Banach Journal of Mathematical Analysis
- Banach J. Math. Anal.
- Volume 5, Number 1 (2011), 181-192.
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).
Banach J. Math. Anal., Volume 5, Number 1 (2011), 181-192.
First available in Project Euclid: 14 August 2011
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Gogic, Ilja. Elementary operators and subhomogeneous C*-algebras II. Banach J. Math. Anal. 5 (2011), no. 1, 181--192. doi:10.15352/bjma/1313362989. https://projecteuclid.org/euclid.bjma/1313362989