Index for 𝐶*-subalgebras

Watatani, Yasuo

doi:10.1090/memo/0424

Cited by 131 publications

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“…The last sentence in the statement of the theorem is an immediate consequence of Proposition 2.10.9 of [6].…”

Section: Index Finitenessmentioning

confidence: 83%

“…An index-finite E always admits a quasi-basis of the form ([6], Lemma 2.1.6). In the special case where B ∩ A is finite dimensional, if one conditional expectation of A onto B is of index-finite type, then every faithful conditional expectation of A onto B is of index-finite type ( [6], Proposition 2.10.2). In the following theorem, C is the C * -subalgebra of A consisting of all scalar multiples of the identity element.…”

Section: Index Finitenessmentioning

confidence: 99%

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Representing conditional expectations as elementary operators

Pereira

2005

Proc. Amer. Math. Soc.

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Abstract. Let A be a C * -algebra and let B be a C * -subalgebra of A. We call a linear operator from A to B an elementary conditional expectation if it is simultaneously an elementary operator and a conditional expectation of A onto B. We give necessary and sufficient conditions for the existence of a faithful elementary conditional expectation of a prime unital C * -algebra onto a subalgebra containing the identity element. We give a description of all faithful elementary conditional expectations. We then use these results to give necessary and sufficient conditions for certain conditional expectations to be index-finite (in the sense of Watatani) and we derive an inequality for the index.

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“…The last sentence in the statement of the theorem is an immediate consequence of Proposition 2.10.9 of [6].…”

Section: Index Finitenessmentioning

confidence: 83%

Section: Index Finitenessmentioning

confidence: 99%

Representing conditional expectations as elementary operators

Pereira

2005

Proc. Amer. Math. Soc.

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“…In the above lemma, Index(E) denotes the index of the conditional expectation E. We refer to [1,3,18] for more details on the index of conditional expectations.…”

Section: Lemma 2 [9]mentioning

confidence: 99%

Operator-valued free Fisher information of random matrices

Meng

2010

Acta Mathematica Scientia

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Abstract. We study the operator-valued free Fisher information of random matrices in an operator-valued noncommutative probability space. We obtain a formula for Φ * M 2 (B) (A, A * , M 2 (B), η), where A ∈ M 2 (B) is a 2 × 2 operator matrix on B, and η is linear operators on M 2 (B). Then we consider a special setting: A is an operator-valued semicircular matrix with conditional expectation covariance, and find that Φ * B (c, c * : B, id) = 2Index(E), where E is a conditional expectation of B onto D and c is a circular variable with covariance E. Introduction and preliminariesFree probability theory is a noncommutative probability theory where the classical concept of independence is replaced by the notion of "freeness". This theory, due to D.Voiculescu, has very important applications on operator algebras (see [6, 21]) Originally, a noncommutative probability space is a pair (A, τ ), where A is a C * − or von Neumann algebra and τ is a state on A. Free independence is defined in terms of reduced free product relation given by τ . This notion was generalized by D.Voiculescu and others to an algebra valued noncommutative probability space where τ is replaced by a conditional expectation E B onto a subalgebra B of A, and freeness is replaced by freeness with amalgamation. Definition 1.[20] Let A be a unital algebra over C, and let B be a subalgebra of A, 1 ∈ B. E B : A −→ B is a conditional expectation, i.e. a linear map such thatan operator-valued (or B− valued) noncommutative probability space and elements in A are called B− random variables.The algebra freely generated by B and an indeterminate X will be denoted bybe subalgebras. The family (A i ) i∈I will be called free with amalgamation over B(or B−free), if E B (a 1 a 2 · · · a n ) = 0 whenever a j ∈ A ij with i 1 = i 2 = · · · = i n and E B (a j ) = 0, 1 ≤ j ≤ n. A sequence {a i } i∈I ⊆ A will be called free with amalgamation over B if the family of subalgebras generated by (B {a i }) i∈I is B−free.2000 Mathematics Subject Classification. Primary 46L54, 42C15.

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“…/´P u i u i 2 B is central, invertible and independent of the choice of .u i / i . For details, see, e.g., [22]. Lemma 7.14.…”

Section: Additional Structure On the Legsmentioning

confidence: 99%

Pseudo-multiplicative unitaries on C-modules and Hopf C-families I

Timmermann¹

2007

J. Noncommut. Geom.

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Abstract. Pseudo-multiplicative unitaries on C*-modules generalize the multiplicative unitaries of Baaj and Skandalis [1], and are analogues of the pseudo-multiplicative unitaries on Hilbert spaces studied by Enock, Lesieur, Vallin [5], [10], [21]. We introduce Hopf C*-families on C*-bimodules and associate to special classes of pseudo-multiplicative unitaries two Hopf C*-families. Furthermore, we discuss dual pairings and counits on these Hopf C*-families, étalé and proper pseudo-multiplicative unitaries, and two classes of examples. In a later article, we will study regularity conditions on pseudo-multiplicative unitaries, coactions of Hopf C*-families on C*-algebras, and duality.Mathematics Subject Classification (2000). 46L55; 46L08, 22A22, 46L05.

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Index for 𝐶*-subalgebras

Cited by 131 publications

References 0 publications

Representing conditional expectations as elementary operators

Representing conditional expectations as elementary operators

Operator-valued free Fisher information of random matrices

Pseudo-multiplicative unitaries on C-modules and Hopf C-families I

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Index for 𝐶*-subalgebras

Cited by 131 publications

References 0 publications

Representing conditional expectations as elementary operators

Representing conditional expectations as elementary operators

Operator-valued free Fisher information of random matrices

Pseudo-multiplicative unitaries on C*-modules and Hopf C*-families I

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Pseudo-multiplicative unitaries on C-modules and Hopf C-families I