The radius of comparison of the tensor product of a C*-algebra with $C(X)$

Abstract

Let $X$ be a compact metric space, let $A$ be a unital AH algebra with large matrix sizes, and let $B$ be a stably finite unital C*-algebra. Then we give a lower bound for the radius of comparison of $C(X)\otimes B$ and prove that the dimension-rank ratio satisfies $\mathrm{drr}(A)=\mathrm{drr}\left(C(X)\otimes A\right)$. We also give a class of unital AH algebras $A$ with $\mathrm{rc}\left(C(X)\otimes A\right)=\mathrm{rc}(A)$. We further give a class of stably finite exact $\mathcal{Z}$-stable unital C*-algebras with nonzero radius of comparison.