New holomorphically closed subalgebras of $C^*$-algebras of hyperbolic groups

TL;DR

This paper constructs dense, holomorphically closed subalgebras within the reduced group $C^*$-algebra of hyperbolic groups, with applications to cyclic cohomology and $L^2$-invariants of negatively curved manifolds.

Contribution

It introduces new subalgebras that are dense, unconditional, and closed under holomorphic calculus, enhancing understanding of hyperbolic group $C^*$-algebras.

Findings

Existence of dense, holomorphically closed subalgebras in hyperbolic group $C^*$-algebras

Applications to cyclic cohomology of group $C^*$-algebras

Implications for delocalized $L^2$-invariants of negatively curved manifolds

Abstract

We construct dense, unconditional subalgebras of the reduced group $C^*$-algebra of a word-hyperbolic group, which are closed under holomorphic functional calculus and possess many bounded traces. Applications to the cyclic cohomology of group $C^*$-algebras and to delocalized $L^2$-invariants of negatively curved manifolds are given.