Completely Bounded Characterization of Operator Algebras with Involution

TL;DR

This paper extends the characterization of operator algebras by exploring completely bounded anti-isomorphisms, linking them to involutions and smooth $C^*$-modules, with implications for $KK$-theory.

Contribution

It generalizes Blecher's theorem to operator $K$-algebras with bounded anti-isomorphisms and connects these structures to smooth $C^*$-modules in $KK$-theory.

Findings

Characterization of operator algebras with bounded anti-isomorphisms.

Extension of Blecher's theorem to operator $K$-algebras.

Connection to smooth $C^*$-modules in $KK$-theory.

Abstract

In this paper we study the completely bounded anti-isomorphisms on operator algebras, that work similarly to the involutions with the exception for the property of being completely isometric. We elaborate the Blecher's characterization theorem for operator algebras to make it applicable to the so-called operator $K$-algebras with completely bounded reflexive anti-isomorphism. We also establish a connection of this result with the notion of smooth $C^*$-modules, that play an important role in Mesland's approach to Baaj-Julg picture of $KK$-theory.