IUMJ

Title: *-Doubles and embedding of associative algebras in $\mathbf{B}(\mathcal{H})$

Authors: Stanislav Popovych

Issue: Volume 57 (2008), Issue 7, 3443-3462

Abstract:

We prove that an associative algebra $\mathcal{A}$ is isomorphic to a subalgebra of a $C^{*}$-algebra if and only if its $*$-double $\mathcal{A} * \mathcal{A}^{*}$ is $*$-isomorphic to a $*$-subalgebra of a $C^{*}$-algebra. In particular each operator algebra is shown to be completely boundedly isomorphic to an operator algebra $\mathcal{B}$ with the greatest $C^{*}$-subalgebra consisting of the multiples of the unit and such that each element in $\mathcal{B}$ is determined by its module up to a scalar multiple. We also study the maximal subalgebras of an operator algebra $\mathcal{A}$ which are mapped into $C^{*}$-algebras under completely bounded faithful representations of $\mathcal{A}$.