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This post is related to an earlier question about Kazhdan property (T). The purpose of the snippet below is to briefly summarize the background for the question in this post.

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Question: Does there exist a unital amenable Banach algebra $A$ without (nonzero) tracial linear functionals such that $Z(A,A\hat{\otimes}_\pi A) = \{0\}$ ?

Here $A\hat{\otimes}_\pi A$ is the projective tensor product of $A$ with itself, $Z(A,E)=\{\texttt{m}\in E: a\texttt{m}=\texttt{m}a\hspace{4mm}\forall a\in A\}$ for any $A$-bimodule $E$, and $f\in A^*\backslash\{0\}$ is tracial if $f(xy-yx) = 0$ for all $x,y\in A$.

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