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

Let $\Sigma \rightarrow G$ be a twist over a locally compact Hausdorff \'{e}tale groupoid $G$. Given $f$ in the reduced C$^*$-algebra $C_r^*(\Sigma;G)$ with open support $U \subseteq G$ we ask when $f$ lies in the closure of the compactly supported sections on $U$. Suppose $G$ satisfies the rapid decay property with respect to a length function $L$. We give a positive answer to our question in two instances: when $L$ is conditionally negative-definite, and when $L$ is the squ...

We study free and compact group actions on unital C*-algebras. In particular, we provide a complete classification theory of these actions for compact Abelian groups and explain its relation to the classical classification theory of principal bundles.

We provide a large class of discrete amenable groups for which the complex group ring has several C*-completions, thus providing partial evidence towards a positive answer to a question raised by Rostislav Grigorchuk, Magdalena Musat and Mikael R{\o}rdam.

We use free probability techniques to compute borders of spectra of non hermitian operators in finite von Neumann algebras which arise as `free sums' of `simple' operators. To this end, the resolvent is analyzed with the aid of the Haagerup inequality. Concrete examples coming from reduced C*-algebras of free product groups and leading to systems of polynomial equations illustrate the approach.

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal B$ generated by $\{x_r\}_{r \in \Lambda}$, which generalizes the strong Haagerup inequalities...

Let $G$ be a locally compact group. It is not always the case that its reduced C*-algebra $C^*_r(G)$ admits a tracial state. We exhibit closely related necessary and sufficient conditions for the existence of such. We gain a complete answer when $G$ compactly generated. In particular for $G$ almost connected, or more generally when $C^*_r(G)$ is nuclear, the existence of a trace is equivalent to amenability. We exhibit two examples of classes of totally disconnected groups fo...

We consider certain positive definite functions on a finitely generated free group G that are defined with respect to a given basis in terms of word length and the number of negative-to-positive generator exponent switches. Some of these functions are eigenfunctions for right convolution by the sum of the generators, and give rise to irreducible unitary representations of G. We show that any state of the reduced C*-algebra of G whose left kernel contains a polynomial in one o...

We study the semigroup C*-algebra of a positive cone P of a weakly quasi-lattice ordered group. That is, P is a subsemigroup of a discrete group G with P\cap P^{-1}=\{e\} and such that any two elements of P with a common upper bound in P also have a least upper bound. We find sufficient conditions for the semigroup C*-algebra of P to be nuclear. These conditions involve the idea of a generalised length function, called a "controlled map", into an amenable group. Here we give ...

We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous $2$-cocycle one associates the reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this algebra admit natural Hilbert space representations. We show the relevance of the orbit closure structure of the unit space of the groupoid in dealing with spectra, norms, numerical ranges and $\epsilon$-pseudospectra of the resulting operators. As an ex...

Let $B$ be a finite dimensional C$^\ast$-algebra equipped with its canonical trace induced by the regular representation of $B$ on itself. In this paper, we study various properties of the trace-preserving quantum automorphism group $\G$ of $B$. We prove that the discrete dual quantum group $\hG$ has the property of rapid decay, the reduced von Neumann algebra $L^\infty(\G)$ has the Haagerup property and is solid, and that $L^\infty(\G)$ is (in most cases) a prime type II$_1$...