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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 Φ:AB, we define Φ n :AM n BM n by Φ n ((a ij ))=(Φ(a ij )). Given a property (P), the authors call Φ completely (P) preserving if each Φ n preserves (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: Φ(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