Commutants modulo the compact operators of certain CSL algebras. II
type: Item
instance of: scholarly article
title: Commutants modulo the compact operators of certain CSL algebras. II
zbMATH Open document ID: 0532.47029
DOI: 10.1007/BF01691902
author: David R. Larson
author: Frank L. Gilfeather
published in: Integral Equations and Operator Theory
publication date: 1983-01-01T00:00:00Z
review text: Let L be a commutative lattice of self-adjoint projection operators acting on a Hilbert space, and write \(C_ L\) for the von Neumann algebra generated by L. The algebra A of operators leaving the range of each projection in L invariant is called a CSL algebra. The object of study of this paper is the essential commutant of A, ess com A, that is, those operators whose commutator with each element of A is compact. In earlier work, the authors showed that ess com A is the sum of the compact operators with a \(C^*\)-subalgebra of \(C_ L\). The present paper refines this result by providing necessary and sufficient conditions for an operator T in \(C_ L\) to belong to the essential commutant of A: writing \(E(\cdot)\) for the spectral measure of T, the criterion is that \(E(F_1)AE(F_2)\) consist entirely of compact operators whenever \(F_ 1\) and \(F_ 2\) are disjoint closed subsets of the complex plane. Several corollaries of the main result are presented. For example, although ess com A is never a von Neumann algebra, the authors show that if 0 is the only compact operator in \(C_ L\), then ess com A is generated by its projections. A different characterization of ess com A, depending on W. Arveson's representation theorem for CSL's has been obtained by \textit{K. R. Davidson} [Indiana Univ. Math. J. 32, 761-771 (1983; Zbl 0498.47003)]. The reader will find that the two papers complement each other well: although the techniques used are quite different, they share several interesting corollaries. [For part I see Topics in modern operator theory, 5th int. Conf., Timişoara and Herculane/Roumania 1980, 105- 120 (1981; Zbl 0477.47028).]
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 3844448
zbMATH Keywords: commutative lattice of self-adjoint projection operators acting on a Hilbert space
zbMATH Keywords: CSL algebra
zbMATH Keywords: essential commutant
reviewed by: Edward A. Azoff
MaRDI profile type: Publication
cites work: Operator algebras and invariant subspaces
cites work: Commutants of nest algebras modulo the compact operators
cites work: On operators commuting with Toeplitz operators modulo the compact operators
cites work: Q3963578
cites work: Q3683060
cites work: Q3935016
cites work: Operators commuting with a von Neumann algebra modulo the set of compact operators
full work available at URL: https://doi.org/10.1007/bf01691902
OpenAlex ID: W3128869815
Recommended article: Q4205968
Recommended article: Q4514945
Recommended article: Compact operators, invariant subspaces, and spectral synthesis
Recommended article: Asymptotic expansions for a model with distinguished “fast” and “slow” variables, described by a system of singularly perturbed stochastic differential equations
Recommended article: A note on lattice ordered ‐algebras and Perron–Frobenius theory
Recommended article: Q3417111
Recommended article: Commutators with the Triangular Projection and Hankel Forms on Nest Algebras
Recommended article: Q3317619
Recommended article: Topological function-theoretic proofs in spectral theory
Recommended article: Unitarily-invariant linear spaces in C*-algebras
description: scientific article; zbMATH DE number 3844448