Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

Skoufranis, Paul

doi:10.1016/j.jfa.2015.09.018

Cited by 9 publications

(18 citation statements)

References 52 publications

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“…We show the equivlance between (b) and (c) in Theorem 1.2 (similarly results for positive operators can be found in [33,63,80]).…”

Section: Hardy-littlewood-pólya Majorization In the Infinite Ssettingsupporting

confidence: 74%

“…Using a result by Day [20], we provide below a similar (but simpler) proof for completeness. We note that the special case for finite factors was described in [36] (see also [80,Theorem 2.18]). The following extends [49] (see also [80,Theorem 2.18] and [33, Theorem 4.7.…”

Section: Alberti-uhlmann Problem In the Setting Of Finite Von Neumann Algebrasmentioning

confidence: 99%

“…Due to the lack of natural meaning of Hardy-Littlewood-Pólya majorization in the infinite case, there are several different extensions of this notion in the literature (see e.g. [2,33,36,51,52,63,80]). Hiai and Nakamura [36] (see also [59, p. 25] and [63,80]) provided a natural definition of Hardy-Littlewood-Pólya majorization in terms of eigenvalue functions in the setting of infinite von Neumann algebras:…”

Section: Introductionmentioning

confidence: 99%

“…[2,33,36,51,52,63,80]). Hiai and Nakamura [36] (see also [59, p. 25] and [63,80]) provided a natural definition of Hardy-Littlewood-Pólya majorization in terms of eigenvalue functions in the setting of infinite von Neumann algebras:…”

Section: Introductionmentioning

confidence: 99%

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Alberti–Uhlmann Problem on Hardy–Littlewood–Pólya Majorization

Huang

Sukochev

2021

Commun. Math. Phys.

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We fully describe the doubly stochastic orbit of a self-adjoint element in the noncommutative L 1 -space affiliated with a semifinite von Neumann algebra, which answers a problem posed by Alberti and Uhlmann (Stochasticity and partial order: doubly stochastic maps and unitary mixing. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982) in the 1980s, extending several results in the literature. It follows further from our methods that, for any σ -finite von Neumann algebra M equipped with a semifinite infinite faithful normal trace τ , there exists a self-adjoint operator y ∈ L 1 (M, τ ) such that the doubly stochastic orbit of y does not coincide with the orbit of y in the sense of Hardy-Littlewood-Pólya, which confirms a conjecture by Hiai (J Math Anal Appl 127:18-48, 1987). However, we show that Hiai's conjecture fails for non-σ -finite von Neumann algebras. The main result of the present paper also answers the (noncommutative) infinite counterparts of problems due to Luxemburg (

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“…We show the equivlance between (b) and (c) in Theorem 1.2 (similarly results for positive operators can be found in [33,63,80]).…”

Section: Hardy-littlewood-pólya Majorization In the Infinite Ssettingsupporting

confidence: 74%

Section: Alberti-uhlmann Problem In the Setting Of Finite Von Neumann Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Alberti–Uhlmann Problem on Hardy–Littlewood–Pólya Majorization

Huang

Sukochev

2021

Commun. Math. Phys.

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“…Therefore, as ǫ was arbitrary, the result follows Proof. If a ∈ conv(U(b)) then a ≺ τ b for all τ ∈ T (A) by [39,Lemma 2.20] (the assumption that τ must be faithful is not required by the same argument as in Theorem 2.3). Suppose that a ≺ τ b for all τ ∈ T (A).…”

Section: The Main Resultsmentioning

confidence: 99%

Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras

Ng¹,

Skoufranis²

2017

Can. j. math.

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Abstract. In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C * -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

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The Dixmier property and tracial states for C⁎-algebras

Archbold

Robert

Tikuisis

2017

Journal of Functional Analysis

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It is shown that a unital C * -algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsidó theorem for simple C * -algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C * -algebras of Powers groups, but not by all C * -algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C * -algebra with unique tracial state to have this uniform property. We give further examples of C * -algebras with the uniform Dixmier property, namely all C * -algebras with the Dixmier property and finite radius of comparison-by-traces. Finally, we determine the distance between two Dixmier sets, in an arbitrary unital C * -algebra, by a formula involving tracial data and algebraic numerical ranges.

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Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

Cited by 9 publications

References 52 publications

Alberti–Uhlmann Problem on Hardy–Littlewood–Pólya Majorization

Alberti–Uhlmann Problem on Hardy–Littlewood–Pólya Majorization

Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras

The Dixmier property and tracial states for C⁎-algebras

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