Group C*-algebras, metrics and an operator theoretic inequality

This is a survey of four recent papers which deal with the relationship of simple C*-algebras to the problem of computing the spectra of self-adjoint operators in the general case, especially when the spectrum is not discrete. It is an expanded version of a talk presented at the 50 year C*-algebra celebration, held at the annual meeting of the AMS in San Antonio during January, 1993.

Let $\Gamma$ be a discrete group. We show that if $\Gamma$ is nonamenable, then the algebraic tensor products $C^*_r(\Gamma)\otimes C^*_r(\Gamma)$ and $C^*(\Gamma)\otimes C^*_r(\Gamma)$ do not admit unique $C^*$-norms. Moreover, when $\Gamma_1$ and $\Gamma_2$ are discrete groups containing copies of noncommutative free groups, then $C^*_r(\Gamma_1)\otimes C^*_r(\Gamma_2)$ and $C^*(\Gamma_1)\otimes C_r^*(\Gamma_2)$ admit $2^{\aleph_0}$ $C^*$-norms. Analogues of these results c...

We introduce two new formulations for the notion of "quantum metric on noncommutative space". For a compact noncommutative space associated to a unital C*-algebra, our quantum metrics are elements of the spatial tensor product of the C*-algebra with itself. We consider some basic properties of these new objects, and state some connections with the Rieffel theory of compact quantum metric spaces. The main gap in our work is the lack of a nonclassical example even in the case o...

In the line of previous work by Naor, we establish new forms of metric $\mathrm{X}_p$ inequalities in group algebras under very general assumptions. Our results' applicability goes beyond the previously known setting in two directions. In first place, we find continuous forms of the $\mathrm{X}_p$ inequality in the $n$-dimensional torus. Second, we consider transferred forms of the sharp scalar valued metric $\mathrm{X}_p$ inequality in the von Neumann algebra $\mathcal{L}(\m...

Let (A,H,D) be a spectral triple, namely: A is a C*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak*-topology on the state space of A. A *-automorphism respe...

In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on Hilbert space. We show that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with decidable word problem. Moreover, we relate the computability of the operator norm on the product of non-abelian free groups to Kirchberg's QWEP Conjecture, a fundamental o...

This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby . We obtain characterizations of radial functions with respect to the $\ell^{2}$ length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the $\ell^{p}$ length on the free real line with infinite generators for $0 < p \leq 2$. We obtain the Levi-Khintchine formulas for ...

Let $G$ be a discrete group. Let $\lambda : G \to B(\ell_2(G),\ell_2(G))$ be the left regular representation. A function $\ph : G \to \comp$ is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined on the linear span $K(G)$ of $\{\lambda(x),x \in G\}$ by $$\sum_{x \in G} f(x) \lambda(x) \to \sum_{x \in G} f(x) \ph(x) \lambda(x)$$ is completely bounded (in short c.b.) on the $C^*$-algebra $C_\lambda^*(G)$ which is generated by $\lambd...

We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry's results in the commutative setting. Our proof is inspired by Pisier's method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in $\mathbb{R}^n$ (where Meyer's conjecture fails) or to the word len...

Let $\Gamma$ be a discrete group. When $\Gamma$ 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 $\Gamma$.