Is there a known example of a countable discrete group G whose full group C\*-algebra C\*(G) is not quasidiagonal?


Let us recall that a separable C\*-algebra A is quasidiagonal if it admits a faithful
\*-representation $\pi:A \to L(H)$ on a separable Hilbert space with the property
 that there is an increasing sequence $(p\_n)$ of finite rank projections in $L(H)$ which converges strongly to $1\_H$
and such that for each $a\in A$, $\lim\_{n\to \infty}\|[\pi(a)p\_n-p\_n\pi(a)]\|=0$.


$G=SL\_3(\mathbb{Z})$ is believed to be a candidate. Bekka has shown
that $C^\*(SL\_3(\mathbb{Z}))$ is not residually finite dimensional.



Discrete groups G whose full C*-algebra C*(G) is not quasidiagonal?
I don't know much about the subject, but isn't this an example? There is a discrete property (T) group $G$ without nontrivial finite dimensional representations [2, Remark in the last page]. Then, its max C\*-algebra contains a non-QD algebra as its direct summand, namely $C^\*(G)(1-z)$ for the Kazhdan projection $z$ (corresponding to the trivial rep) [1, Corollary 17.2.6].


[1] N. P. Brown and N. Ozawa. -algebras and finite-dimensional approximations, volume 88 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008.


[2] P. de la Harpe, A. G. Robertson, and A. Valette. On the spectrum of the sum of generators for a finitely generated group. Israel J. Math., 81(1-2):65–96, 1993.


