Polynomials of almost-normal arguments in $C^*$-algebras

TL;DR

This paper investigates how polynomial functions of elements with nearly normal properties in $C^*$-algebras behave, showing that many features of the functional calculus are preserved approximately when the self-commutator is small.

Contribution

It demonstrates that for elements with small self-commutator norm, polynomial calculus retains key properties approximately, extending the functional calculus to almost-normal elements.

Findings

Properties of the functional calculus are approximately preserved for elements with small self-commutator.

Error in properties is proportional to the self-commutator norm $ orm{[a,a^*]}$.

Results provide a quantitative understanding of near-normal elements in $C^*$-algebras.

Abstract

The functional calculus for normal elements in $C^*$-algebras is an important tool of analysis. We consider polynomials $p(a,a^*)$ for elements $a$ with small self-commutator norm $\|[a,a^*]\| \le \delta$ and show that many properties of the functional calculus are retained modulo an error of order $\delta$.