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Elementary operators and subhomogeneous C*-algebras (II)

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).

C*-algebra ; subhomogeneous ; elementary operator ; 2-primal ideal ; Glimm ideal

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