Miscellaneous on Commutants mod Normed Ideals and Quasicentral Modulus I

TL;DR

This paper explores the structure of commutants mod normed ideals linked to compact manifolds with boundary, including K-theory results and properties of quasicentral modulus for specific operator classes.

Contribution

It introduces new definitions of commutants mod normed ideals for manifolds with boundary and derives an exact sequence for their K-theory, expanding understanding of operator algebra structures.

Findings

Derived an exact sequence for K-theory of commutants mod normed ideals.

Made remarks on bicommutants mod normed ideals.

Analyzed quasicentral modulus for p-Schatten classes with 0 < p < 1.

Abstract

We define commutants mod normed ideals associated with compact smooth manifolds with boundary. The results about the K-theory of these operator algebras include an exact sequence for the connected sum of manifolds, derived from the Mayer-Vietoris sequence. We also make a few remarks about bicommutants mod normed ideals and about the quasicentral modulus for the quasinormed p-Schatten-von Neumann classes 0 < p < 1.