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Zbl 1181.47036
Hou, Jinchuan; Huang, Li
(Hou, Jin-chuan)
Characterizing isomorphisms in terms of completely preserving invertibility or spectrum.
(English)
[J] J. Math. Anal. Appl. 359, No. 1, 81-87 (2009). ISSN 0022-247X

Let $A$ and $B$ be algebras. Given a map $\Phi:A\to B$, we define $\Phi_n:A\otimes M_n\to B\otimes M_n$ by $\Phi_n((a_{ij})) = (\Phi(a_{ij}))$. Given a property {\bf (P)}, the authors call $\Phi$ completely {\bf (P)} preserving if each $\Phi_n$ preserves {\bf (P)}. The main result characterizes surjective maps between standard operator algebras on infinite-dimensional Banach spaces that preserve invertibility in both directions completely. The result is standard: $\Phi(A) = TAS$ for some bounded linear or conjugate-linear operators $S$ and $T$ acting on appropriate spaces. A result on maps that are completely spectrum preserving follows easily.
[Matej Brešar (Maribor)]
MSC 2000:
*47B48 Operators on Banach algebras
46H05 General theory of topological algebras

Keywords: standard operator algebra; isomorphism; invertibility; spectrum; preserver problem

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