Ranges of bimodule projections and reflexivity

Eleftherakis, George; Todorov, Ivan G.

doi:10.1016/j.jfa.2012.03.001

Cited by 6 publications

(34 citation statements)

References 18 publications

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“…, x n )}. If ω : G → R + is a continuous homomorphism, let E ω,t = {t ∈ G : ω(t) ≤ t}; we call the subsets of G of this form level sets (see [9]). where ∆ is the modular function.…”

Section: Sets Of Finite Widthmentioning

confidence: 99%

Sets of multiplicity and closable multipliers on group algebras

Shulman

Todorov

Turowska

2015

Journal of Functional Analysis

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Abstract. We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L 2 (G)) of bounded linear operators on L 2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E ⊆ G is a set of multiplicity if and only if the set E * = {(s, t) ∈ G × G : ts −1 ∈ E} is a set of operator multiplicity. Analogous results are established for M1-sets and M0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function ψ : G → C defines a closable multiplier on the reduced C*-algebra C * r (G) of G if and only if Schur multiplication by the function N (ψ) : G × G → C, given by N (ψ)(s, t) = ψ(ts −1 ), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L 2 (G). Similar results are obtained for multipliers on VN(G).

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Section: Sets Of Finite Widthmentioning

confidence: 99%

Sets of multiplicity and closable multipliers on group algebras

Shulman

Todorov

Turowska

2015

Journal of Functional Analysis

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“…Proof. By [8,Corollary 3.4], the algebraic sum W = U + Ran Φ is weak* closed. Given projections P, Q ∈ D, let Σ Q,P be the (contractive) Schur idempotent given by Σ Q,P (T ) = QT P , and…”

Section: Hyperreflexivity and Spansmentioning

confidence: 99%

“…We next introduce a hyperreflexivity analogue of approximately I-injective masa-bimodules defined in [8]. Let us say that a uniformly bounded sequence (Φ n ) n∈N ⊆ I decreases to a subspace V ⊆ B(H) if Φ 1 ≥ Φ 2 ≥ .…”

Section: Proof Let T ∈ B(h)mentioning

confidence: 99%

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Schur idempotents and hyperreflexivity

2016

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Abstract. We show that the set of Schur idempotents with hyperreflexive range is a Boolean lattice which contains all contractions. We establish a preservation result for sums which implies that the weak* closed span of a hyperreflexive and a ternary masa-bimodule is hyperreflexive, and prove that the weak* closed span of finitely many tensor products of a hyperreflexive space and a hyperreflexive range of a Schur idempotent (respectively, a ternary masa-bimodule) is hyperreflexive.

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“…The weak* closed masa-bimodules are precisely the weak* closed invariant subspaces of Schur multipliers or, equivalently, of weak* continuous (completely) bounded masa-bimodule maps. The projections in the algebra of all Schur multipliers, called henceforth Schur idempotents, were at the core of the methods developed in [8] in order to address the union problem, as well as the closely related problem of the reflexivity of weak* closed spans.…”

Section: Introductionmentioning

confidence: 99%

Operator synthesis and tensor products

Eleftherakis

Todorov

2015

Trans. Amer. Math. Soc.

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Abstract. We show that Kraus' property Sσ is preserved under taking weak* closed sums with masa-bimodules of finite width, and establish an intersection formula for weak* closed spans of tensor products, one of whose terms is a masabimodule of finite width. We initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form κ × λ, where κ is a set of finite width, while λ is operator synthetic, is, under a necessary restriction on the sets λ, again operator synthetic. We show that property Sσ is preserved under spatial Morita subordinance.

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Ranges of bimodule projections and reflexivity

Cited by 6 publications

References 18 publications

Sets of multiplicity and closable multipliers on group algebras

Sets of multiplicity and closable multipliers on group algebras

Schur idempotents and hyperreflexivity

Operator synthesis and tensor products

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