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Elementary operators and subhomogeneous ${C}^{*}$-algebras. II. (English) Zbl 1213.46045

Summary: 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 [I. Gogić, Proc. Edinb. Math. Soc., II. Ser. 54, No. 1, 99–111 (2011; Zbl 1213.46046)] we showed that if $A$ satisfies that (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 satisfies neither (a) nor (b).

However, if the primitive spectrum of a unital SFT algebra $A$ is Hausdorff, we show that such an $A$ satisfies both (a) and (b).

##### MSC:
 46L05 General theory of ${C}^{*}$-algebras 46L07 Operator spaces and completely bounded maps 47B47 Commutators, derivations, elementary operators, etc. 46H10 Ideals and subalgebras of topological algebras