A property for locally convex $^*$-algebras related to Property $(T)$ and character amenability

TL;DR

This paper introduces property (FH) for locally convex *-algebras, linking it to Property (T) and character amenability, and provides new characterizations of group properties like amenability and compactness via cohomological and fixed point methods.

Contribution

It defines property (FH) for locally convex *-algebras, establishes its equivalence with Property (T) for certain function algebras, and offers new cohomological characterizations of group properties.

Findings

Property (FH) characterizes Property (T) for certain function algebras.

Many Banach algebras with canonical characters have property (FH).

Group properties like amenability and compactness are characterized via cohomology and fixed points.

Abstract

For a locally convex $^*$-algebra $A$ equipped with a fixed continuous $^*$-character $\varepsilon$, we define a cohomological property, called property $(FH)$, which is similar to character amenability. Let $C_c(G)$ be the space of continuous functions on a second countable locally compact group $G$ with compact supports, equipped with the convolution $^*$-algebra structure and a certain inductive topology. We show that $(C_c(G), \varepsilon_G)$ has property $(FH)$ if and only if $G$ has property $(T)$. On the other hand, many Banach algebras equipped with canonical characters have property $(FH)$ (e.g., those defined by a nice locally compact quantum group). Furthermore, through our studies on both property $(FH)$ and character amenablility, we obtain characterizations of property $(T)$, amenability and compactness of $G$ in terms of the vanishing of one-sided cohomology of certain topological algebras, as well as in terms of fixed point properties. These characterizations can be regarded as analogues of one another. Moreover, we show that $G$ is compact if and only if the normed algebra $\big\{f\in C_c(G): \int_G f(t)dt =0\big\}$ (under $\|\cdot\|_{L^1(G)}$) admits a bounded approximate identity with the supports of all its elements being contained in a common compact set.