A property for locally convex $^*$-algebras related to property $(T)$ and character amenability
Volume 227 / 2015
Abstract
For a locally convex $^*$-algebra $A$ equipped with a fixed continuous $^*$-character $\varepsilon$ (which is roughly speaking a generalized $F^*$-algebra), we define a cohomological property, called property $(FH)$, which is similar to character amenability. Let $C_c(G)$ be the space of continuous functions with compact support on a second countable locally compact group $G$ 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 three sets of characterizations can be regarded as analogues of one another. Moreover, we show that $G$ is compact if and only if the normed algebra $\{f\in C_c(G): \int_G f(t)\,dt =0\}$ (under $\|\cdot\|_{L^1(G)}$) admits a bounded approximate identity with the supports of all its elements being contained in a common compact set.
