Commutants modulo the compact operators of certain CSL algebras. II
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).]
commutative lattice of self-adjoint projection operators acting on a Hilbert space
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