Rank-one decomposition of operators and construction of frames

Kornelson, Keri; Larson, David R.

doi:10.1090/conm/345/06249

Cited by 38 publications

(60 citation statements)

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“…They proved that a sufficient condition for a positive bounded operator A ∈ B(H) to be a (possibly infinite) sum of projections converging in the strong operator topology is that its essential norm A ess is >1 (see [5], Theorem 2). This result served as a basis for further work by Kornelson and Larson [13] and then by Antezana et al [1] on decompositions of positive operators into strongly converging sums of rank one positive operators with preassigned norms. In [12], the necessary and sufficient condition for a positive bounded operator to be a strongly converging sum of projections was obtained by the three authors of this article for the B(H) case and for the case of a countably decomposable type III von Neumann factor, and for the "diagonalizable" case of type II von Neumann factors.…”

Section: Introduction and The Main Resultsmentioning

confidence: 69%

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Projection decomposition in multiplier algebras

Kaftal¹,

Ng²,

Zhang³

2011

Math. Ann.

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In this paper we present new structural information about the multiplier algebra Mult (A) of a sigma-unital purely infinite simple C*-algebra A, by characterizing the positive elements a in Mult(A) that are strict sums of projections belonging to A. If a is not in A and is not a projection, then the necessary and sufficient condition for a to be a strict sum of projections belonging to A is that the norm ||a||>1 and that the essential norm ||a||_ess >=1. Based on a generalization of the Perera-Rordam weak divisibility of separable simple C*-algebras of real rank zero to all sigma-unital simple C*-algebras of real rank zero, we show that every positive element of A with norm greater than 1 can be approximated by finite sums of projections. Based on block tri-diagonal approximations, we decompose any positive element a in Mult(A) with ||a||>1 and ||a||_ess >=1 into a strictly converging sum of positive elements in A with norm greater than 1.Comment: To appear in Mathematische Annale

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Section: Introduction and The Main Resultsmentioning

confidence: 69%

“…Acknowledgments V. Kaftal [5,13] that stimulated this project. V. Kaftal and S. Zhang were partially supported by grants from the Charles Phelps Taft Research Center.…”

mentioning

confidence: 99%

Projection decomposition in multiplier algebras

Kaftal¹,

Ng²,

Zhang³

2011

Math. Ann.

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“…Some papers dealing with infinite frames which relate directly or indirectly to this article are [15,13,9,1,10,20,19,21].…”

Section: Introductionmentioning

confidence: 99%

Operator-valued frames

Kaftal

Larson

Zhang

2009

Trans. Amer. Math. Soc.

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Abstract. We develop a natural generalization of vector-valued frame theory, which we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, disjointedness, complementarity, and composition of operator-valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operatorvalued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe generators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra generated by the group representation, and the resulting norm pathconnectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we generalize this multiplicity one result to operator-valued frames. However, both the parametrization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parametrization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.

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“…The theory of majorization started more than a century ago motivated by wealth distribution (Lorenz [32]), inequalities involving convex functions and the study of stochastic matrices (Hardy, Littlewood and Pólya [18]), convex combinations of permutation matrices (Birkhoff [6]), and the relation between diagonals of selfadjoint matrices and their eigenvalue lists (Schur [40] and Horn [19], and others [35]- [36], [33], [15]). In more recent years majorization theory and also its extension to infinite sequences has turned up prominently in several areas of operator theory and frame theory ( [20]- [21], [5], [9], [2], [30]). We contributed to this study in [28], where we extended many classical facts on (finite) majorization to infinite sequences decreasing to zero and used these extensions to prove an infinite dimensional Schur-Horn theorem.…”

Section: Introductionmentioning

confidence: 99%

Majorization and arithmetic mean ideals

Kaftal¹,

Weiss²

2011

Indiana Univ. Math. J.

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Abstract. Following An infinite dimensional Schur-Horn theorem and majorization theory [28], this paper further studies majorization for infinite sequences. It extends to the infinite case classical results on "intermediate sequences" for finite sequence majorization. These and other infinite majorization properties are then linked to notions of infinite convexity and invariance properties under various classes of substochastic matrices to characterize arithmetic mean closed operator ideals and arithmetic mean at infinity closed operator ideals.

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Rank-one decomposition of operators and construction of frames

Cited by 38 publications

References 9 publications

Projection decomposition in multiplier algebras

Projection decomposition in multiplier algebras

Operator-valued frames

Majorization and arithmetic mean ideals

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