Mathematische Zeitschrift

, Volume 266, Issue 3, pp 693–705 | Cite as

Ideals of a C*-algebra generated by an operator algebra

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Abstract

In this paper, we consider ideals of a C*-algebra \({C^*(\mathcal{B})}\) generated by an operator algebra \({\mathcal{B}}\) . A closed ideal \({J\subseteq C^*(\mathcal{B})}\) is called a K-boundary ideal if the restriction of the quotient map on \({\mathcal{B}}\) has a completely bounded inverse with cb-norm equal to K−1. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from \({C^*(\mathcal{B})}\) onto \({\mathcal{B}}\) which is a projection on \({\mathcal{B}}\) is completely bounded. Moreover, we prove that Kadison’s similarity problem is decided on one particular C*-algebra which is a completion of the *-double of \({M_2(\mathbb{C})}\) .

Mathematics Subject Classification (2000)

46L05 46L07 47L55 47L07 47L30 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute of MathematicsChalmers University of TechnologyGöteborgSweden

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