Strong Morita equivalence of operator spaces

Eleftherakis, George; Kakariadis, Evgenios T. A.

doi:10.1016/j.jmaa.2016.09.042

Cited by 14 publications

(28 citation statements)

References 23 publications

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“…We expect that the above notion of stable equivalence is related to a notion of Morita equivalence for operator systems, similar to what happens in the case of C * -algebras [13], operator algebras [9] and operator spaces [28]…”

Section: Stable Equivalence For Operator Systemsmentioning

confidence: 68%

Spectral Truncations in Noncommutative Geometry and Operator Systems

Connes

Suijlekom

2020

Commun. Math. Phys.

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In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by $$C^*$$ C ∗ -algebras is taken over by operator systems. As part of the techniques we treat $$C^*$$ C ∗ -envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a $$C^*$$ C ∗ -algebra. We analyze this in the context of the finite Fourier transform on the cyclic group. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation $$d(x,y)< \varepsilon $$ d ( x , y ) < ε between two points, or more generally a tolerance relation which naturally gives rise to an operator system.

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Section: Stable Equivalence For Operator Systemsmentioning

confidence: 68%

Spectral Truncations in Noncommutative Geometry and Operator Systems

Connes

Suijlekom

2020

Commun. Math. Phys.

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“…Even though ∆ ∼ is originally defined on approximately unital operator algebras in [19], the elements we record here still hold for non-unital cases. This is exhibited in [20] where ∆ ∼ is considered for operator spaces. As noted ∆ ∼ and SME ∼ coincide with the usual strong Morita equivalence when restricted to C*-algebras.…”

Section: Suppose Thatmentioning

confidence: 97%

“…Theorem 4.4 implies that the operator algebras are equivalent by the same TRO that gives E SME ∼ F . By[20, Theorem 4.6] if two operator spaces X and Y are strongly ∆-equivalent by a σ-TRO then they are stably isomorphic. Hence we get the following corollary.Corollary 5.1.…”

mentioning

confidence: 99%

Morita equivalence of C*-correspondences passes to the related operator algebras

2017

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We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong ∆-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an application we show that strong Morita equivalence coincides with strong ∆-equivalence for tensor algebras of aperiodic C*-correspondences. IntroductionIntroduced by Rieffel in the 1970's [45,46], Morita theory provides an important equivalence relation between C*-algebras. In the past 25 years there have been fruitful extensions to cover more general (possibly nonselfadjoint) spaces of operators. These directions cover (dual) operator algebras and (dual) operator spaces, e.g. [3,4,6,7,[16][17][18][19][20][21][22]34]. There are two main streams in this endeavour. Blecher, Muhly and Paulsen [6] introduced a strong Morita equivalence SME ∼ , along with a Morita Theorem I, where the operator algebras A and B are symmetrically decomposed by two bimodules M and N , i.e. A ≃ M ⊗ B N and B ≃ N ⊗ A M. Morita Theorems II and III for SME ∼ were provided by Blecher [3]. On the other hand Eleftherakis [16] introduced a strong ∆-equivalence ∆ ∼ that is given by a generalized similarity under a TRO M , i.e. A ≃ M ⊗ B ⊗ M * and B ≃ M * ⊗ A ⊗ M. Although they coincide in the case of C*-algebras, relation ∆ ∼ is strictly stronger than relation SME ∼ . Indeed SME ∼ does not satisfy a Morita Theorem IV, even when X and Y are unital [6, Example 8.2]. However a Morita Theorem IV holds for ∆ ∼ on σ-unital operator algebras [19, Theorem 3.2].

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“…Theorem 4.6 in [11] implies that there exists a completely isometric onto linear map K ∞ (A) → K ∞ (B). By using the Banach-Stone theorem for operator algebras, we may assume that this map is also a homomorphism [1, 4.5.13].…”

Section: 5 the Space T = [Ndm]mentioning

confidence: 99%

On stable maps of operator algebras

Eleftherakis

2019

Journal of Mathematical Analysis and Applications

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We define a strong Morita-type equivalence ∼ σ∆ for operator algebras. We prove that A ∼ σ∆ B if and only if A and B are stably isomorphic. We also define a relation ⊂ σ∆ for operator algebras. We prove that if A and B are C * -algebras, then A ⊂ σ∆ B if and only if there exists an onto * -homomorphism θ : B ⊗ K → A ⊗ K, where K is the set of compact operators acting on an infinite dimensional separable Hilbert space. Furthermore, we prove that if A and B are C * -algebras such that A ⊂ σ∆ B and B ⊂ σ∆ A, then there exist projections r,r in the centers of A * * and B * * , respectively, such that Ar ∼ σ∆ Br and A(id A * * − r) ∼ σ∆ B(id B * * −r).

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Strong Morita equivalence of operator spaces

Abstract: This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International licence Newcastle University ePrints-eprint.ncl.ac.uk Eleftherakis GK, Kakariadis ETA. Strong Morita equivalence of operator spaces.

Cited by 14 publications

References 23 publications

Spectral Truncations in Noncommutative Geometry and Operator Systems

Spectral Truncations in Noncommutative Geometry and Operator Systems

Morita equivalence of C*-correspondences passes to the related operator algebras

On stable maps of operator algebras

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