Examples of non connective C*-algebras

TL;DR

This paper provides examples of two infinite families of non-connective groups, extending the 3-dimensional Hantzsche-Wendt group, to illustrate properties of non-connective C*-algebras.

Contribution

It introduces new examples of non-connective groups based on generalizations of the Hantzsche-Wendt group, enriching the understanding of non-connective C*-algebras.

Findings

Two infinite families of non-connective groups are constructed.

These groups generalize the 3-dimensional Hantzsche-Wendt group.

The examples illustrate properties of non-connective C*-algebras.

Abstract

We present an example of two infinite families of not connective groups. Both of them are generalized of the 3-dimensional Hantzsche-Wendt group.

Equations

$$\Phi :A\to \prod _{n}CL(\mathcal{H})/\underset{n}{⨁}CL(\mathcal{H})$$

$$[[I(G),\mathcal{K}]]\to KK(I(G),\mathcal{K})\cong K^0(I(G))$$

$$0\to {\mathbb{Z}}^n\to \Gamma \to H\to 0,$$

$$G_{2}\cong\text{gen}\{A=(\left[\begin{array}[]{clcr}1&0&0\\ 0&-1&0\\ 0&0&-1\end{array}\right],(1/2,1/2,0)),B=(\left[\begin{array}[]{clcr}-1&0&0\\ 0&1&0\\ 0&0&-1\end{array}\right],(0,1/2,1/2))\}.$$

$$G_n=\{x_1,x_2,...,x_n\mid x_i^{-1}{x}_j^2{x}_i{x}_j^2,\forall i\ne j\}$$

Full text

1 Introduction

For a Hilbert space ${\cal H},$ we denote by $L({\cal H})$ the $C^{\star}$-algebra of bounded and linear operators on ${\cal H}.$ The ideal of compact operators is denoted by ${\cal K}\subset L({\cal H}).$ For the $C^{\star}$ - algebra $A,$ the cone over $A$ is defined as $CA=C_{0}[0,1)\otimes A,$ the suspension of $A$ as $SA=C_{0}(0,1)\otimes A.$

Definition 1**.**

Let $A$ be a $C^{\star}$-algebra and $n\in{\mathbb{N}},n\geq 1.$ $A$ is connective if there is a $\star$-monomorphism

[TABLE]

which is liftable to a completely positive and contractive map $\phi:A\to\prod_{n}CL({\cal H}).$

For a discrete group $G,$ we define $I(G)$ to be the augmention ideal, i.e. the kernel of the trivial representation $C^{\star}(G)\to{\mathbb{C}}.$ $G$ is called connective if $I(G)$ is a connective $C^{\star}$-algebra. From definition (see [5, p. 4921]) connectivity of $G$ may be viewed as a stringent topological property that accounts simultaneously for the quasidiagonality of $C^{\star}(G)$ and the verification of the Kadison-Kaplansky conjecture for certain classes of groups. Here we can referring to conjecture from 2014 [2, p. 166]. If $G$ is a discrete, countable, torsion-free, amenable group, then the natural map

[TABLE]

is an isomorphism of groups. Where $KK(I(G),\mathcal{K})$ is the Kasparov group and $[[I(G),\mathcal{K}]]$ is a group of the homotopy classes of asymptotic morphisms. In 2017 M. Dadarlat found an amenable and not connective group $G_{2}$ for which the above conjecture fails [4, Cor. 3.2].

Connective groups must be torsion-free, [3, Remark 2.8 and 4.4]. Here is a short list of such groups:

a countable torsion free nilpotent groups, [3, Th.4.3];

• 2.

let $0\to N\to G\to H\to 0$ be a central extension of discrete countable amenable groups where $N$ is torsion-free. If $H$ is connected then so does $G$; [3, Th. 4.1];

• 3.

wreath product of connected groups is a connected group [7, Th.3.2];

• 4.

a torsion-free crystallographic group is connective if and only if is locally indicable if and only if is diffuse (see below) and [4].

A discrete group $G$ is called locally indicable if every finitely generated non-trivial subgroup $L$ of $G$ has an infinite abelianization. The group $G$ is called diffuse if every non-empty finite subset $A$ of $G$ has an element $a\in A$ such that for any $g\in G,$ either $ga$ or $g^{-1}a$ is not in $A,$ [4], [8]. More examples of nonabelian connective groups were exhibited in [4], [5], [7]. The above group $G_{2}$ is a 3-dimensional, torsion-free crystallographic group, where a crystallographic group $\Gamma,$ of dimension $n$ is a cocompact and discrete subgroup of the isometry group $E(n)=O(n)\ltimes{\mathbb{R}}^{n}$ of the Euclidean space ${\mathbb{R}}^{n}$. $\Gamma$ is cocompact if and only if the orbit space $E(n)/\Gamma$ is compact. From Bieberbach theorems (see [10, Chapter 1]) any crystallographic group $\Gamma$ defines a short exact sequence

[TABLE]

where a free abelian group ${\mathbb{Z}}^{n}$ is a maximal abelian subgroup and $H$ is a finite group. $H$ is sometimes called a holonomy group of $\Gamma.$ The above group $G_{2}$ is isomorphic to the subgroup $E(3)$ and is generated by

[TABLE]

A torsion-free crystallographic group is called a Bieberbach group. The orbit space ${\mathbb{R}}^{n}/\Gamma$ of a Bieberbach group is a $n$-dimensional closed flat Riemannian manifold $M$ with holonomy group isomorphic to $H.$ A general characterization of connective Bieberbach groups is given in [4]. The following two theorems give us a landscape of them.

Theorem 1**.**

([4, Theorem 1.2])* Let $\Gamma$ be a Bieberbach group. The following assertions are equivalent.*

$\Gamma$* is connective*

• 2.

Every nontrivial subgroup of $\Gamma$ has a nontrivial center.

• 3.

$\Gamma$* is a poly-${\mathbb{Z}}$ group*

• 4.

$\widehat{G}\setminus\{\iota\}$* has no nonempty compact open subsets.*

The unitary dual $\widehat{G}$ of $G$ consists of equivalence classes of irreducible unitary representations of $G.$ $\iota$ denotes the trivial representation.

Theorem 2**.**

([4, Theorem 1.1])* A Bieberbach group with a finite abelianization is not connective.*

In our note we give an example of two infinite families of not connective groups. Both of them are generalization of the 3-dimensional Hantzsche-Wendt group $G_{2}.$

2 Examples

Example 1**.**

([10, Definition 9.1])Let $n\geq 3$. By generalized Hantzsche-Wendt (GHW for short) group we shall understand any torison-free crystallographic groups of rank $n$ with a holonomy group $({\mathbb{Z}}_{2})^{n-1}.$

Example 2**.**

([1, Definition], [11, Definition 1])* Let $n\geq 0.$ A group*

[TABLE]

we shall call a combinatorial Hantzsche-Wendt group.

For the properties of GHW groups we refer to [10, Chapter 9]. We have $G_{0}=1$ and $G_{1}\simeq{\mathbb{Z}}.$ Combinatorial Hantzsche-Wendt groups are torsion-free, see [1, Theorem 3.3] and for $n\geq 2$ are nonunique product groups. A group $G$ is called a unique product group if given two nonempty finite subset $X,Y$ of $G,$ there exists at least one element $g\in G$ which has a unique representation $g=xy$ with $x\in X$ and $y\in Y.$ We are ready to present our main result.

Proposition 1**.**

Generalized Hantzsche-Wendt groups with trivial center and nonabelian, combinatorial Hantzsche-Wendt groups are not connective.

Proof: From [3, Remark 2.8 (i)] the connectivity property is inherited by subgroups. Let $G$ be any group from family of GHW groups or family of combinatorial Hanzsche-Wendt groups. In both cases a group $G_{2}$ is a subgroup of $G$. In the first case it follows from [10, Proposition 9.7]. In the second case it follows from definition, see [1, Prop. 3.4]. Note that in the case of GHW groups we can also use Theorem 2, since all these groups have a finite abelianizations.

$\Box$

Remark 1**.**

From [11], for $n\geq 3,G_{n}$ has a non-abelian free subgroup. Hence is not amenable.

Remark 2**.**

The counterexample to the Kaplansky unit conjecture was given in 2021 by G. Gardam [9]. It was found in the group ring ${\mathbb{F}}_{2}[G_{2}].$ The Kaplansky unit conjecture states that every unit in $K[G]$ is of the form $kg$ for $k\in K\setminus\{0\}$ and $g\in G.$

Acknowledgements We thank the referee for a number of suggestions that improved the exposition.