## Banach Journal of Mathematical Analysis

### Elementary operators and subhomogeneous C*-algebras II

Ilja Gogic

#### Abstract

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).

#### Article information

Source
Banach J. Math. Anal., Volume 5, Number 1 (2011), 181-192.

Dates
First available in Project Euclid: 14 August 2011

https://projecteuclid.org/euclid.bjma/1313362989

Digital Object Identifier
doi:10.15352/bjma/1313362989

Mathematical Reviews number (MathSciNet)
MR2738529

Zentralblatt MATH identifier
1213.46045

#### Citation

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

#### References

• S.D. Allen, A.M. Sinclair and R.R. Smith, The ideal structure of the Haagerup tensor product of $C^*$-algebras, J. reine angew. Math. 442 (1993), 111–148.
• R.J. Archbold, Density theorems for the centre of a $C^*$-algebra, J. London Math. Soc. (2) 10 (1975), 189–197.
• R. J. Archbold and C. J. K. Batty, On Factorial States of Operator Algebras III, J. Operator Theory 15 (1986), 53–81.
• R.J. Archbold, D. W.B. Somerset and R.M. Timoney, Completely bounded mappings and simplicial complex structure in the primitive ideal space of a $C^*$-algebra, Trans. Amer. Math. Soc. 361 (2009), 1397–1427.
• R.J. Archbold, D. W.B. Somerset and R.M. Timoney, On the central Haagerup tensor product and completely bounded mappings of a $C^*$-algebra, J. Funct. Anal. 226 (2005), 406–428.
• D. Bakić and B. Guljaš, On a class of module maps of Hilbert $C^*$-modules, Math. Commun. 7 (2002), 177–192.
• D.P. Blecher and C. Le Merdy, Operator algebras and Their modules, Clarendon Press, Oxford, 2004.
• J.M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280.
• I. Gogić, Derivations which are inner as completely bounded maps, Oper. Matrices 4 (2010), 193–211.
• I. Gogić, Elementary operators and subhomogeneous $C^*$-algebras, Proc. Edin. Math. Soc. (to appear).
• B. Magajna, Uniform approximation by elementary operators, Proc. Edin. Math. Soc. 52/03 (2009) 731–749.
• G.K. Pedersen, $C^*$-algebras and their automorphism groups, Academic Press, London, 1979.
• N.C. Phillips, Recursive subhomogeneous algebras, Trans. Amer. Math. Soc. 359 (2007), 4595–4623.
• D.W. Somerset, The central Haagerup tensor product of a $C^*$-algebra, J. Operator Theory 39 (1998), 113–121.
• R.M. Timoney, Computation versus formulae for norms of elementary operators, preprint.
• J. Vesterstrøm, On the homomorphic image of the center of a $C^*$-algebra, Math. Scand. 29 (1971) 134–136.
• N.E. Wegge-Olsen, K-theory and $C^*$-algebras, Oxford University Press, Oxford, 1993.