Some Results and a K-theory Problem about Threshold Commutants mod Normed Ideals

TL;DR

This paper extends previous results on the structure of commutants mod ideals for Hermitian operators, computes K-theory groups for certain operator algebras, and poses open problems in this area.

Contribution

It generalizes the essential centre result to threshold ideals and computes the K_0-group for the commutant mod trace-class, introducing new K-theoretic problems.

Findings

Extended the essential centre result to threshold ideals.

Computed the K_0-group of the commutant mod trace-class.

Posed an open problem for the K_1-group in a simple case.

Abstract

We extend to the case of a threshold ideal our result with J. Bourgain about the essential centre of the commutant mod a diagonalization ideal for a n-tuple of commuting Hermitian operators . We also compute the $K_0$-group of the commutant mod trace-class of a unitary operator with spectrum equal to its essential spectrum. We present the problem of computing the $K_1$-group for a commutant mod trace-class in its simplest case.