A quasidiagonal $C^{\ast }$-algebra with a nonquasidiagonal maximal tensor square

Abstract

We construct a separable unital residually finite-dimensional $C^{\ast }$-algebra $A$ whose maximal tensor square contains a proper isometry. In particular, $A$ is quasidiagonal whereas $A\tmax A$ is not. The algebra is generated by finite regular representations obtained from congruence quotients of an explicit countable affine group. A property~$(T)$ subgroup determines a diagonal Kazhdan projection in the maximal tensor square. Conjugation by a dilation makes this projection equivalent to a proper subprojection. Strictness is detected by a quasi-regular representation whose two factor restrictions descend to the specified algebra $A$. We give the norm estimate needed for this descent and construct the proper isometry explicitly.

arXiv comment8 pages, mostly generated by an LLM. The Author claims no novelty, but improved readability