Complete isometries—an illustration of noncommutative functional analysis

Abstract

Abstract. This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from 'noncommutative functional analysis', more specifically the new field of operator spaces. In our illustration we show how the classical characterization of (possibly non-surjective) isometries between function algebras generalizes to operator algebras. We give some variants of this characterization, and a new proof which has some advantages.

Logmodularity and isometries of operator algebras

“…This gives (iii). The proofs of (iv) and (v) are then similar to the matching items of (vi) in 3.1 of [12]. We will simply prove the assertions in (iv) that were not in the original version of [12].…”

“…Set N = Ker ρ, and obtain a surjective triple isomorphism ξ : C * e (A) → Z/N . Appealing to Lemma 2.10 in [12] almost immediately gives (vi). We may then follow the proof of (i) ⇒ (iii) of 3.1 in [12], replacing B there by C, and so on, to obtain a completely isometric partial triple morphism θ of C * e (A) into C/(J + K).…”

“…The first set of characterizations will be collected in the following theorem. We apologize in advance to the reader-since this is quite similar to the analogous result for selfadjoint operator algebras (Theorem 3.1 in [12]), we cannot justify repeating all the ideas and notation established there. Thus we must refer the reader to that paper for further details regarding our notation and proof.…”

“…We will simply prove the assertions in (iv) that were not in the original version of [12]. Note that the partial isometry u coming from the proof of 3.1 in [12] may be defined to be the weak* limit of T (e α )(1 − p), for a c.a.i. {e α } of A.…”

“…which is the formula given in [12] for the homomorphism θ we are looking for. Also, by the second last displayed formula above, we have u * uθ(a) = lim α θ(e α a) = θ(a), and similarly for θ(a)u * u.…”