Gary Weiss

Commutator structure of operator ideals

Commutator structure of operator ideals Ken Dykema, ,1 Tadeusz Figiel, Gary Weiss, and Mariusz Wodzicki Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA b Instytut Matematyczny Polskiej Akademii Nauk, 81-825 Sopot, Poland Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

Quasitriangular subalgebras of semifinite von Neumann algebras are closed

Arveson has shown in [ 1 ] that if A is a nest algebra of operators acting on a separable Hilbert space H with nest of order type the extended natural numbers and if K(H) is the ideal of compact operators on H, then the quasitriangular algebra A + K(H) is norm closed. In [6] Fall, Arveson, and Muhly extended Arveson’s result proving that in B(H), general quasitriangular algebras (i.e., compact perturbations of nest algebras) are always norm closed. A key step in their proof, and a result of independent interest, first obtained by Erdos in [S], is the a-weak density of A n K(H) in A. From this fact they deduced that quasitriangular algebras were norm closed by using an argument depending on the duality K(H)** = B(H). Every semifinite von Neumann algebra M has an ideal of compact operators that behaves like K(H), namely the norm-closed two-sided ideal K generated by the finite projections of the algebra. It is a natural question,

Matrix norm inequalities and the relative Dixmier property

AbstractIf x is a selfadjoint matrix with zero diagonal and non-negative entries, then there exists a decomposition of the identity into k diagonal orthogonal projections {pm} for which

Traces, ideals, and arithmetic means

\parallel \sum p_m xp_m \parallel \leqslant (1/k)\parallel x\parallel

Majorization and a Schur-Horn Theorem for positive compact operators, the nonzero kernel case

From this follows that all bounded matrices with non-negative entries satisfy the relative Dixmier property or, equivalently, the Kadison Singer extension property. This inequality fails for large Hadamard matrices. However a similar inequality holds for all matrices with respect to the Hilbert-Schmidt norm with constant k−1/2 and for Hadamard matrices with respect to the Schatten 4-norm with constant 21/4k−1/2.

B(H)-Commutators: A Historical Survey

ABsTRAcr. Let H denote a separable, infinite-dimensional complex Hilbert space. A two-sided ideal I of operators on H possesses the generalized Fuglede property (GFP) if, for every normal operator N and every X E L(H), NX XN E I implies N*X XN* E I. Fugledes Theorem says that I = {O} has the GFP. It is known that the class of compact operators and the class of Hilbert-Schmidt operators have the GFP. We prove that the class of finite rank operators. and the Schatten p-classes for O < p < 1 fail to have the GFP. The operator we use as an example in the case of the Schatten p-classes is multiplication by z + w on L2 of the torms.

Diagonality and idempotents with applications to problems in operator theory and frame theory

This article grew out of recent work of Dykema, Figiel, Weiss, and Wodzicki (Commutator structure of operator ideals) which inter alia characterizes commutator ideals in terms of arithmetic means. In this paper we study ideals that are arithmetically mean (am) stable, am-closed, am-open, soft-edged and soft-complemented. We show that many of the ideals in the literature possess such properties. We apply these notions to prove that for all the ideals considered, the linear codimension of their commutator space (the “number of traces on the ideal”) is either 0, 1, or ∞. We identify the largest ideal which supports a unique nonsingular trace as the intersection of certain Lorentz ideals. An application to elementary operators is given. We study properties of arithmetic mean operations on ideals, e.g., we prove that the am-closure of a sum of ideals is the sum of their am-closures. We obtain cancellation properties for arithmetic means: for principal ideals, a necessary and sufficient condition for first order cancellations is the regularity of the generator; for second order cancellations, sufficient conditions are that the generator satisfies the exponential Δ2-condition or is regular. We construct an example where second order cancellation fails, thus settling an open question. We also consider cancellation properties for inclusions. And we find and use lattice properties of ideals associated with the existence of “gaps.”

Unitarily invariant trace extensions beyond the trace class

Let L(H) be the ring of bounded operators on a separable Hilbert space. Assuming the continuum hypothesis, we prove that in L(H) every two-sided ideal that contains an operator of infinite rank is the sum of two smaller two-sided ideals. The proof involves a new combinatorial description of ideals of L(H). This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice. 1. Introduction and notational conventions. Let H be a separable, infinite-di- mensional, complex Hilbert space, and let L(H) be the ring of bounded linear operators on H. Assuming the continuum hypothesis, we shall prove that every two-sided ideal I of L(H) that properly includes the ideal of finite-rank operators can be decomposed as the sum I = J, + J2 of two-sided ideals Ji properly included in I. The question whether such a decomposition exists for the ideal K(H) of compact operators was raised by Brown, Pearcy and Salinas (1). We shall also show that the decomposition of K(H) is necessarily nonconstructive, in the sense that there need not be any definable ideals c K(H) whose sum is K(H). Our proof is in two parts. The first part develops a new characterization of ideals in terms of sequences of natural numbers and thereby reduces the problem to a combinatorial one. The second part uses the continuum hypothesis to solve this combinatorial problem. The two parts are presented in ??2 and 3, respectively. ?4 contains some additional comments. Before starting the proof, we adopt some notational conventions. By the sum of two sequences of real numbers, we mean the sequence obtained by adding corresponding terms; similarly, one sequence is < another if every