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.
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 $x,y\in A$.