On the projection of norm one in $W^ *$-algebras

Tomiyama, Jun

doi:10.3792/pja/1195524885

Cited by 237 publications

(172 citation statements)

References 13 publications

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“…According to Tomyiama's Theorem ( [7]), such a map E is completely positive and also satisfies the condition…”

Section: Resultsmentioning

confidence: 99%

On spectra of Lüders operations

Nagy

2008

Journal of Mathematical Physics

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Abstract. We show that the all eigenvalues of certain generalized Lüders operations are non-negative real numbers, in two cases of interest. In particular, given a commuting n-tuple A = (A 1 , . . . , An) consisting of positive operators on a Hilbert space H, satisfying n j=1 A j = I, we show that the spectrum of the Lüders operation:is contained in [0, ∞), so the only solution of the equation Λ A (X) = I − X is the "expected" one X = 1 2 I.

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“…According to Tomyiama's Theorem ( [7]), such a map E is completely positive and also satisfies the condition…”

Section: Resultsmentioning

confidence: 99%

On spectra of Lüders operations

Nagy

2008

Journal of Mathematical Physics

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“…(iii) We say (A, E) is a B-valued C * -noncommutative probability space if, in addition, A is a unital C * -algebra, B a C * -subalgebra of A and E is a projection of norm 1 onto B. (It follows from [20] that then E is positive and a B-bimodule map.) (iv) We say (A, E) is a B-valued W * -noncommutative probability space if, in addition, A is a unital W * -algebra, B a W * -subalgebra of A and E is a normal projection of norm 1 onto B.…”

Section: B-circular Elementsmentioning

confidence: 99%

“…From (20) we have x 1 = z+z * Notation 4.8. Let (A, E) be a B-valued Banach * -noncommutative probability space.…”

Section: B-circular Elementsmentioning

confidence: 99%

Hyperinvariant subspaces for some B–circular operators

Dykema

Tucci

2005

Math. Ann.

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Abstract. We show that if A is a Hilbert-space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra vN (A) that is generated by A, is independent of the representation of vN (A), thought of as an abstract W * -algebra.We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras.We introduce the B-circular operators as a special case of Speicher's B-Gaussian operators in free probability theory, and we prove several results about a B-circular operator z, including formulas for the B-valued Cauchy-and R-transforms of z * z. We show that a large class of L ∞ ([0, 1])-circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F 3 ). These results generalize some of what is known about the quasinilpotent DT-operator.

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“…We note that it follows that M must be positive, and satisfy M(fh) = fM(h) if / E L°°(Y) [12]. It is clear that if p is a relatively invariant measure over y, then M(f)(y) = ¡fdpy defines an invariant conditional mean.…”

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confidence: 99%

Amenable Pairs of Groups and Ergodic Actions and the Associated von Neumann Algebras

Zimmer¹

1978

Transactions of the American Mathematical Society

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Abstract. If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair (X, Y). This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.

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On the projection of norm one in $W^ *$-algebras

Cited by 237 publications

References 13 publications

On spectra of Lüders operations

On spectra of Lüders operations

Hyperinvariant subspaces for some B–circular operators

Amenable Pairs of Groups and Ergodic Actions and the Associated von Neumann Algebras

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