Polynomials of almost normal arguments in $C^{\ast }$-algebras

Abstract

The functional calculus for normal elements in $C^{\ast }$-algebras is an important tool of analysis. We suggest an approximate substitute for such calculus for elements $a$ with the small self-commutator norm $\Vert [a,a^{\ast }]\Vert \le \delta$. We show that many properties of the functional calculus are conserved up to order $\delta$.