Finite sums of projections in von Neumann algebras

Halpern, Herbert; Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang

doi:10.1090/s0002-9947-2013-05683-8

Cited by 14 publications

(16 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equivalently, there is a unitary operator U ∈ M with U N = I andAs the first application, we prove that a positive operator A ∈ M can be written as a finite sum of projections in M if and only if τ (A) ≥ τ (RA), where RA is the range projection of A. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if X ∈ M, X = X * and τ (X) = 0, then there exists a nilpotent element Z ∈ M such that X is the real part of Z.…”

supporting

confidence: 59%

“…In [9], H. Halpern, V. Kaftal, P. Ng and S. Zhang provided some sufficient conditions and also a necessary condition for a positive operator can be written as a finite sum of projections. For a type II 1 factor, they showed a sufficient condition for a positive diagonal operator A to be a finite sum of projections is τ (A + ) > τ (A − ) or equivalently τ (A) > τ (R A ).…”

Section: Introductionmentioning

confidence: 99%

“…To prove this theorem, we develop some new techniques in this paper. On the other hand, several results, e.g., Proposition 2.3, Corollary 2.5, Lemma 2.8 and Lemma 6.1, in [9] play important roles in the proof.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On finite sums of projections and Dixmier's averaging theorem for type ${\rm II}_1$ factors

Cao¹,

Fang²,

Yao³

2020

Preprint

View full text Add to dashboard Cite

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given an X ∈ M, X = X * , then there is a decomposition of the identity into N ∈ N mutually orthogonal nonzero projections Ej ∈ M, I = N j=1 Ej, such that EjXEj = τ (X)Ej for all j = 1, • • • , N . Equivalently, there is a unitary operator U ∈ M with U N = I andAs the first application, we prove that a positive operator A ∈ M can be written as a finite sum of projections in M if and only if τ (A) ≥ τ (RA), where RA is the range projection of A. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if X ∈ M, X = X * and τ (X) = 0, then there exists a nilpotent element Z ∈ M such that X is the real part of Z. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that let X1, • • • , Xn ∈ M. Then there exist unitary operators U1,This result is a stronger version of Dixmier's averaging theorem for type II1 factors.

show abstract

supporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On finite sums of projections and Dixmier's averaging theorem for type ${\rm II}_1$ factors

Cao¹,

Fang²,

Yao³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Fong and Murphy [13] characterized operators which are positive combinations of projections. Analogous results were recently investigated for C- * algebras and von Neumann algebras, see [16,18]. However, the most relevant results for us are due to Kruglyak, Rabanovich, and Samoȋlenko [25,24] who characterized the set of all (α, N) such that αI is the sum of K orthogonal projections.…”

Section: Introductionmentioning

confidence: 77%

A combinatorial characterization of tight fusion frames

2015

View full text Add to dashboard Cite

Abstract. In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza et al. [8]. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques [5,6,8]. We end the paper by giving a complete list of maximal TFF sequences in dimensions ≤ 9.

show abstract

“…19. (a) A C * -algebra A is of type A if and only if there is an abelian hereditary C * -subalgebra of A that generates an essential closed ideal of A.…”

mentioning

confidence: 99%

A Murray–von Neumann Type Classification of C*-algebras

Wong

2015

Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics

View full text Add to dashboard Cite

We define type A, type B, type C as well as C * -semi-finite C *algebras.It is shown that a von Neumann algebra is a type A, type B, type C or C * -semi-finite C * -algebra if and only if it is, respectively, a type I, type II, type III or semi-finite von Neumann algebra. Any type I C * -algebra is of type A (actually, type A coincides with the discreteness as defined by Peligrad and Zsidó), and any type II C * -algebra (as defined by Cuntz and Pedersen) is of type B. Moreover, any type C C * -algebra is of type III (in the sense of Cuntz and Pedersen). Conversely, any separable purely infinite C * -algebra (in the sense of Kirchberg and Rørdam) with either real rank zero or stable rank one is of type C.We also prove that type A, type B, type C and C * -semi-finiteness are stable under taking hereditary C * -subalgebras, multiplier algebras and strong Morita equivalence. Furthermore, any C * -algebra A contains a largest type A closed ideal J A , a largest type B closed ideal J B , a largest type C closed ideal J C as well as a largest C * -semi-finite closed ideal J sf . Among them, we have J A + J B being an essential ideal of J sf , and J A + J B + J C being an essential ideal of A. On the other hand, A/J C is always C * -semi-finite, and if A is C * -semi-finite, then A/J B is of type A. This paper is dedicated to Charles Batty on the occasion of his 60th birthday.

show abstract

Finite sums of projections in von Neumann algebras

Cited by 14 publications

References 38 publications

On finite sums of projections and Dixmier's averaging theorem for type ${\rm II}_1$ factors

On finite sums of projections and Dixmier's averaging theorem for type ${\rm II}_1$ factors

A combinatorial characterization of tight fusion frames

A Murray–von Neumann Type Classification of C*-algebras

Contact Info

Product

Resources

About