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**1 - 3**of**3**### A double commutant theorem for operator algebras

- J. Operator Theory
, 2004

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### A NOTE ON EXTENSIONS OF HILBERT C-MODULES AND THEIR MORPHISMS

"... Abstract. The aim of this paper is to connect the results of D. Bakic and B. Guljas about C-extensions of Hilbert C-modules with results of D.P. Blecher about Hilbert C-extensions of operator spaces. In the rst part, we give conditions on a completely bounded linear operator between Hilbert C-module ..."

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Abstract. The aim of this paper is to connect the results of D. Bakic and B. Guljas about C-extensions of Hilbert C-modules with results of D.P. Blecher about Hilbert C-extensions of operator spaces. In the rst part, we give conditions on a completely bounded linear operator between Hilbert C-modules for the possibility of extending the operator to a "corner-preserving " C-morphism of the corresponding linking-algebras (or, equivalently, for the operator being a Hilbert C-morphism). The second part provides an order preserving bijection between the sets of C-extensions of a Hilbert C-module and its Hilbert C-extensions, the latter being a generalized version of Blecher's Hilbert C-extensions of operator spaces dened in [5]. 1.

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

, 2009

"... 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∗-s ..."

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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.