Some examples of lifting problems from quotient algebras

TL;DR

This paper investigates lifting problems for unitaries and extremal partial isometries from quotient C*-algebras to their parent algebras, providing examples and counterexamples to these questions.

Contribution

It introduces specific constructions of ideals in C*-algebras that serve as examples or counterexamples for lifting unitaries and extremal partial isometries.

Findings

Certain ideals allow lifting of unitaries from quotient to algebra

Some ideals prevent lifting of extremal partial isometries

Counterexamples demonstrate limitations of lifting properties in C*-algebras

Abstract

We consider three lifting questions: Given a $C\sp{*}$-algebra $I$, if there is a unital $C\sp{*}$-algebra $A$ contains $I$ as an ideal, is every unitary from $A/I$ lifted to a unitary in $A$? is every unitary from $A/I$ lifted to an extremal partial isometry? is every extremal partial isometry from $A/I$ lifted to an extremal partial isometry? We show several constructions of $I$ which serve as working examples or counter-examples for above questions.