Completely bounded isomorphisms of operator algebras and similarity to complete isometries

Abstract

A well-known theorem of Paulsen says that if A is a unital operator algebra and ϕ : A → B(H) is a unital completely bounded homomorphism, then ϕ is similar to a completely contractive map ϕ ′ . Motivated by classification problems for Hilbert space contractions, we are interested in making the inverse ϕ ′−1 completely contractive as well whenever the map ϕ has a completely bounded inverse. We show that there exist invertible operators X and Y such that the map XaX −1 → Y ϕ(a)Y −1 is completely contractive and is "almost" isometric on any given finite set of elements from A with non-zero spectrum. Although the map cannot be taken to be completely isometric in general, we show that this can be achieved if A is completely boundedly isomorphic to either a C * -algebra or a uniform algebra. In the case of quotient algebras of H ∞ , we translate these conditions in function theoretic terms and relate them to the classical Carleson condition.2010 Mathematics Subject Classification. Primary:47L30, 46L07, 47L55.

Residually finite-dimensional operator algebras

“…A classical theorem of Paulsen [34], [33] says that completely bounded homomorphisms on operator algebras are necessarily similar to completely contractive ones. In [15], the possibility of obtaining a "two-sided" version of Paulsen's theorem for completely bounded isomorphisms was investigated. More precisely, the question is this: given a completely bounded isomorphism Φ : A → B, do there exist two invertible operators X ∈ B(H 1 ) and Y ∈ B(H 2 ) such that the map…”

“…is a complete isometry? It was shown in [15] that in general the answer is no. We show next that a weaker statement always hold.…”

Completely bounded subcontexts of a Morita context of unital $C^*$-algebras

“…For unital C * -algebras, any cb isomorphism is similar to a complete isometric (ci) isomorphism, but this is not true for unital operator algebras in general [9,Thm. 2.6,Sec.…”

“…3,resp.]. In [9], Clouâtre studies how close a cb isomorphism is to factoring as a complete isometric isomorphism composed with similarities. We would like to pursue a similar line of inquiry, but for cb Morita equivalences.…”

Compact ideals and rigidity of representations for amenable operator algebras

“…Part of our motivation stems from the fact that this latter class of maps is very rigid, as they lift to * -isomorphisms of the associated C * -envelopes; this is sometimes referred to as Arveson's implementation theorem [2]. The question of determining when a representation is similar to a complete isometry was considered by the first author in [7], prompted by classification problems for certain Hilbert space contractions, and therein it was shown to have an affirmative answer in various contexts. We will show here that this question also has an affirmative answer for the operator algebra constructed by Choi, Farah and Ozawa.…”