(\pi_{u},\operatorname{\mathcal{H}}_{u})$ of $A$. This is the unique
$*$-representation satisfying the following universal property: Any
$*$-representation of $A$ uniquely factors through $C^{*}_{\pi_{u}}(G)$. The
$C^{*}$-algebra $C^{*}(A)=C^{*}_{\pi_{u}}(G)$ is called the universal
$C^{*}$-enveloping algebra of $A$. Since $A$ is $*$-semisimple, we can see $A$
as a dense $*$-subalgebra of $C^{*}(G)$. If $A$ is Hermitian, then the
inclusion $A\subseteq C^{*}(A)$ is spectral, i.e.

$\displaystyle\operatorname{Sp}(a,A)=\operatorname{Sp}(a,C^{*}(A))\quad\text{for
all}\quad a\in A.$

The converse is rather trivial.

The main tool applied in [SW20] was the notion of spectral interpolation which
was inspired by [Pyt82]. Let us explain it more in detail. We say that
$A\subseteq B$ is a nested pair of ($*$-semisimple) Banach $*$-algebras if $A$
and $B$ are ($*$-semisimple) Banach $*$-algebras and $A$ embeds continuously
into $B$ as a dense $*$-subalgebra. A nested triple of ($*$-semisimple) Banach
algebras is defined similarly. Let $S$ be a $*$-subalgebra of $A$. We say that
$S_{h}$ has invariant spectral radius in $(A,B)$ if $r(a,A)=r(a,B)$ for all
$a\in S$. We say that $S$ is spectral subalgebra of $(A,B)$ if
$\operatorname{Sp}(a,A)=\operatorname{Sp}(a,B)$ for all $a\in S$. Suppose that
$A\subseteq B\subseteq C$ is a nested triple of Banach $*$-algebras and $S$ is
a dense $*$-subalgebra of $A$. We say that $(A,B,C)$ is a spectral
interpolation of triple Banach $*$-algebras relative to $S$ if there is a
constant $0<\theta<1$ such that

$\displaystyle r(a,B)\leq r(a,A)^{1-\theta}r(a,C)^{\theta}\quad\text{for
all}\quad a\in S_{h}.$

The following two results are crucial to our main result.

######  Proposition 2 ([SW20, Proposition 2.6]).

Let $A\subseteq B$ be a nested pair of $*$-semisimple Banach $*$-algebras, and
$S$ be a dense $*$-subalgebra of $A$. Suppose that $S_{h}$ has an invariant
spectral radius in $(A,B)$. Then $A$ and $B$ have the same universal
$C^{*}$-enveloping algebra. In particular, if $B$ is a $C^{*}$-algebra, then
$B$ is the universal $C^{*}$-enveloping algebra of $A$.

######  Theorem 3 ([SW20, Theorem 3.4]).

If $(A,B,C)$ is a spectral interpolation of triple $*$-semisimple Banach
$*$-algebras relative to a quasi-Hermitian dense $*$-subalgebra $S$ of $A$,
then $S_{h}$ has invariant spectral radius in $(B,C)$. In particular, we have
the canonical $*$-isomorphism $C^{*}(B)\cong C^{*}(C)$.

###  2.2. Group representation

Let $G$ be a discrete group. A unitary representation of $G$ is a pair
$(\pi,\operatorname{\mathcal{H}})$ where $\operatorname{\mathcal{H}}$ is a
Hilbert space and $\pi:G\rightarrow\mathcal{U}(\operatorname{\mathcal{H}})$ is
a group homomorphism into the group of unitary operators on
$\operatorname{\mathcal{H}}$. It can be extended linearly to the
$*$-representation
$\pi:\ell^{1}(G)\rightarrow\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$.
Conversely, any $*$-representation of $\ell^{1}(G)$ restricts to a unitary
representation of $G$. Thus the representation theory of $G$ and $\ell^{1}(G)$
coincide.

Let $H\leq G$ be a subgroup. Consider the regular representation

$\displaystyle\lambda_{G/H}:\ell^{1}(G)\rightarrow\operatorname{\mathcal{B}}(\ell^{2}(G/H))$

defined by

$\displaystyle\lambda_{G/H}(f)g(xH)=(f*g)(xH)=\sum_{y\in G}f(y)g(y^{-1}xH)$

for all $f\in\ell^{1}(G)$, $g\in\ell^{2}(G/H)$, and $x\in G$. The kernel of
$\operatorname{Ker}=\ker(\lambda_{G/H})$ consists of the functions
$g\in\ell^{1}(G)$ such that $\sum_{h\in H}g(xhy)=0$ for all $x,y\in G$. Let us
denote

$\displaystyle\ell^{1}(G:H)=\ell^{1}(G)/Ker.$

Then the regular representation gives rise to the faithful $*$-representation

$\displaystyle\widetilde{\lambda}_{G/H}:\ell^{1}(G:H)\rightarrow\operatorname{\mathcal{B}}(\ell^{2}(G/H)).$

For $f\in\ell^{1}(G)$, denote by $[f]\in\ell^{1}(G:H)$ the corresponding
class, and by $\widetilde{f}\in\ell^{1}(G/H)$ the function defined by
$\widetilde{f}(xH)=\sum_{h\in H}f(xh)$ for all $x\in G$. Put

$\displaystyle[\operatorname{\mathbb{C}}G]=\\{[f]\in\ell^{1}(G:H):f\in\operatorname{\mathbb{C}}G\\}.$

######  Definition 4.

A $(G,H)$-representation is a unitary representation
$(\pi,\operatorname{\mathcal{H}})$ of $G$ such that $\pi(f)=0$ for all $f\in
Ker$.

Remark that $(\pi,\operatorname{\mathcal{H}})$ is a $(G,H)$ representation if
and only if $(\pi,\operatorname{\mathcal{H}})$ is a $*$-representation of
$\ell^{1}(G)$ such that $\pi(Ker)=\\{0\\}$ if and only if
$(\pi,\operatorname{\mathcal{H}})$ induces a $*$-representation of
$\ell^{1}(G:H)$ if and only if $(\pi,\operatorname{\mathcal{H}})$ induces a
$*$-representation of $[\operatorname{\mathbb{C}}G]$. We will denote by
$C^{*}(G:H)$ the universal $C^{*}$-enveloping algebra of $\ell^{1}(G:H)$ (or
equivalently $[\operatorname{\mathbb{C}}G]$). If we denote by
$(\pi_{u},\operatorname{\mathcal{H}}_{u})$ the largest $(G:H)$ representation
of $G$ (in the sense of weak containment), then we have
$C^{*}_{\pi_{u}}(G)\cong C^{*}(G:H)$.

###  2.3. Pseudo-function algebras

We will now construct an analogue of pseudo-function $*$-algebras for the pair
$(G,H)$. Let $1\leq p\leq q\leq\infty$ be conjugate numbers, i.e.
$p^{-1}+q^{-1}=1$. Consider the regular representation

$\displaystyle\lambda^{p}_{G/H}:\ell^{1}(G)\rightarrow\operatorname{\mathcal{B}}(\ell^{p}(G/H))$

defined by

$\displaystyle\lambda^{p}_{G/H}(f)g(xH)=(f*g)(xH)\sum_{y\in G}f(y)g(y^{-1}xH)$

for all $f\in\ell^{1}(G)$, $g\in\ell^{p}(G/H)$, and $x\in G$. Since its kernel
is independent of the choice of $1\leq p\leq\infty$, we get the faithful
representation

$\displaystyle\widetilde{\lambda}^{p}_{G/H}:\ell^{1}(G:H)\rightarrow\operatorname{\mathcal{B}}(\ell^{p}(G/H)).$

We denote by

$\displaystyle[f]\in\ell^{1}(G:H)\mapsto\|f\|_{PF_{p}}:=\|\lambda^{p}_{G/H}(f)\|$

the induced norm on $\ell^{1}(G:H)$. For $[f]\in\ell^{1}(G:H)$, its
$PF_{p}^{*}$-norm is given by

$\displaystyle\|f\|_{PF_{p}^{*}}=\max\\{\|f\|_{PF_{p}},\|f\|_{PF_{q}}\\}.$

Denote by $PF_{p}^{*}(G:H)$ the completion of $\ell^{1}(G:H)$ with respect to
$PF_{p}^{*}$-norm. Note that $\|f\|_{PF_{p}}=\|f^{*}\|_{PF_{q}}$ for all
$f\in\ell^{1}(G)$, and consequently $PF_{p}^{*}(G:H)$ becomes a unital Banach
$*$-algebra. For instance, when $p=2$, we have $PF_{2}^{*}(G:H)\cong
C^{*}_{\lambda_{G/H}}(G)$. The following lemma helps to understand the image
of the algebras $PF_{p}^{*}(G:H)$.

######  Lemma 5.

Let $1\leq p_{1}\leq p_{2}\leq 2$. The identity map on $\ell^{1}(G:H)$ extends
to the norm decreasing injective $*$-homomophisms

$\displaystyle\ell^{1}(G:H)\rightarrow PF_{1}^{*}(G:H)\rightarrow
PF_{p_{1}}^{*}(G:H)\rightarrow PF_{p_{2}}^{*}(G:H)\rightarrow
C^{*}_{\lambda_{G/H}}(G).$

In particular, these Banach $*$-algebras are $*$-semisimple.

###### Proof.

Let $q_{i}$ be the conjugate of $p_{i}$, i.e. $p_{i}^{-1}+q_{i}^{-1}=1$. For
any $f\in\ell^{1}(G)$, we have by complex interpolation

$\displaystyle\|f\|_{PF^{*}_{p_{1}}}\geq\|\lambda_{G/H}^{p_{1}}(f)\|^{1-\theta}\|\lambda^{q_{1}}_{G/H}(f)\|^{\theta}\geq\|\lambda^{p_{2}}_{G/H}(f)\|$

where $\theta\in[0;1]$ is such that $p_{2}^{-1}=(1-\theta)p_{1}^{-1}+\theta
q_{1}^{-1}$. Using the same inequality for $q_{2}$, we get
$\|f\|_{PF^{*}_{p_{1}}}\geq\|f\|_{PF^{*}_{p_{2}}}$. In other words, the
identity on $\ell^{1}(G:H)$ extends to a norm decreasing $*$-homomorphism
$i_{p_{1},p_{2}}:PF_{p_{1}}^{*}(G:H)\rightarrow PF_{p_{2}}^{*}(G:H)$. Take any
$T\in\ker(i_{p_{1},p_{2}})$. Then there is a sequence of functions
$f_{n}\in\operatorname{\mathbb{C}}G$ such that $[f_{n}]\rightarrow T$ in
$PF_{p_{1}}^{*}$-norm and $[f_{n}]\rightarrow 0$ in $PF_{p_{2}}^{*}$-norm.
Seeing $T$ as an operator on $\ell^{p_{1}}(G/H)$, we get

$\displaystyle\langle
T\delta_{yH},\delta_{xH}\rangle_{\ell^{p_{1}},\ell^{q_{1}}}$
$\displaystyle=\lim_{n}\langle
f_{n}*\delta_{yH},\delta_{xH}\rangle_{\ell^{p_{1}},\ell^{q_{1}}}$
$\displaystyle=\lim_{n}\sum_{h\in H}f_{n}(xhy)$ $\displaystyle=\lim_{n}\langle
f_{n}*\delta_{yH},\delta_{xH}\rangle_{\ell^{p_{2}},\ell^{q_{2}}}=0$

for all $x,y\in G$. Since $\\{\delta_{xH}:x\in G\\}$ is total in
$\ell^{p}(G/H)$ for any $1\leq p<\infty$, we get $T=0$. This proves that
$i_{p_{1},p_{2}}$ is injective. ∎

##  3\. Main section

###  3.1. Quasi-Hermitian pair

We are ready to give the definition of quasi-Hermitian pair.

######  Definition 6.

Let $G$ be a discrete group, and let $H$ be its subgroup. The pair $(G,H)$ is
called quasi-Hermitian (resp. quasi-symmetric) if
$[\operatorname{\mathbb{C}}G]$ is quasi-Hermitian (resp. quasi-symmetric) in
$PF^{*}_{1}(G:H)$. The group $G$ is Hermitian (resp. symmetric) if the pair
$(G,\\{e\\})$ is quasi-Hermitian (resp. quasi-symmetric).

######  Example 7.

If $H$ has finite index in $G$, then the Banach $*$-algebras $PF^{*}_{p}(G:H)$
are finite dimensional and Hermitian. In particular, the pair $(G,H)$ is
quasi-Hermitian.

######  Example 8.

Suppose that $H$ is a non-amenable normal subgroup of $G$ such that $Q=G/H$ is
quasi-Hermitian. Then $G$ is not quasi-Hermitian, and $(G,H)$ is quasi-
Hermitian. Indeed, the quotient map $p:G\rightarrow Q$ induces $*$-isomorphism
$\ell^{1}(G:H)\cong\ell^{1}(Q)$ that is stable on
$[\operatorname{\mathbb{C}}G]\cong\operatorname{\mathbb{C}}Q$. Since
$\ell^{1}(Q)$ is isometrically embedded in
$\operatorname{\mathcal{B}}(\ell^{1}(G/H))$ and
$\operatorname{\mathcal{B}}(\ell^{\infty}(G/H))$, we have
$\ell^{1}(G:H)\cong\ell^{1}(Q)\cong PF^{*}_{1}(G:H)$. Thus, $(G:H)$ is quasi-
Hermitian if and only if $[\operatorname{\mathbb{C}}G]$ is quasi-Hermitian in
$PF^{*}_{1}(G:H)$ if and only if $\operatorname{\mathbb{C}}Q$ is quasi-
Hermitian in $\ell^{1}(Q)$ if and only if $Q$ is quasi-Hermitian.

######  Proposition 9.

Let $1\leq p_{1}<p_{2}<p_{3}\leq 2$. Then

$\displaystyle(PF^{*}_{p_{1}}(G:H),PF^{*}_{p_{2}}(G:H),PF^{*}_{p_{3}}(G:H))$

is a spectral interpolation of triple Banach $*$-algebras relative to
$PF^{*}_{p_{1}}(G:H)$.

###### Proof.

The proof is a verbatim of [SW20, Proposition 4.5]. The spectral radius
inequality part is a well known consequence of complex interpolation, and
Lemma 5 shows that $PF^{*}_{p_{1}}(G:H)\subseteq PF^{*}_{p_{2}}(G:H)\subseteq
PF^{*}_{p_{3}}(G:H)$ is a nested triple of $*$-subalgebras.

∎

######  Theorem 10.

Let $G$ be a discrete group, and let $H\leq G$ be a subgroup. The following
are equivalent.

  1. (i)

$(G,H)$ is quasi-symmetric.

  2. (ii)

$(G,H)$ is quasi-Hermitian.

  3. (iii)

For all $f=f^{*}\in\operatorname{\mathbb{C}}G$, we have
$r([f],PF^{*}_{1}(G:H))=\|\lambda_{G/H}(f)\|$.

  4. (iv)

For all $f=f^{*}\in\operatorname{\mathbb{C}}G$, $r([f],PF^{*}_{p}(G:H))$ is
independent of $p\in[1,2]$.

  5. (v)

For all $f\in\operatorname{\mathbb{C}}G$, we have
$\operatorname{Sp}([f],PF^{*}_{1}(G:H))=\operatorname{Sp}([f],C^{*}_{\lambda_{G/H}}(G))$.

  6. (vi)

For all $f\in\operatorname{\mathbb{C}}G$,
$\operatorname{Sp}([f],PF^{*}_{1}(G:H))$ is independent of $p\in[1,2]$.

###### Proof.

The direction $(i)\Rightarrow(ii)$ is the easier part of Shirali-Ford’s
theorem (see [BD73, Remark p.225]). The directions $(iii)\Rightarrow(iv)$,
$(v)\Rightarrow(vi)$, $(vi)\Rightarrow(i)$ are clear. The direction
$(iv)\Rightarrow(v)$ is thanks to Barnes-Hulanicki’s Theorem [Bar90, p.329].
It only remains to prove $(ii)\Rightarrow(iii)$. The proof is an adaptation of
[SW20, Proposition 4.7]. Assume $(ii)$. Then
$(PF^{*}_{1}(G:H),PF^{*}_{p}(G:H),C^{*}_{\lambda_{G/H}}(G))$ is a triple of
spectral interpolation relative to the quasi-Hermitian $*$-subalgebra
$[\operatorname{\mathbb{C}}G]\subseteq PF^{*}_{1}(G:H)$ for any $1<p<2$. Thus,

$\displaystyle r([f],PF^{*}_{p}(G:H))=\|\lambda_{G/H}(f)\|\quad\text{for
all}\quad f\in\operatorname{\mathbb{C}}G$

by Theorem 3. Take any $f=f^{*}\in\operatorname{\mathbb{C}}G$. For any
$n\in\operatorname{\mathbb{N}}$ and $\varepsilon>0$, there exists
$g\in\operatorname{\mathbb{C}}G$ such that
$\|\widetilde{g}\|_{\ell^{1}(G/H)}\leq 1$ and

$\displaystyle(1-\varepsilon)\|f^{(n)}\|_{PF^{*}_{1}(G:H)}\leq\|\lambda_{G/H}^{1}(f^{(n)})\widetilde{g}\|_{\ell^{1}(G/H)}=\|\lambda_{G/H}^{1}(f^{(n-1)})\widetilde{f*g}\|_{\ell^{1}(G/H)}.$

Let $S$ be a symmetric finite subset of $G$ containing the support of
$f^{(n)}*g$. Then $\lambda^{1}_{G/H}(f^{(n)})\widetilde{g}$ is supported on
$[S]=\\{sH\in G/H:s\in S\\}$ and

$\displaystyle(1-\varepsilon)\|f^{(n)}\|_{PF^{*}_{1}(G:H)}$
$\displaystyle\leq\|\lambda_{G/H}^{1}(f^{(n-1)})(\widetilde{f*g})\operatorname{\mathbbm{1}}1_{[S]}\|_{\ell^{1}(G/H)}$
$\displaystyle\leq\|f^{(n-1)}\|_{PF_{p}^{*}(G:H)}\|\widetilde{f*g}\|_{\ell^{p}(G/H)}\|\operatorname{\mathbbm{1}}1_{[S]}\|_{\ell^{q}(G/H)}$
$\displaystyle\leq\|f^{(n-1)}\|_{PF_{p}^{*}(G:H)}\|\widetilde{f*g}\|_{\ell^{p}(G/H)}|S|^{1/q}$
$\displaystyle\leq\|f^{(n-1)}\|_{PF_{p}^{*}(G:H)}\|f\|_{PF^{*}_{1}(G:H)}|S|^{1/q}.$

Taking the $n$th root on both sides and $\limsup$ over $p\rightarrow 1$, we
get

$\displaystyle(1-\varepsilon)^{1/n}\|f^{(n)}\|_{PF^{*}_{1}(G:H)}^{1/n}$
$\displaystyle\leq\limsup_{p\rightarrow
1}\|f^{(n-1)}\|_{PF_{p}^{*}(G:H)}^{1/n}\|f\|_{PF^{*}_{1}(G:H)}^{1/n}$
$\displaystyle\leq\|f^{(n-1)}\|_{PF_{p_{0}}^{*}(G:H)}^{1/n}\|f\|_{PF^{*}_{1}(G:H)}^{1/n}+\varepsilon$

for some $1<p_{0}$. Now letting $n\rightarrow\infty$ and then
$\varepsilon\rightarrow 0$, we get

$\displaystyle r([f],PF_{1}^{*}(G:H))\leq
r([f],PF_{p_{0}}^{*}(G:H))=\|\lambda_{G/H}(f)\|.$

Since the converse inequality is always true, the above is equality.

∎

###  3.2. Co-amenable subgroup

Co-amenability was introduced in [Eym72] for a pair $(G,H)$ of locally compact
group $G$ and its closed subgroup $H$. It extends the usual amenability. Basic
examples are when the quotient $G/H$ is compact, and when $H$ is normal and
the quotient group $G/H$ is amenable. We recall the definition.

######  Definition 11.

Let $G$ be a discrete group, and let $H\leq G$ be a subgroup. We say that $H$
is co-amenable in $G$ if there exists a left $G$-invariant mean on
$\ell^{\infty}(G/H)$.

The usual amenability is characterized in many different ways. One of them is
the canonical equality of the full group $C^{*}$-algebra $C^{*}(G)$ and the
reduced group $C^{*}$-algebra $C^{*}_{\lambda}(G)$. A partial analogue of this
result is observed for co-amenability.

######  Proposition 12.

Let $G$ be a discrete group and let $H\leq G$ be a subgroup. Consider the
following statements.

  1. (i)

The canonical map $C^{*}(G:H)\twoheadrightarrow C^{*}_{\lambda_{G/H}}(G)$ is
injective.

  2. (ii)

The canonical map $C^{*}(PF^{*}_{1}(G:H))\twoheadrightarrow
C^{*}_{\lambda_{G/H}}(G)$ is injective.

  3. (iii)

The subgroup $H$ is co-amenable in $G$.

We have $(i)\Rightarrow(ii)\Rightarrow(iii)$. If $H$ is normal, we have
$(iii)\Rightarrow(i)$.

###### Proof.

The implication $(i)\Rightarrow(ii)$ follows from the surjective
$*$-homomorphisms

$\displaystyle C^{*}(G:H)\twoheadrightarrow
C^{*}(PF^{*}_{1}(G:H))\twoheadrightarrow C^{*}_{\lambda_{G,H}}(G).$

Let us prove $(ii)\Rightarrow(iii)$. Suppose that the map
$C^{*}(PF^{*}_{1}(G:H))\rightarrow C^{*}_{\lambda_{G/H}}(G)$ is injective,
hence $*$-isomorphism. Consider the character
$\sigma:[f]\in\ell^{1}(G:H)\mapsto\sum_{x\in G}f(x)$. Since we have

$\displaystyle|\sigma([f])|=|\sum_{x\in G}f(x)|\leq\sum_{xH\in G/H}|\sum_{h\in
H}f(xh)|=\|\lambda_{G/H}^{1}(f)\delta_{H}\|_{1}\leq\|f\|_{PF^{*}_{1}(G:H)}.$

$\sigma$ extends to a character on
$C^{*}(PF^{*}_{1}(G:H))=C^{*}_{\lambda_{G/H}}(G)$. It also extends to a state
on $\operatorname{\mathcal{B}}(\ell^{2}(G/H))$ which we also denote by
$\sigma$. Observe that $\sigma(\lambda_{G/H}(x))=\sigma([\delta_{x}])=1$ for
all $x\in G$. In particular, the left translation operators
$\\{\lambda_{G/H}(x):x\in G\\}$ are in the multiplicative domain of $\sigma$,
that is
$\sigma(\lambda_{G/H}(x)^{*}\lambda_{G/H}(x))=\sigma(\lambda_{G/H}(x))^{*}\sigma(\lambda_{G/H}(x))$
and
$\sigma(\lambda_{G/H}(x)\lambda_{G/H}(x)^{*})=\sigma(\lambda_{G/H}(x))\sigma(\lambda_{G/H}(x))^{*}$
for all $x\in G$. It follows from Bimodule Property (cf. [BO08, Proposition
1.5.7]) that

$\displaystyle\sigma(\lambda_{G/H}(x)T\lambda_{G/H}(x^{-1}))=\sigma(\lambda_{G/H}(x))\sigma(M)\sigma(\lambda_{G/H}(x^{-1}))=\sigma(T)$

for all $T\in\operatorname{\mathcal{B}}(\ell^{2}(G/H))$. Recall that the
pointwise multiplication operators give faithful $*$-representation

$\displaystyle f\in\ell^{\infty}(G/H)\mapsto
M_{f}\in\operatorname{\mathcal{B}}(\ell^{2}(G/H))$

and we have $M_{\lambda_{x}f}=\lambda_{G/H}(x)M_{f}\lambda_{G/H}(x^{-1})$.
Now, it is clear that $\sigma$ gives a $G$-invariant state on
$\ell^{\infty}(G/H)$.

For the direction $(iii)\Rightarrow(i)$, when $H$ is normal and co-amenable in
$G$, the quotient group $Q=G/H$ is amenable. Moreover, we have the canonical
isomorphisms $\ell^{1}(Q)\cong\ell^{1}(G:H)$, $C^{*}(Q)\cong C^{*}(G:H)$, and
$C^{*}_{\lambda}(Q)\cong C^{*}_{\lambda_{G/H}}(G)$. Thus $(iii)\Rightarrow(i)$
follows. ∎

Now, we are ready to prove Theorem 1.

###### Proof of Theorem 1.

Assume that $(G,H)$ is a quasi-Hermitian pair. Then
$[\operatorname{\mathbb{C}}G]$ is spectral in
$(PF^{*}_{1}(G:H),C^{*}_{\lambda_{G/H}}(G))$ by Theorem 10. Proposition 2
yields that we have canonical $*$-isomorphism $C^{*}(G:H)\rightarrow
C^{*}_{\lambda_{G/H}}(G)$. Then $H$ is co-amenable in $G$ by Proposition 12. ∎

######  Corollary 13.

Quasi-Hermitian discrete groups are amenable.

###### Proof.

Choose $H=\\{e\\}$ and apply Theorem 1. ∎

## References

  * [Bar90] Bruce A. Barnes, _When is the spectrum of a convolution operator on $L^{p}$ independent of $p$?_, Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 2, 327–332. MR 1057760
  * [BD73] Frank F. Bonsall and John Duncan, _Complete normed algebras_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
  * [BO08] Nathanial P. Brown and Narutaka Ozawa, _$C^{*}$ -algebras and finite-dimensional approximations_, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387
  * [Boy16] Adrien Boyer, _Quasi-regular representations and rapid decay_ , Potential Anal. 44 (2016), no. 2, 355–372. MR 3460029
  * [Eym72] Pierre Eymard, _Moyennes invariantes et représentations unitaires_ , Lecture Notes in Mathematics, Vol. 300, Springer-Verlag, Berlin-New York, 1972\. MR 0447969
  * [Pal13] Rui Palma, _Quasi-symmetric group algebras and $C^{*}$-completions of Hecke algebras_, Operator algebra and dynamics, Springer Proc. Math. Stat., vol. 58, Springer, Heidelberg, 2013, pp. 253–271. MR 3142041
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  * [SW20] Ebrahim Samei and Matthew Wiersma, _Quasi-Hermitian locally compact groups are amenable_ , Adv. Math. 359 (2020), 106897, 25. MR 4031113


# FedOS: using open-set learning to stabilize training in federated learning

Mohamad Mohamad
<EMAIL_ADDRESS>Julian Neubert
<EMAIL_ADDRESS>Juan Segundo Argayo
<EMAIL_ADDRESS>
###### Abstract

Federated Learning is a recent approach to train statistical models on
distributed datasets without violating privacy constraints. The data locality
principle is preserved by sharing the model instead of the data between
clients and the server. This brings many advantages but also poses new
challenges. In this report, we explore this new research area and perform
several experiments to deepen our understanding of what these challenges are
and how different problem settings affect the performance of the final model.
Finally, we present a novel approach to one of these challenges and compare it
to other methods found in literature. The code is available on our github
111https://github.com/mohamad-m2/federated-learning.

##  1 Introduction

Nowadays, phones are the primary and sometimes the only computing device for
many people. As such, the interaction with them and their powerful sensors
leave an unprecedented amount of rich data, much of it private in nature.
Cameras, microphones, GPS, and tactile sensors are all examples of this. In
the field of Big Data and Machine Learning applications, this data regains
importance as models learned with it hold the promise of greatly improving
usability by powering more intelligent applications, but the sensitive nature
of the data means there are risks and responsibilities to storing it in a
centralized location [13].

Federated learning (FL) is a privacy-preserving framework, originally
introduced by McMahan _et al_. [14], for training models from decentralized
user data residing on devices at the edge, the one mentioned before. Models
are trained iteratively across many federated rounds. For each round, every
participating device (a.k.a. client), receives an initial model from a central
server, performs stochastic gradient descent (SGD) on its local training data,
and sends back the gradients. The server then aggregates all gradients from
the participating clients and updates the starting model [7]. In FL, raw
client data is never shared with the server or other clients. This
distinguishes FL from traditional distributed optimization and requires
dealing with heterogeneous data

##  2 Related Work

lr | 0.2 | 0.15 | 0.1 | 0.08 | 0.06 | 0.05 | 0.04 | 0.02 | 0.01 | 0.005
---|---|---|---|---|---|---|---|---|---|---
raw | 47.67 | 44.7 | 49.26 | 47.75 | 49.98 | 48.61 | 47.46 | 44.13 | 43.83 | 39.92
scaled | 51.08 | 49.88 | 53.33 | 52.29 | 48.09 | 50.77 | 51.27 | 43.80 | 37.21 | 28.71
standardized | 53.42 | 54.07 | 54.27 | 53.53 | 53.85 | 52.32 | 52.74 | 47.54 | 40.56 | 34.81
Table 1: Hyper-parameter tuning on centralized setting. Learning rate and pre-
processing.

FedProx. There are two key challenges in Federated Learning that differentiate
it from traditional distributed optimization: (1) significant variability in
terms of the system’s characteristics on each device in the network (systems
heterogeneity), and (2) non-identically distributed data across the network
(statistical heterogeneity). FedProx (Tian Li _et al_. [12]) can be viewed as
a generalization and re-parametrization of FedAvg. In this work, the authors
have theoretically proven convergence guarantees when learning over data from
non-identical distributions (satistical heterogeneity), while also adhering to
the constraints each device-level system has by allowing each participating
device to perform a variable amount of work (system heterogeneity). They
demonstrate that in highly heterogeneous settings, FedProx has significantly
more stable and accurate convergence behavior relative to FedAvg, improving
absolute test accuracy by 22% on average.

FedIR. As well as [12], the authors of this paper (Hsu _et al_. [6])
recognized that one of the challenges in terms of data diversity relies on the
fact that data at the source is far from independent and identically
distributed (IID). In addition, they also found that differing quantities of
data are typically available at each device (imbalance). In their work, they
characterized the effect these real-world data distributions have on
distributed learning, using FedAvg as a benchmark. They proposed two new
algorithms, FedVC and FedIR. The latter one addresses the issue of the non-
identical class distribution shift present in the federated clients. If we
consider a target distribution $p(x,y)$ of images $x$ and class labels $y$ on
which a model is supposed to perform well, and a predefined loss function
$l(x,y)$, the objective of learning is to minimize the expected loss
$E_{p}[l(x,y)]$ with respect to the target distribution $p$. Instead, training
examples on a federated client $k$ are sampled from a client-specific
distribution $q_{k}(x,y)$. This implies that the empirical loss being
optimized on every client is a biased estimator of the loss with respect to
the target distribution, since $E_{q_{k}}[l(x,y)]\neq E_{p}[l(x,y)]$.
Therefore, this algorithm presents an importance reweighting scheme that
applies importance weights $w_{k}(x,y)$ to every client’s local objective.
With those in place, an unbiased estimator of loss with respect to the target
distribution can be obtained using training examples from the client
distribution. The paper shows a consistent improvement versus the experiments
run over the FedAvg baseline.

##  3 Baseline

In this section, we will describe the baseline we compare our experiments to
and give a brief justification for some of the initial hyperparameter choices.

###  3.1 Centralized Setting

Following the project proposal, we choose LeNet-5 [11] as the architecture for
our experiments, the only change is that our model takes the 3 input channels
from RGB color images. This model is not the state of the art for the CIFAR-10
dataset [9], but it is sufficient to show the relative performance of our
experiments. Additionally, using such a compact model in the federated setting
brings several benefits due to the potentially limited performance of client
devices and the need to share the model and its updates between the server and
clients for each round.

Figure 1: FedAvg ablation study. Figure (a) shows how classes are distributed
for different values of $\alpha$ taking a random client as an example. Figure
(c) illustrates the relative validation accuracy of FedAvg during the training
process for different values of $\alpha$. Figures (b) and (d) show the
relative validation accuracy after 120 rounds of training with different
hyperparameter settings hyperparameters by varying local epochs and reporting
client fraction respectively.

We perform a small hyperparameter search over the learning rate and pre-
processing to find an optimal setting to conduct our later experiments. The
results of which can be seen in Table 1. We notice, that standardizing the
data yields the best results. Interestingly, however, the model performed
equally well over a wide range of learning rates. The decreased performance of
models trained with lower learning rates is probably due to the short training
process. They would likely converge after more epochs to similarly high values
of accuracy, if not more. A learning rate schedule could help us circumvent
this problem. However, for simplicity, we choose a learning rate of $0.1$ for
all following experiments. In this setting, we achieve a final validation
accuracy of $54.03\%$ after 120 epochs of training.

###  3.2 Federated Setting

Similar to Hsu _et al_. [6] we draw the data distributions for our clients
from a Dirichlet distribution. This parametric distribution allows us to
compare how different federated approaches handle different severities of
unbalanced client data by varying $\alpha$. For very large $\alpha$, the
clients will have an almost equal distribution among the classes. The smaller
the value for $\alpha$, the more extreme the difference between the clients.
Having $\alpha=0$ will lead to each client only having samples from a single
class. For now, we will set $\alpha=1$ and further explore the effect of
different client distributions in our ablation studies in Sec. 4. Having
clients with diverse data is important for our experiments, as this difference
in data availability on each client device is one of the major limiting
factors for federated learning when compared to a centralized baseline.
Additionally, it mirrors how data is available in the real world, with
different users having different interests, therefore gathering data about
different topics, in different quantities and qualities. As we can not emulate
all of these parameters with the given dataset, we focus our experiments on
the distribution of classes among clients.

Furthermore, we keep our baseline simple by implementing a synchronous
communication scheme and using federated averaging to aggregate the clients’
updates. Section 7 shows some experiments with more advanced aggregation
schemes. By default, we distribute the dataset among $20$ clients, all with
equal number of samples ($2000$). For each communication round, $20\%$ of them
are chosen to train the network for one epoch on their local data and report
their new weights back to the server. We evaluate our results by comparing the
final model’s accuracy on a centralized validation set held by the server.
This allows us to analyze how well the model was able to train on the
distributed data and provides a more accurate comparison to its centralized
counterpart. This is functionally equivalent to comparing the weighted
accuracy between clients if the total test set distributed among the clients
is equal to the central test set and all clients report their local accuracy.

In this setting, we achieve a relative validation accuracy of $45.98\%$ after
120 communication rounds and $\alpha=1$.

##  4 Ablation study

Since we choose the hyperparameters for our baseline without any factual
basis, we will use this section to explore the effect each parameter has on
the final result of the model.

###  4.1 Data distribution

Probably the most important factor affecting validation accuracy is the data
distribution on each client device. Most machine learning methods assume the
data they train on to be independent and identically distributed (IID). This
means all data samples are drawn from the same distribution and independently
from each other. Violating this assumption often leads to decreased
performance of the model.

Figure 1 (a) shows the number of samples a typical client has access to given
different values of $\alpha$. We observe that the smaller $\alpha$ is, the
stronger the class unbalance on each client. For very small values such as
$\alpha=0.01$, each client typically only has access to samples of a single
class. This violation of the IID assumption leads to strong performance
decreases. Figure 1 (c) shows the relative validation accuracy during training
over the number of communication rounds. For $\alpha=100$ the data is more or
less evenly distributed among the clients, so the model can achieve results
very similar to those of the centralized case, albeit after a longer training
process. For $\alpha=1$ and lower values, the model does not only produce much
worse results, but we also observe that the training is highly unstable for
these hard cases.

Figure 2: Normalization effect. we show in these graphs how different
normalization layers perform compared to the raw model for various $\alpha$
values. We present both mean and best validation accuracy.

###  4.2 Client settings

Since the homogeneity of the data distribution on each client device is
typically dictated by the task, it is worth analyzing which other parameters
exist, that affect the predictive quality of the final model, and that can be
influenced by the application. Here we analyze the effects of how many epochs
the local modal is trained for on each selected client device during each
round of communication and the fraction of clients chosen to participate in
each round of communication relative to the total number of clients. In
response to the inconsistent validation accuracy during the training process,
that we discussed in the previous subsection, we will report both the average
and the best validation accuracy each model achieved over the training
process.

Figure 1 (b) shows what happens when we increase the number of local epochs.
We observe that this helps for large values of $\alpha$, while it seems to
have no effect for smaller values. An intuitive explanation for this could be,
that increasing the number of local epochs results in a longer effective
training process, since the number of communication rounds does not change.
However, this effect only helps in the case of more homogeneous data
distributions on the clients, since each model is expected to have similar
gradients. If the clients’ data grows more heterogeneous, each model will
converge towards a different local optimum, rendering the weight updates, that
are sent back to the server incompatible.

Another parameter of the training process we experiment with is the relative
amount of clients, that are chosen to train on their local data for each
round. Figure 1 (d) shows the results of this study. While the results do not
differ significantly from each other, we do see a slight improvement for
larger client fractions. This remains true over all values of $\alpha$, but
more pronounced for larger values. Choosing a larger fraction of clients to
train each round most likely stabilizes training, as the average update step
will more likely point towards the true local minimum of the loss function.
Again, this technique is more effective for larger values of $\alpha$, as the
weight updates of the clients are more compatible with each other.

##  5 Normalization

Since normalization techniques proved their capabilities in helping deep
networks train faster and attain better results, they have been widely used in
many benchmark models [5, 18]. While batch normalization [8] helps to minimize
internal covariate shift by batch statistics, it runs into problems with small
batches. This issue is addressed with an alternative normalization method
known as Group normalization [19]. Group Normalization uses channel instead of
batch statistics, which makes it more robust to the challenges and limitations
that come with small batch sizes. Considering how widely both techniques used
are, we discuss here how they perform in the federated scenario.

###  5.1 Batch normalization

As mentioned earlier, batch normalization uses batch statistics to compute a
mean and a variance used for standardizing the batch, and then applying an
affine transformation with learnable parameters. Usually, the shortcoming of
this technique is its inaccurate estimation when having batches of
insufficient sizes. In our study, we notice that Batch Normalization boosts
the model’s learning capabilities while $\alpha$ is large but these
improvements aren’t achievable anymore when $\alpha$ decreases below a
specific threshold. In fact for low $\alpha$ values Batch Normalization hurts
the training of the model (Figure 2) and attains worse results than models
with no normalization layers. This is mainly due to its biased estimation but
in this case, the problem isn’t the batch size, instead, it is the highly
heterogeneous batches between clients. This results in highly biased
statistics to be aggregated on the server’s side.

###  5.2 Group normalization

Group normalization was introduced as a simple alternative to Batch
Normalization. Group Normalization normalizes layer outputs based on channel
statistics. Specifically for a number of groups $G$ ($G$ is a hyperparameter),
it standardizes each group of channels’ features. Group Normalization shares a
similar concept as other normalization methods (layer normalization [1], and
instance normalization [17]). Actually, in the extreme cases of $G=1$, and
$G=N$ (where $N$ is the number of channels) it is equivalent to Layer
Normalization and Instance Normalization respectively. We performed our
analysis with (2 groups for the 1 Convolutional layer with 6 channels, 4
groups for the 2 one with 16 channels, and 30 groups for the last one with 120
channels), as shown in 2, Group Normalization outperforms Batch Normalization
for lower $\alpha$. This shows that group norm isn’t as limited as Batch
Normalization with the batch samples’ distribution, still the model suffer
from the non-identicalness of the client’s data issue.

(a)

(b)

Figure 3: FedOS. Figure (a) shows the structure of the model during training
and some samples of the unknown class generated by the GAN. In (b) we show the
results of the ablation study performed on our proposed FedOS method, in
particular we show the max and mean relative accuracy over 60 rounds. The mean
represents model learning stability while the max shows goodness. We have 4
hyperparameters in total with the default values (local epochs: 1, client
fraction: 20%, unknown class loss weight: 1, unknown class fraction: 60%). For
each study we change one parameter while fixing the others to their default
values.

##  6 Our method

As we saw in our ablation studies in Sec. 4, one of the biggest challenges in
federated learning is the heterogeneity of the client devices. During all of
our experiments, the model struggled to achieve meaningful results in settings
with highly unbalanced data distributions on client devices. We hypothesize
that an important factor is, that most clients are only aware of a fraction of
the classes the model is designed to recognize. In fact, in the case of
$\alpha=0.01$, many clients only have access to a single class and are not
aware of features that could be necessary to recognize other classes. This
does not only pose an issue for local training on the client devices. Since
each client trains only on a small subset of classes, their weights will
likely diverge from each other, making the aggregation of client updates even
more challenging. In this section, we present a novel approach that is
designed to address these shortcomings. During the local training step on each
client device, we add artificial samples from an unknown class [2] that
represent those classes that the client might not be aware of. Ideally, these
artificial samples share the same feature space as the real images in the
dataset. This allows the local model on each client to improve its feature
extraction capabilities while simultaneously learning which features are
important to discriminate those classes, which the local model has access to.

Let $\mathcal{D}$ denote our full dataset, $\mathcal{D_{C}}$ is the fraction
of $\mathcal{D}$ belonging to Client $\mathcal{C}$,
$\mathcal{\overline{D_{C}}}$ represents $\mathcal{D\setminus D_{C}}$, and
$\mathcal{M_{C}}$ is the model currently training on $\mathcal{D_{C}}$.
Suppose a global dataset containing all classes as $\mathcal{D_{G}}$. by
definition, a class belonging to $\mathcal{D}$ belongs to $\mathcal{D_{G}}$
which implies $\mathcal{\overline{D_{C}}\subset D\subset D_{G}}$. What we want
to achieve is $\mathcal{M_{C}}$ predicting unknown for
$\mathcal{\overline{D_{C}}}$ for all clients so that at the aggregation step
all models will distinguish their classes from those of other clients. We
estimate $\mathcal{D_{G}}$ using a relatively large dataset containing a fair
amount of different classes and a GAN [4] learning its general distribution.
We also modify our model to accept one extra class ”unknown” which introduces
a new term to the loss function that symbolizes the loss of unknown samples’
misclassification, this term is weighted using a hyperparameter
$\mathcal{W_{U}}$.

$L(C)=L(D_{C})+W_{U}*L(\text{unknown})$ (1)

###  6.1 Pre-training

Inspired by [15, 3] we train our GAN for generating and modeling the unknown
class. We differ from them in the final objective we have for training the
GAN. In the context of open-set recognition, the model at the end should be
able to recognize high-resolution unknown samples for which it has never been
trained, so they train their adversarial network to model the out-of-
distribution space using the classes at hand. On the other hand, our objective
is to take into account the existence of numerous different classes that don’t
appear within the local data during training. We try to model the general
space features by training the network on many classes to capture the most
repeated features. We train our GAN on a random subset of the Tiny Imagenet
dataset [10] containing samples from different classes. this step has a low
cost since we don’t need a specific class to train on, which makes the
training almost unsupervised. The only requirement is to train on many
classes.

###  6.2 Training

First, the trained GAN is sent to the client-side, this step accounts only for
half a round since the adversarial network doesn’t need to be sent back to the
server-side and its size is comparable to the model used for classification.
The clients generate a set of random samples proportional to the amount of
data they have locally. This is controlled by our second hyper parameter
$\mathcal{F_{U}}$. Suppose $\mathcal{N}$ is the total number of local samples,
and the quantity of unknown samples generated is $\mathcal{F_{U}*N}$. We add
an extra output class to classify ”unknown” samples (see Figure 3(a)) and the
training proceeds with the normal FedAvg scheme. At the end of training and
before inference, we disable the extra output forcing the model to output one
of the known classes since our final objective is a closed set classification
task.

| Relative accuracy @N rounds
---|---
| Alpha=1 | Alpha=0.01
Methods | @125 | @250 | @375 | @500 | @125 | @250 | @375 | @500
FedAvg | 65.94 | 82.17 | 83.03 | 88.34 | 40.41 | 41.92 | 51.19 | 60.75
FedIR | 77.66 | 82.98 | 89.03 | 89.36 | 51.75 | 59.56 | 64 | 69.03
FedProx | 63.85 | 82.07 | 86.51 | 91.4 | 28.81 | 37.64 | 48.58 | 48.58
FedOS | 77.92 | 84.09 | 94.98 | 95.85 | 46.6 | 56.13 | 64.7 | 69.11
Table 2: Performance comparison of different methods

###  6.3 FedOS ablation study

To study how the different hyperparameters affect our model behavior we
conduct a detailed ablation study on all 4 parameters (local epochs $E$,
reporting client fraction $CL$, unknown class weight $\mathcal{W_{U}}$, and
unknown class fraction $\mathcal{F_{U}}$). For each of them, we perform tests
varying only one value while keeping all other parameters constant with
default settings (refer to 3 for the default values). For high homogeneity
levels ($\alpha=100$) the method behaves similarly to FedAvg Fig. 1 where
increasing local epochs or client fraction boosts performance. On the other
side varying the parameters controlling the unknown class doesn’t have the
same effect simply since at high values of $\alpha$ the new unknown class
isn’t strictly needed. It is still interesting to point out that for
$\mathcal{W_{U}}=0.5$ the accuracy is higher than FedAvg with the same
settings, which shows that the additional class could have a regularizing
effect. Moving down to $\alpha=1$ we still have the same effect for $E$ and
$CL$, however, for $\mathcal{W_{U}}=1.5$ we have better performance and
stability (max and mean accuracy). Similarly for increasing $\mathcal{F_{U}}$
we obtain better results. Setting $\mathcal{F_{U}}=0.8$ is overkill. Even if
it has a slightly better performance than 0.4, it has decreased training
stability. The most interesting result is at $\alpha=0.01$ where our method is
at its full potential. The method outperforms FedAvg results for
hyperparameter settings. Performing a few additional local epochs further
enhances performance, it does, however, drop drastically at $E=5$. Similar
behavior occurs with $CL$, but contrary to $E$, model stability keeps
improving. Changing $\mathcal{W_{U}}$ seems to have minor effects for this
$\alpha$, however, choosing the highest $\mathcal{F_{U}}$ boosts both
stability and performance in agreement with our hypothesis, where we mentioned
that those unknown classes help the model learn discriminative features better
for very low alphas.

##  7 Experiments

Finally, we want to evaluate the model’s accuracy on the test set. We will
compare the baseline FedAvg algorithm to our extension proposal, named FedOS
in this context, and two alternative client update aggregation schemes from
literature, FedIR, and FedProx. Our experiments are limited to the more
challenging values of $\alpha$, namely $\alpha=1$ and $\alpha=0.01$. All other
hyperparameters remain unchanged. We split the dataset among $20$ clients
according to the distribution defined by $\alpha$ and use this dataset for all
$4$ algorithms. Training is done over $125$, $250$, $375$, and $500$ rounds
respectively, picking $20\%$ of the clients each round to train for $1$ local
epoch. The model’s weights used for evaluation are those that achieved the
highest accuracy on a validation dataset during the training process. We
choose this approach due to the aforementioned instability while training for
low values of $\alpha$. Our extension is implemented with DCGAN [16], down-
scaling the GAN’s width so that it is close in size to our inference model,
LeNet-5. The GAN network is trained for 90 epochs on 50% of randomly selected
Tiny Imagenet samples. For our extension we set $\mathcal{W_{U}}=1.5$,
$\mathcal{F_{U}}=0.4$ for $\alpha=1$ and $\mathcal{W_{U}}=1$,
$\mathcal{F_{U}}=0.8$ for $\alpha=0.01$. For FedProx we use the standard
parameter setting of $\mu=1$. The results of these experiments are reported in
Tab. 2.

FedOS Outperforms all methods for $\alpha=1$. Sometimes even using much fewer
training rounds, as in the case of $375$ rounds for FedOS and $500$ rounds for
all other methods. In the case of $\alpha=0.01$, the FedOS takes more rounds
to reach good results, however, it does surpass FedIR at 375 and 500 rounds.
Note that since for each round we are training on only 20% out of a total of
20 clients, only 4 clients are chosen to report their weight updates. In the
case where each client only has a single class, the entire model trains on 4
each round. Thus, an interesting alternative study would be to increase the
total number of clients, with each client having fewer samples.

##  8 Conclusion

In this work, we explored Federated Learning by having a look at some of the
challenges that come with it, specifically training on non-IID client
datasets. Furthermore, we discussed several techniques to tackle these
challenges and compared their relative performance. Finally, we present our
own idea, to add unknown class samples during the training process to help the
local model focus on learning to distinguish features from those classes, that
are present on the client device. We show that our approach performs well for
highly heterogeneous clients, as well as slightly boosts performance in more
homogeneous test cases. Since both the quantity and quality of our experiments
were limited by the computational power provided by Colab, we would like to
both further validate our hypotheses in the future, as well as compare our
novel approach to more state of the art methods.

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