Norming Algebras and Automatic Complete Boundedness of Isomorphisms of Operator Algebras
نویسنده
چکیده
Abstract. We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if A1 and A2 are operator algebras, then any bounded epimorphism of A1 onto A2 is completely bounded provided that A2 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 λ-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 Ai of C ∗-diagonals (Ci,Di) (i = 1, 2) satisfying Di ⊆ Ai ⊆ Ci extends uniquely to a ∗isomorphism of the C∗-algebras generated by A1 and A2; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.
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