Hyperreflexivity constants of the bounded $n$-cocycle spaces of group algebras and C$^*$-algebras

TL;DR

This paper establishes a new property for Banach algebras, applies it to group and C$^*$-algebras, and derives explicit bounds for hyperreflexivity constants of cocycle spaces and convolution operators, extending previous results.

Contribution

Introduces the strong property $( ext{B})$ with a constant for Banach algebras and applies it to derive explicit hyperreflexivity bounds for cocycle spaces and convolution operators.

Findings

All C$^*$-algebras and group algebras have the strong property $( ext{B})$ with a specific constant.

Provides a concrete upper bound for the hyperreflexivity constant of bounded $n$-cocycles.

Shows hyperreflexivity of convolution operators on $L^p(G)$ for amenable groups with explicit bounds.

Abstract

We introduced the concept of strong property $(\mathbb{B})$ with a constant for Banach algebras and, by applying certain analysis on the Fourier algebra of a unit circle, we show that all C$^*$-algebras and group algebras have the strong property $(\mathbb{B})$ with a constant given by $288\pi(1+\sqrt{2})$. We then use this result to find a concrete upper bound for the hyperreflexivity constant of $\mathcal{C}^n(A,X)$, the space of bounded $n$-cocycles from $A$ into $X$, where $A$ is a C$^*$-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$-bimodule for which $\mathcal{H}{n+1}(A,X)$ is a Banach space. As another application, we show that for a locally compact amenable group $G$ and $1<p<\infty$, the space $CV_P(G)$ of convolution operators on $L^p(G)$ are hyperreflexive with a constant given by $288\pi(1+\sqrt{2})$. This is the generalization of a well-known result of E. Christiensen for $p=2$.