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Zbl 1213.46045
Gogić, Ilja
Elementary operators and subhomogeneous $C^*$-algebras. II.
(English)
[J] Banach J. Math. Anal. 5, No. 1, 181-192, electronic only (2011). ISSN 1735-8787/e

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 [{\it 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 2000:
*46L05 General theory of C*-algebras
46L07 Operator spaces etc.
47B47 Derivations and linear operators defined by algebraic conditions
46H10 Ideals and subalgebras of topological algebras

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

Citations: Zbl 1213.46046

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