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

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 $\delta$.