Isomorphisms of tensor algebras arising from weighted partial systems

Dor-On, Adam

doi:10.1090/tran/7045

Cited by 8 publications

(13 citation statements)

References 43 publications

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“…In case is a graded (not necessarily completely) isometric isomorphism, because is an isometric isomorphism it is automatically a *-isomorphism by invoking [Gar65, Corollary 4.2]. Hence, the proof of [Dor18, Theorem 4.3 item (2)] can be carried out to show that the -correspondences and are unitarily isomorphic. Thus, items and in the above theorem are both equivalent to the existence of a graded isometric isomorphism between and .…”

Section: Hierarchy Of Isomorphism Problemsmentioning

confidence: 99%

“…Non-self-adjoint classification started with a paper of Arveson [Arv67] on classification of operator algebras associated to measure-preserving automorphisms. It was later realized that classification problems for many algebras can be put in the unified context of tensor algebras of -correspondences [MS00], and in many concrete cases such problems were resolved [DK08, DK11a, Dor18, KK04].…”

Section: Introductionmentioning

confidence: 99%

“…As the tensor algebra is naturally a subalgebra of (see § 2.2), it is invariant under the circle action as a subalgebra of , and there are spectral subspaces defined for as well. For each , the -spectral subspace for is given by We say that is the grading for , and by [Dor18, Proposition 4.2] we have concretely, when is universal, that …”

Section: Introductionmentioning

confidence: 99%

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Classification of irreversible and reversible Pimsner operator algebras

Dor-On¹,

Eilers²,

Geffen³

2020

Compositio Math.

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Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$ -algebras with additional $C^{*}$ -algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$ -correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

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Section: Hierarchy Of Isomorphism Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of irreversible and reversible Pimsner operator algebras

Dor-On¹,

Eilers²,

Geffen³

2020

Compositio Math.

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“…The origins of this framework are traced back to the seminal work of Arveson [1]. Since then, a number of rigidity results have appeared in the literature for tensor algebras of graphs or dynamical systems as in the work of Katsoulis-Kribs [26] and Solel [40], Davidson-Katsoulis [11,12] that supersedes the work of previous authors [3,19,37,38], Davidson-Roydor [13], Davidson-Ramsey-Shalit [14,15], Dor-On [16], Katsoulis-Ramsey [28], and the work of the second author with Davidson [10] and Katsoulis [23]. In this endeavour Davidson-Katsoulis [12] developed the notion of piecewise conjugacy for classical systems as the essential level of equivalence obtained from tensor algebras.…”

Section: Introductionmentioning

confidence: 97%

On the Quantized Dynamics of Factorial Languages

Barrett

Kakariadis

2017

The Quarterly Journal of Mathematics

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We study local piecewise conjugacy of the quantized dynamics arising from factorial languages. We show that it induces a bijection between allowable words of same length and thus it preserves entropy. In the case of sofic factorial languages we prove that local piecewise conjugacy translates to unlabeled graph isomorphism of the follower set graphs. Moreover it induces an unlabeled graph isomorphism between the Fischer covers of irreducible subshifts. We verify that local piecewise conjugacy does not preserve finite type nor irreducibility; but it preserves soficity. Moreover it implies identification (up to a permutation) for factorial languages of type 1 if, and only if, the follower set function is one-to-one on the symbol set.2010 Mathematics Subject Classification. 58F03, 47L75, 46L55, 54H20.

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“…We remark that Proposition 3.1 does not hold for bounded bimodule maps. Dor-On [15] illustrates this by examining a particular class of C*-correspondences related to weighted partial systems. In particular it is shown that unitary equivalence and bounded isomorphisms correspond to different notions of equivalences of the original data.…”

Section: Lemma 32 Let X a Be A Right Hilbert Module Then We Have Thatmentioning

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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Isomorphisms of tensor algebras arising from weighted partial systems

Cited by 8 publications

References 43 publications

Classification of irreversible and reversible Pimsner operator algebras

Classification of irreversible and reversible Pimsner operator algebras

On the Quantized Dynamics of Factorial Languages

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

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