Almost normal operators mod Hilbert-Schmidt and the K-theory of the Banach algebras $E\Lambda(\Omega)$

TL;DR

This paper explores the K-theory of specific Banach algebras related to Lipschitz and Dirichlet functions, investigating a potential analogue of the BDF-theorem involving the Pincus g-function.

Contribution

It establishes a connection between the K-theory of these Banach algebras and a mod Hilbert-Schmidt version of the BDF-theorem, highlighting their duality and corona algebra properties.

Findings

Partially answers the mod Hilbert-Schmidt analogue of the BDF-theorem.

Shows that these Banach algebras have duality properties and corona algebras that are C*-algebras.

Links the Pincus g-function to the K-theory framework in this context.

Abstract

Is there a mod Hilbert-Schmidt analogue of the BDF-theorem, with the Pincus g-function playing the role of the index ? We show that part of the question is about the K-theory of certain Banach algebras. These Banach algebras, related to Lipschitz functions and dirichlet algebras have nice Banach space duality properties. Moreover their corona algebras are C*-algebras.