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Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

Author: David R. Pitts
Journal: Proc. Amer. Math. Soc. 136 (2008), 1757-1768
MSC (2000): Primary 47L30, 46L07, 47L55
Published electronically: December 3, 2007
MathSciNet review: 2373606
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Abstract: We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if $ \mathcal{A}_1$ and $ \mathcal{A}_2$ are operator algebras, then any bounded epimorphism of $ \mathcal{A}_1$ onto $ \mathcal{A}_2$ is completely bounded provided that $ \mathcal{A}_2$ contains a norming $ C^*$-subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a $ C^*$-algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a $ C^*$-algebra is similar to a $ *$-representation precisely when the image operator algebra $ \lambda$-norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras $ \mathcal{A}_i$ of $ C^*$-diagonals $ (\mathcal{C}_i, \mathcal{D}_i)$ ($ i=1,2$) satisfying $ \mathcal{D}_i\subseteq\mathcal{A}_i\subseteq \mathcal{C}_i$ extends uniquely to a $ *$-isomorphism of the $ \mathcal{C}^*$-algebras generated by $ \mathcal{A}_1$ and $ \mathcal{A}_2$; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

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David R. Pitts
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130

Received by editor(s): September 18, 2006
Received by editor(s) in revised form: March 29, 2007
Published electronically: December 3, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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