37 research outputs found
Majorization and arithmetic mean ideals
Following "An infinite dimensional Schur-Horn theorem and majorization
theory", Journal of Functional Analysis 259 (2010) 3115-3162, 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.Comment: To appear in Indiana University Mathematics Journa
Projection decomposition in multiplier algebras
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
Traces, ideals, and arithmetic means
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 Δ(2)-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.