Examples of non connective C*-algebras
Anna G\k{a}sior, Andrzej Szczepa\'nski

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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Holomorphic and Operator Theory · Advanced Topics in Algebra
Examples of non connective -algebras
A. Ga̧sior, A. Szczepański The first author is supported by the Polish National Science Center grant DEC2017/01/X/ST1/00062.
Abstract
We give an example of two infinite families of not connective groups. Both of them are generalization of the 3-dimensional Hantzsche - Wendt group. Key words. connective - algebras, crystallographic groups, combinatorial and generalized Hantzsche-Wendt groups, Mathematics Subject Classification: 46L05, 20H15, 46L80
1 Introduction
For a Hilbert space we denote by the -algebra of bounded and linear operators on The ideal of compact operators is denoted by For the - algebra the cone over is defined as the suspension of as
Definition 1**.**
Let be a -algebra and is connective if there is a -monomorphism
[TABLE]
which is liftable to a completely positive and contractive map
For a discrete group we define to be the augmention ideal, i.e. the kernel of the trivial representation is called connective if is a connective -algebra. From definition (see [5, p. 4921]) connectivity of may be viewed as a stringent topological property that accounts simultaneously for the quasidiagonality of 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 is a discrete, countable, torsion-free, amenable group, then the natural map
[TABLE]
is an isomorphism of groups. Where is the Kasparov group and is a group of the homotopy classes of asymptotic morphisms. In 2017 M. Dadarlat found an amenable and not connective group 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 be a central extension of discrete countable amenable groups where is torsion-free. If is connected then so does ; [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 is called locally indicable if every finitely generated non-trivial subgroup of has an infinite abelianization. The group is called diffuse if every non-empty finite subset of has an element such that for any either or is not in [4], [8]. More examples of nonabelian connective groups were exhibited in [4], [5], [7]. The above group is a 3-dimensional, torsion-free crystallographic group, where a crystallographic group of dimension is a cocompact and discrete subgroup of the isometry group of the Euclidean space . is cocompact if and only if the orbit space is compact. From Bieberbach theorems (see [10, Chapter 1]) any crystallographic group defines a short exact sequence
[TABLE]
where a free abelian group is a maximal abelian subgroup and is a finite group. is sometimes called a holonomy group of The above group is isomorphic to the subgroup and is generated by
[TABLE]
A torsion-free crystallographic group is called a Bieberbach group. The orbit space of a Bieberbach group is a -dimensional closed flat Riemannian manifold with holonomy group isomorphic to 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 be a Bieberbach group. The following assertions are equivalent.*
* is connective*
- 2.
Every nontrivial subgroup of has a nontrivial center.
- 3.
* is a poly- group*
- 4.
* has no nonempty compact open subsets.*
The unitary dual of consists of equivalence classes of irreducible unitary representations of 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
2 Examples
Example 1**.**
([10, Definition 9.1])Let . By generalized Hantzsche-Wendt (GHW for short) group we shall understand any torison-free crystallographic groups of rank with a holonomy group
Example 2**.**
([1, Definition], [11, Definition 1])* Let 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 and Combinatorial Hantzsche-Wendt groups are torsion-free, see [1, Theorem 3.3] and for are nonunique product groups. A group is called a unique product group if given two nonempty finite subset of there exists at least one element which has a unique representation with and 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 be any group from family of GHW groups or family of combinatorial Hanzsche-Wendt groups. In both cases a group is a subgroup of . 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.
Remark 1**.**
From [11], for 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 The Kaplansky unit conjecture states that every unit in is of the form for and
Acknowledgements We thank the referee for a number of suggestions that improved the exposition.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] W. Craig, P. Linnell, Unique product groups and congruence subgroups, to appear in J.Pure and Apply Algebra
- 2[2] M. Dadarlat, Group quasi-representations and almost flat bundles, J. Noncommut. Geom. 8(1) (2014) 163 – 178.
- 3[3] M. Dadarlat, Deformations of nilpotent groups and homotopy symmetric ℂ ⋆ superscript ℂ ⋆ {\mathbb{C}}^{\star} -algebras. Math. Ann., 367, (2017) 121-134
- 4[4] M. Dadarlat and E. Weld, Connective Bieberbach groups, Internat J. Math., 31(2020), no.6 20050047, 13 pp.
- 5[5] M. Dadarlat; U. Pennig, Connective C ⋆ superscript 𝐶 ⋆ C^{\star} -algebras. J. Funct. Anal. 272 (2017), no. 12, 4919 – 4943.
- 6[6] M. Dadarlat. On the asymptotic homotopy type of inductive limit ℂ ⋆ superscript ℂ ⋆ {\mathbb{C}}^{\star} -algebras. Math. Ann., 297(4):671 – 676, 1993.
- 7[7] M. Dadarlat, U. Pennig, A. Schneider, Deformations of wreath products, Bull. Lond. Math. Soc. 49(1) (2017) (ISSN 1469-2120) 23 – 32, http://dx.doi.org/10.1112/blms.12008
- 8[8] A. Ga̧sior, R. Lutowski, A. Szczepański, A short note about diffuse Bieberbach groups, J. Algebra, 494, 2018, 237 - 245
