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Article overview
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Generalized spectral radius formula and Olsen's question | | Terry Loring
; Tatiana Shulman
; | | Date: |
27 Jul 2010 | | Abstract: | Let $A$ be a $C^*$-algebra and $I$ be a closed ideal in $A$. For $xin A$,
its image under the canonical surjection $A o A/I$ is denoted by $dot x$, and
spectral radius of $x$ is denoted by $r(x)$. It is proved that $$max{r(x),
|dot x|} = inf |(1+i)^{-1}x(1+i)|$$ (where infimum is taken over all
$iin I$ such that $1+i$ is invertible), which generalizes spectral radius
formula of Murphy and West cite{MurphyWest} (Rota for $mathcal{B(H)}$
cite{Rota}).
Using this formula we give a partial answer to an open question of C. Olsen:
if $p$ is a polynomial then for an operator $T$ from an open dense subset of
$B(H)$ we show there is a compact perturbation $T+K$ of $T$ such that
$$|p(T+K)| = |p(T)|_e.$$ We also apply this formula to find new examples of
semiprojective $C^*$-algebras. | | Source: | arXiv, 1007.4655 | | Services: | Forum | Review | PDF | Favorites | |
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