Perturbations of intermediate C∗-subalgebras for simple C∗-algebras

Ino, Shoji; Watatani, Yasuo

doi:10.1112/blms/bdu001

Cited by 6 publications

(25 citation statements)

References 27 publications

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“…Let η : D → E be the natural inclusion map which is injective as E B was assumed to be faithful. Let B(E) denote the C*-algebra of adjointable operator on E. We now provide row versions of the estimates from [12] starting with a 'row version' of [12, Lemma 3.2].…”

Section: Universal Constants For Ino and Watatani's Theroemmentioning

confidence: 99%

“…We now modify [12,Lemma 3.4] to work with the row metric obtaining universal constants independent of the inclusion C ⊆ D. The argument is based on techniques developed by Christensen in [4] and [5].…”

Section: Universal Constants For Ino and Watatani's Theroemmentioning

confidence: 99%

“…is in the unit ball of M 1,n (A) since M M * = 1 A . By modifying the estimates in the last displayed equation on page 6 of [12] to work with rows we have…”

Section: Universal Constants For Ino and Watatani's Theroemmentioning

confidence: 99%

“…In [5] Christensen shows that sufficiently close von Neumann subalgebras of a finite von Neumann algebra arise from small unitary perturbations, and his work gives uniform estimates valid for all finite von Neumann algebras. In their recent paper Ino and Watatani [12] prove an analogous theorem in the context of intermediate C * -subalgebras [12,Proposition 3.6]: given an inclusion of C*-algebras C ⊆ D with a finite index conditional expectation, and intermediate subalgebras A, B with C ⊆ A, B ⊆ D, if A and B are sufficiently close then they are necessarily small unitary perturbations of each other. In contrast to Christensen's work, the estimates obtained by Ino and Watatani are given in terms of the basis for C ⊆ D and so depend on the inclusion C ⊆ D. The main purpose of this paper (Theorem 3.7) is to obtain uniform estimates valid for all finite index inclusions.…”

Section: Introductionmentioning

confidence: 99%

“…) and by following the argument in[12, Lemma 3.4] we may find a projection q ∈ λ(A)′ ∩ C * (λ(A), e B , 1 E ) with q − e B ≤ 2δ < 1. (53)and a unitary w ∈ C * (λ(A), e B , 1 E ) such that wqw * = e B and…”

mentioning

confidence: 99%

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A Kadison–Kastler row metric and intermediate subalgebras

Dickson

2014

Int. J. Math.

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In this paper we introduce a row version of Kadison and Kastler's metric on the set of C*-subalgebras of B(H). By showing C*-algebras have row length (in the sense of Pisier) of at most 2 we show that the row metric is equivalent to the original Kadison Kastler metric. Ino and Watatani have recently proved that in certain circumstances sufficiently close intermediate C*-algebras occur as small unitary perturbations. By adjusting their arguments to work with the row metric we are able to obtain universal constants independent of inclusions.

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Section: Universal Constants For Ino and Watatani's Theroemmentioning

confidence: 99%

Section: Universal Constants For Ino and Watatani's Theroemmentioning

confidence: 99%

“…is in the unit ball of M 1,n (A) since M M * = 1 A . By modifying the estimates in the last displayed equation on page 6 of [12] to work with rows we have…”

Section: Universal Constants For Ino and Watatani's Theroemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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A Kadison–Kastler row metric and intermediate subalgebras

Dickson

2014

Int. J. Math.

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On Possible Values of the Interior Angle Between Intermediate Subalgebras

Gupta¹,

Sharma

2023

J. Aust. Math. Soc.

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We show that all values in the interval $[0,{\pi }/{2}]$ can be attained as interior angles between intermediate subalgebras (as introduced by Bakshi and the first named author [‘Lattice of intermediate subalgebras’, J. Lond. Math. Soc. (2)104(2) (2021), 2082–2127]) of a certain inclusion of simple unital $C^*$ -algebras. We also calculate the interior angles between intermediate crossed product subalgebras of any inclusion of crossed product algebras corresponding to any action of a countable discrete group and its subgroups on a unital $C^*$ -algebra.

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Lattice of intermediate subalgebras

Bakshi

Gupta

2021

Journal of London Math Soc

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Analogous to subfactor theory, employing Watatani's notions of index and C * -basic construction of certain inclusions of C * -algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital C * -algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate C * -subalgebras of any inclusion of unital C *algebras admitting a finite index conditional expectation. Then, on the lines of Bakshi et al. (Trans. Amer. Math. Soc. 371 (2019) 5973-5991), we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate C * -subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an irreducible inclusion of type III factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite. Contents

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Perturbations of intermediate C∗-subalgebras for simple C∗-algebras

Cited by 6 publications

References 27 publications

A Kadison–Kastler row metric and intermediate subalgebras

A Kadison–Kastler row metric and intermediate subalgebras

On Possible Values of the Interior Angle Between Intermediate Subalgebras

Lattice of intermediate subalgebras

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