[Paper Review] Exotic group C*-algebras

Matthew Wiersma|arXiv (Cornell University)|Mar 20, 2014

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper introduces new constructions of exotic group C*-algebras for nonamenable discrete groups, particularly focusing on $SL_n(\mathbb{Z})$ and free groups, by combining ideal completions with lattice structures of C*-algebras. It proves that $C^*_{\mathcal{F}_0}(SL_n(\mathbb{Z})) \vee C^*_{\ell^p}(SL_n(\mathbb{Z}))$ forms a new class of intermediate C*-algebras not dominated by any $C^*_{\ell^q}$, establishing uncountably many distinct exotic C*-algebras for $SL_n(\mathbb{Z})$.

ABSTRACT

Let $Γ$ be a discrete group. When $Γ$ is nonamenable, the reduced and full group $C$*-algebras differ and it is generally believed that there should be many intermediate $C$*-algebras, however few examples are known. In this paper we give new constructions and compare existing constructions of intermediate group $C$*-algebras for both generic and specific groups $Γ$.

Motivation & Objective

• Address the long-standing open problem of constructing explicit intermediate C*-algebras between the reduced and full group C*-algebras for nonamenable groups.
• Compare and unify existing constructions of exotic C*-algebras, including those from ideal completions ($\ell^p$-representations), Bekka's arithmetic group constructions, and the $\lambda_{G_d}$-based method.
• Develop a lattice structure on the set of group C*-algebras to systematically generate new exotic C*-algebras.
• Establish the existence of uncountably many distinct exotic C*-algebras for $SL_n(\mathbb{Z})$ by combining $C^*_{\ell^p}$ and $C^*_{\mathcal{F}_0}$ algebras.
• Provide a new class of exotic C*-algebras for $\mathbb{F}_\infty$ that lie outside the $\ell^p$-chain, expanding the known landscape of exotic group C*-algebras.

Proposed method

• Utilize ideal completions via translation-invariant ideals $D \subset \ell^\infty(\Gamma)$, particularly $D = \ell^p(\Gamma)$, to define C*-seminorms and complete $\mathbb{C}[\Gamma]$ into $C^*_D(\Gamma)$.
• Introduce a lattice structure on the set of group C*-algebras to generate new C*-algebras as joins and intersections of existing ones.
• Apply the Fell topology and weak containment techniques to analyze convergence of representations and detect trivial subrepresentations.
• Use induction and restriction of representations between subgroups, particularly embedding $\mathbb{F}_2 \subset SL_2(\mathbb{Z})$, to transfer properties from free groups to $SL_n(\mathbb{Z})$.
• Apply results from Bekka on congruence subgroups and trivial representation isolation to show that certain induced representations do not weakly contain the trivial representation.
• Construct a new class of exotic C*-algebras via the join $C^*_{\mathcal{F}_0}(\Gamma) \vee C^*_{\ell^p}(\Gamma)$, proving they are not dominated by any $C^*_{\ell^q}$ for $q > p$.

Experimental results

Research questions

• RQ1 Can new exotic group C*-algebras be systematically constructed for nonamenable groups beyond the known $\ell^p$-completions?
• RQ2 Are there exotic C*-algebras for $SL_n(\mathbb{Z})$ that lie outside the $\ell^p$-chain and are not dominated by any $C^*_{\ell^q}$?
• RQ3 Does the join $C^*_{\mathcal{F}_0}(SL_n(\mathbb{Z})) \vee C^*_{\ell^p}(SL_n(\mathbb{Z}))$ yield a new, distinct exotic C*-algebra for each $p$?
• RQ4 Can representations associated to $\ell^p$-completions be shown not to weakly contain the trivial representation in $SL_n(\mathbb{Z})$?
• RQ5 Is it possible to construct exotic C*-algebras for $\mathbb{F}_\infty$ that are not part of the $\ell^p$-family?

Key findings

• The paper constructs a new class of exotic group C*-algebras for $SL_n(\mathbb{Z})$ via the join $C^*_{\mathcal{F}_0}(SL_n(\mathbb{Z})) \vee C^*_{\ell^p}(SL_n(\mathbb{Z}))$, which are not dominated by any $C^*_{\ell^q}$ for $q > p$.
• For every $p \in [1, \infty)$, there exists $q > p$ such that $C^*_{\mathcal{F}_0}(SL_n(\mathbb{Z})) \vee C^*_{\ell^p}(SL_n(\mathbb{Z})) \not\succeq C^*_{\ell^q}(SL_n(\mathbb{Z}))$, proving the join is not contained in any $C^*_{\ell^q}$ algebra.
• An uncountable family of distinct exotic C*-algebras is constructed for $SL_n(\mathbb{Z})$, extending beyond the $\ell^p$-chain and the $\lambda_{G_d}$-based constructions.
• New exotic C*-algebras are constructed for $\mathbb{F}_\infty$, the free group on countably many generators, that lie outside the $\ell^p$-family, demonstrating a broader class of exotic algebras.
• Representations associated to $\ell^p$-completions for $SL_n(\mathbb{Z})$ do not weakly contain the trivial representation, a key property used to distinguish the new C*-algebras from $C^*_{\ell^q}$ algebras.