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

Volume 227 / 2015

Xiao Chen, Anthony To-Ming Lau, Chi-Keung Ng Studia Mathematica 227 (2015), 259-286 MSC: Primary 22D15, 22D20, 46K10; Secondary 22D10, 22D12, 16E40, 43A07, 46H15. DOI: 10.4064/sm227-3-5

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.

Authors

  • Xiao ChenChern Institute of Mathematics
    Nankai University
    Tianjin 300071, China
    e-mail
  • Anthony To-Ming LauDepartment of Mathematical and
    Statistical Sciences
    University of Alberta
    Edmonton, Alberta
    Canada T6G-2G1
    e-mail
  • Chi-Keung NgChern Institute of Mathematics
    Nankai University
    Tianjin 300071, China
    e-mail
    e-mail

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