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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 ΘA 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 ΘA 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).

46L05General theory of C * -algebras
46L07Operator spaces and completely bounded maps
47B47Commutators, derivations, elementary operators, etc.
46H10Ideals and subalgebras of topological algebras