On stable maps of operator algebras

Eleftherakis, George

doi:10.1016/j.jmaa.2018.12.031

Cited by 4 publications

(4 citation statements)

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“…Proof. Since A and B are stably isomorphic, we have that they are also σ∆ equivalent, that is, A ∼ σ∆ B (according to [8,Theorem 3.3]). So, there exist Hilbert spaces H , K and completely isometric homomorphisms a : A → B(H) and β : B → B(K) and also a σ-TRO M ⊆ B(H, K) such that…”

Section: Definition 38 Let a Be An Abstract Approximately Unital Oper...mentioning

confidence: 99%

“…In [7,8,9,10], a new Morita type equivalence between operator algebras and operator spaces was developed: σ∆ equivalence. It was proved that two operator spaces X, Y are σ∆ equivalent, in which case we write X ∼ σ∆ Y , if and only if X and Y are stably isomorphic.…”

Section: Introductionmentioning

confidence: 99%

“…For further details about the notion of σ∆ equivalence of operator algebras and operator spaces, we refer the reader to [7,8,9,10]. If X , Y are operator spaces, then X ∼ σ∆ Y if and only if X and Y are stably isomorphic, that is, K ∞ (X) ∼ = K ∞ (Y ) (similarly for operator algebras).…”

Section: Introductionmentioning

confidence: 99%

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Hilbert modules, rigged modules and stable isomorphism

Eleftherakis,

Papapetros

2021

Preprint

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Every Hilbert module is a module of Morita equivalence between certain C * -algebras A and B. We present a new subcategory of Hilbert modules, the σ∆-Hilbert modules. Every σ∆-Hilbert module implements a stable isomorphism between A and B. Conversely, if the C * -algebras A and B are stably isomorphic, there exists a σ∆-Hilbert module which is a module of Morita equivalence between them. We generalize the above theory in the context of rigged modules over nonselfadjoint algebras. We develop a theory of Morita equivalence for rigged modules.

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Section: Definition 38 Let a Be An Abstract Approximately Unital Oper...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hilbert modules, rigged modules and stable isomorphism

Eleftherakis,

Papapetros

2021

Preprint

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“…The idea of taking into account the selfadjoint structure, as far as it is present in any nonselfdjoint algebra, inspired the notion of TRO-equivalence [27], which led to an equivalence of Morita type, called ∆-equivalence, generalising Rieffel's strong Morita equivalence of C*algebras [53], and examined for operator spaces, operator algebras, and their dual counterparts in [26,28,29,30,31,32,33,34]. It turns out, in particular, that ∆-equivalence is wellsuited to describe stable isomorphism of operator algebras and operator spaces (that is, the isomorphism of their amplified copies obtained by tensoring with -depending on the setting -either the C*-algebra of compact operators or the von Neumann algebra of all bounded linear operators on a Hilbert space).…”

Section: Introductionmentioning

confidence: 99%

Morita equivalence for operator systems

Eleftherakis¹,

Kakariadis²,

Todorov³

2021

Preprint

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We define ∆-equivalence for operator systems and show that it is identical to stable isomorphism. We define ∆-contexts and bihomomorphism contexts and show that two operator systems are ∆-equivalent if and only if they can be placed in a ∆-context, equivalently, in a bihomomorphism context. We show that nuclearity for a variety of tensor products is an invariant for ∆-equivalence and that function systems are ∆-equivalent precisely when they are order isomorphic. We prove that ∆-equivalent operator systems have equivalent categories of representations. As an application, we characterise ∆-equivalence of graph operator systems in combinatorial terms.

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A Morita Characterisation for Algebras and Spaces of Operators on Hilbert Spaces

Eleftherakis

Papapetros

2020

Integr. Equ. Oper. Theory

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The similarity problem is one of the most famous open problems in the theory of C * -algebras. We say that a C * -algebra A satisfies the similarity property ((SP) for short) if every bounded homomorphism u : A → B(H) is similar to a * -homomorphism and that a von Neumann algebra A satisfies the weak similarity property ((WSP) for short) if every w * -conitnuous unital and bounded homomorphism u : A → B(H), where H is a Hilbert space, is similar to a * -homomorphism. We prove that a von Neumann algebra A satisfies (WSP) if and only if the algebras A ′ ⊗B(ℓ 2 (I)) are hyperreflexive for all cardinals I. In the case in which A is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra A ′ ⊗B(ℓ 2 (N)) is hyperreflexive. We also introduce the hypothesis (CHH): Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under (CHH), all C * -algebras satisfy (SP). Finally, we prove that the spatial tensor product A ⊗B, where A is an injective von Neumann algebra and B is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the w * -similarity degree d * (A ⊗B).

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On stable maps of operator algebras

Cited by 4 publications

References 8 publications

Hilbert modules, rigged modules and stable isomorphism

Hilbert modules, rigged modules and stable isomorphism

Morita equivalence for operator systems

A Morita Characterisation for Algebras and Spaces of Operators on Hilbert Spaces

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