Completely contractive projections on operator algebras

Abstract

The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer, Robertson and Youngson, Youngson, and others, concerning projections on C*-algebras and their ranges. (See papers of these authors referenced in the bibliography.) In particular we investigate the `bicontractive projection problem' and related questions in the category of operator algebras. To do this, we will add the ingredient of `real positivity' from recent papers of the first author with Read.Comment: To appear Pacific J Math; several corrections and small improvements. Keywords Operator algebra, completely contractive map, projection, conditional expectation, bicontractive projection, real positive, noncommutative Banach-Stone theore

Jordan operator algebras: basic theory

“…Again we may assume that A and B are unital, since by the usual arguments is a real positive isometric surjection between unital Jordan operator algebras. Then the result follows from [, Proposition 6.6]. If T is a completely isometric surjective Jordan algebra homomorphism then by Proposition , T extends to a unital completely isometric surjection between the unitizations, which then extends by Arveson's lemma e.g.…”

“…More examples will be considered elsewhere, e.g. in we show that the ranges of various natural classes of contractive projections on operator algebras are Jordan operator algebras, and several other examples are given in .…”

Generalization of 𝐶*-algebra methods via real positivity for operator and Banach algebras

“…We give two or three applications from [21] of Theorem 3.3. The first is related to Kadison's Banach-Stone theorem for C * -algebras [50], and uses our Banach-Stone type theorem [16,Theorem 4.5.13].…”

“…Theorem 3.6. [21] The range of a completely contractive projection P : A → A on an approximately unital operator algebra is again an operator algebra with product P (xy) and cai (P (e t )) for some cai (e t ) of A, iff P is real completely positive.…”

“…This section also discusses some of the origins of the theory of positivity in our sense in the setting of algebras, which the later parts of our paper developes further. In the remainder of the paper we illustrate our real-positivity theory by showing how it relates to these results of Kadison, and also give some small extensions and additional details for our recent paper with Ozawa [22], and for [21] with Neal.…”