@MISC{Popovych09∗-doublesand, author = {Stanislav Popovych}, title = {∗-Doubles and embedding of associative algebras in B(H)}, year = {2009} }

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Abstract

We prove that an associative algebra A is isomorphic to a subalgebra of a C∗-algebra if and only if its ∗-double A∗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 B with the greatest C∗-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∗-algebras under completely bounded faithful representations of A.