A Generalized Spectral Radius Formula and Olsen's Question

Abstract

Let $A$ be a $C^{\ast }$-algebra and $I$ be a closed ideal in $A$. For $x\in A$, its image under the canonical surjection $A\to A/I$ is denoted by $\dot{x}$, and the spectral radius of $x$ is denoted by $r(x)$. We prove that $$\max \{r(x),\Vert \dot{x}\Vert \}=\inf \Vert (1+i{)}^{-1}x(1+i)\Vert$$ (where infimum is taken over all $i\in I$ such that $1+i$ is invertible), which generalizes spectral radius formula of Murphy and West \cite{MurphyWest} (Rota for $\mathcal{B}(\mathcal{H})$ \cite{Rota}). Moreover if $r(x)<\Vert \dot{x}\Vert$ then the infimum is attained. A similar result is proved for commuting family of elements of a $C^{\ast }$-algebra. Using this we give a partial answer to an open question of C. Olsen: if $p$ is a polynomial then for "almost every" operator $T\in B(H)$ there is a compact perturbation $T+K$ of $T$ such that $$\Vert p(T+K)\Vert =\Vert p(T){\Vert }_e.$$ We show also that if operators $A,B$ commute, $A$ is similar to a contraction and $B$ is similar to a strict contraction then they are simultaneously similar to contractions.

In new version we added the case of arbitrary many polynomials and added some new cases of operators for which Olsen's question has a positive answer. New version also contains a result concerning commuting operators similar to contractions. We removed a section on semiprojective C*-alegbras, it will be written somewhere else

References in corpus (1)

• Semiprojectivity of universal C*-algebras generated by algebraic elements

Cited by in corpus (1)

• Noncommutative Semialgebraic Sets in Nilpotent Variables