Semiprojectivity of universal C*-algebras generated by algebraic elements

TL;DR

This paper proves that certain universal C*-algebras generated by algebraic elements with specific root multiplicity conditions are semiprojective, with additional residual finite-dimensionality in the multiple root case, impacting polynomially compact operators.

Contribution

It establishes semiprojectivity of universal C*-algebras generated by algebraic elements with roots of uniform multiplicity and shows residual finite-dimensionality for multiple roots.

Findings

Universal C*-algebras with roots all multiple are residually finite-dimensional.

Universal C*-algebras with roots all simple are semiprojective.

Applications to polynomially compact operators are provided.

Abstract

Let $p$ be a polynomial in one variable whose roots either all have multiplicity more than 1 or all have multiplicity exactly 1. It is shown that the universal $C^*$-algebra of a relation $p(x)=0$, $\|x\| \le 1$ is semiprojective. In the case of all roots multiple it is shown that the universal $C^*$-algebra is also residually finite-dimensional. Applications to polynomially compact operators are given.