One-sided M-Ideals and multipliers in operator spaces, I

Blecher, David P.; Effros, Edward G.; Zarikian, Vrej

doi:10.2140/pjm.2002.206.287

Cited by 27 publications

(86 citation statements)

References 34 publications

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“…1) X is completely M -embedded if and only if X is a complete M -ideal (in the sense of [ER1]) in its bidual (see e.g. [BEZ,Lemma 3.1] and [BZ2,Chapter 7]). …”

Section: One-sided M-embedded Spacesmentioning

confidence: 99%

“…Then by [BEZ,Theorem 5.1], P ∈ Ball(M ℓ (X (4) )) and Q ∈ Ball(M r (X (4) )), which implies that P Q = QP . Hence by [BZ2,Theorem 5.30 (ii)], J ∩X is a right M -ideal in X * * .…”

Section: Theorem 34 Suppose That X Is a Left M -Embedded Operator Smentioning

confidence: 99%

“…Later, Blecher, Effros, and Zarikian defined two varieties of M -ideals for the noncommutative setting, the "left M -ideals" and the "right M -ideals", and developed the one-sided M -ideal theory in a series of papers (see e.g. [BEZ,BZ1,BZ2] and [BSZ] with Smith). The intention was to create a tool for "non-commutative functional analysis".…”

Section: Introductionmentioning

confidence: 99%

“…If X is a right M -ideal in X * * , then by Proposition 2.3, π X * is a complete right L-projection with kernel X ⊥ . By [BEZ,Theorem 3.10(b)], it is the only completely contractive projection with kernel X ⊥ . Hence X satisfies all the equivalent conditions in Proposition 3.6.…”

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

Sharma

2010

Studia Math.

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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.

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“…1) X is completely M -embedded if and only if X is a complete M -ideal (in the sense of [ER1]) in its bidual (see e.g. [BEZ,Lemma 3.1] and [BZ2,Chapter 7]). …”

Section: One-sided M-embedded Spacesmentioning

confidence: 99%

“…Then by [BEZ,Theorem 5.1], P ∈ Ball(M ℓ (X (4) )) and Q ∈ Ball(M r (X (4) )), which implies that P Q = QP . Hence by [BZ2,Theorem 5.30 (ii)], J ∩X is a right M -ideal in X * * .…”

Section: Theorem 34 Suppose That X Is a Left M -Embedded Operator Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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

Sharma

2010

Studia Math.

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“…There is such a result in [7], but it makes reference to the containing O-algebra in the hypotheses. [7] One-sided ideals and approximate identities in operator algebras 431…”

Section: Introduction and Notationmentioning

confidence: 99%

One-sided ideals and approximate identities in operator algebras

Blecher¹

2004

J. Aust. Math. Soc.

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A left ideal of any C-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.)-Indeed, left ideals in C'-algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C-algebra. We study left ideals from the perspective of 'Hamana theory' and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approximate identity, including a Banach-Stone theorem for these algebras, and an analysis of the 'multiplier operator algebra'.2000 Mathematics subject classification: primary 46L05, 46L07,47L30; secondary 46H10, 47L75.

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Multipliers and Dual Operator Algebras

Blecher

2001

Journal of Functional Analysis

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In a previous paper we showed how the main theorems characterizing operator algebras and operator modules, fit neatly into the framework of the``noncommutative Shilov boundary,'' and more particularly via the left multiplier operator algebra of an operator space. As well as giving new characterization theorems, the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated. Recent progress of the author with Effros and Zarikian now enables weak*-versions of these characterization theorems. For example, we prove a result analogous to Sakai's famous characterization of von Neumann algebras as the C*-algebras with predual, namely, that the _-weakly closed unital (not-necessarilyselfadjoint) subalgebras of B(H ) for a Hilbert space H, are exactly the unital operator algebras which possess an operator space predual. This removes one of the hypotheses from an earlier characterization due to Le Merdy. We also show that the multiplier operator algebras of dual operator spaces are dual operator algebras. Using this we refine several known results characterizing dual operator modules. Academic Press

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One-sided M-Ideals and multipliers in operator spaces, I

Cited by 27 publications

References 34 publications

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

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

One-sided ideals and approximate identities in operator algebras

Multipliers and Dual Operator Algebras

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