中科院数学与系统科学研究院
数学研究所
学术报告会
报告人：吴志强（南开大学）
题 目：Property T of unital C*-algebra crossed products
时 间：2016.07.21（星期四）, 13:30-15:00
地 点：数学院南楼N913室
摘 要：
Let $\Gamma$ be a discrete group that acts on a unital C*-algebra A through an action $\beta$. We study property T as well as its stronger variant of the reduced C*-crossed product $A\rtimes_{\beta, r}\Gamma$.
If $\Gamma$ is amenable and A is nuclear and has enough tracial states, then $A\rtimes_{\beta, r}\Gamma$ having property T will imply that A is finite dimensional and $\Gamma$ is a finite group. If $\Lambda$ is a countable group acting ergodicly on an infinite $\sigma$-finite measure space $(\Omega, \mu)$, then there exists a $\Lambda$-invariant mean on $L^\infty(\Omega, \mu)$ if and only if the corresponding reduced crossed product does not have property T. Moreover, if $\Gamma$ is an ICC group, then $\Gamma$ is inner amenable if and only if $\ell^\infty(\Gamma\setminus {e\})\rtimes_{\mathbf{i},r} \Gamma$ does not have property T.
On the other hand, $A\rtimes_{\beta, r} \Gamma$ has strong property T if and only if A admits no $\beta$-invariant tracial state, or $\Gamma$ has property T and $(A\rtimes_{\beta, r} \Gamma, A)$ has strong property T. If $\Gamma$ has the Haagerup property, the strong property T of $(A\rtimes_{\beta, r} \Gamma, A)$ can be replaced by that of A. Furthermore, a non-compact locally compact group G is amenable if and only if $C_b(G)\rtimes_{\mathbf{lt}, r} G_\mathrm{d}$ does not have strong property T (respectively, $L^\infty(G)\rtimes_{\mathbf{lt}, r} G_\mathrm{d}$ does not have property T), where $G_\mathrm{d}$ is the group G equipped with the discrete topology.
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