*-Doubles and embedding of associative algebras in B(H)

TL;DR

This paper investigates the *-double functor linking associative and involutive algebras, establishing conditions for embedding associative algebras into $C^*$-algebras and exploring applications in operator algebra theory.

Contribution

It proves that an associative algebra embeds into a $C^*$-algebra iff its *-double is *-isomorphic to a *-subalgebra, and analyzes maximal subalgebras in operator algebras.

Findings

Associative algebra embeds into $C^*$-algebra iff its *-double is *-isomorphic to a *-subalgebra.

Every operator algebra is completely boundedly isomorphic to one with a maximal $C^*$-subalgebra of scalar multiples.

Identification of maximal subalgebras mapped into $C^*$-algebras under faithful representations.

Abstract

We study the *-double functor between the categories of associative and involutive algebras. It is proved that an associative algebra is isomorphic to a subalgebra of a $C\sp*$-algebra if and only if its *-double is *-isomorphic to a *-subalgebra of a $C\sp*$-algebra. Some applications in the theory of operator algebras are presented. In particular each operator algebra is shown to be completely boundedly isomorphic to an operator algebra $B$ with the greatest $C\sp*$-subalgebra consisting of the multiples of the unit and such that each element in $B$ is determined by its module up to a scalar multiple. We also study the maximal subalgebras of an operator algebra $A$ which are mapped into $C\sp*$-algebras under completely bounded faithful representations of $A$.