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Characterizing isomorphisms in terms of completely preserving invertibility or spectrum. (English) Zbl 1181.47036
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.

47B48Operators on Banach algebras
46H05General theory of topological algebras
Full Text: DOI
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