Elementary operators and subhomogeneous C*-algebras II

Abstract

Let $A$ be a separable unital $C^*$-algebra and let $\Theta_A$ be the canonical contraction from the Haagerup tensor product of $A$ with itself to the space of completely bounded maps on $A$. In our previous paper we showed that if $A$ satisfies (a) the lengths of elementary operators on $A$ are uniformly bounded, or (b) the image of $\Theta_A$ equals the set of all elementary operators on $A$, then $A$ is necessarily SFT (subhomogeneous of finite type). In this paper we extend this result; we show that if $A$ satisfies (a) or (b) then the codimensions of $2$-primal ideals of $A$ are also finite and uniformly bounded. Using this, we provide an example of a unital separable SFT algebra which does not satisfy (a) nor (b). However, if the primitive spectrum of a unital SFT algebra $A$ is Hausdorff, we show that such an $A$ satisfies (a) and (b).