Operator spaces which are one-sided M-ideals in their bidual

Abstract

Abstract. We generalize an important class of Banach spaces, namely the Membedded Banach spaces, to the non-commutative setting of operator spaces. The one-sided M -embedded operator spaces are the operator spaces which are one-sided M -ideals in their second dual. We show that several properties from the classical setting, like the stability under taking subspaces and quotients, unique extension property, Radon Nikodým Property and many more, are retained in the non-commutative setting. We also discuss the dual setting of one-sided L-embedded operator spaces.

Real operator algebras and real completely isometric theory

“…The following result follows from above the lemma, [15,Proposition 2.12], and the fact that a real operator space X is right M -embedded if and only if X c is right M -embedded.…”

“…We also infer that a real operator space X is M -embedded if and only if X c is M -embedded. This facilitates in generalizing results in one-sided M -embedded theory from [15] to real operator spaces. For instance, we show that real one-sided M -embedded TRO are of the form A ∼ = ⊕ • i,j K(H i , H j ) completely isometrically, for some real Hilbert spaces H i , H j .…”

Ideals and hereditary subalgebras in operator algebras

“…Remark. Similarly, if A is an approximately unital ideal in its bidual then so is any closed subalgebra, or quotient by a closed ideal (see [40]).…”

“…In Section 5 we study operator algebras A which are hereditary subalgebras of their bidual, which as we said above is equivalent to the multiplication x → axb being weakly compact on A for all a, b ∈ A. Some of the properties of such algebras are similar to operator algebras which are one-sided ideals in their bidual, which were studied in [40,3]. We also study the more general class of algebras that we call nc-discrete, which means that all the open projections are also closed (or equivalently lie in the multiplier algebra M (A)).…”

Ideals and structure of operator algebras

“…The present paper is a continuation of a program (see e.g. [7,8,9,15,25,37]) studying the structure of operator algebras and operator spaces using 'one-sided ideals'. We shall have nothing to say about general one-sided ideals in an operator algebra A, indeed not much is known about general closed ideals in some of the simplest classical function algebras.…”