Lifting Normal Elements in Nonseparable Calkin Algebras

TL;DR

This paper proves that in nonseparable Hilbert spaces, normal elements in certain quotient algebras can always be lifted to normal elements in the original algebra, expanding understanding of operator algebra structures.

Contribution

It introduces a novel lifting result for normal elements in nonseparable Calkin algebras using advanced distance estimates.

Findings

Normal elements in B(H)/K can be lifted to B(H) for nonseparable H.

The result applies to any closed ideal not equal to the compact operators.

Utilizes the distance estimate of Kachkovskiy and Safarov.

Abstract

We use the remarkable distance estimate of Ilya Kachkovskiy and Yuri Safarov, to show that if $H$ is a nonseparable Hilbert space and $K$ is any closed ideal in $B(H)$ that is not the ideal of compact operators, then any normal element of $B(H)/K$ can be lifted to a normal element of $B(H)$.