Algebra isomorphisms between standard operator algebras

Thomas Tonev ; Aaron Luttman

Studia Mathematica, Tome 192 (2009), p. 163-170 / Harvested from The Polish Digital Mathematics Library

Résumé

If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set $σ_{π}(A) = {λ ∈ σ(A): |λ| = max_{z∈σ(A)} |z|}$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation $σ_{π}(φ(A)∘φ(B)) = σ_{π}(AB)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be linear nor continuous, is a bijective bounded linear operator such that either φ or -φ is multiplicative or anti-multiplicative. This holds in particular for the algebras of finite rank operators or of compact operators on X and Y and extends earlier results of Molnár. If, in addition, $σ_{π}(φ(A₀)) ≠ -σ_{π}(A₀)$ for some A₀ ∈ then φ is either multiplicative, in which case X is isomorphic to Y, or anti-multiplicative, in which case X is isomorphic to Y*. Therefore, if X ≇ Y* then φ is multiplicative, hence an algebra isomorphism, while if X ≇ Y, then φ is anti-multiplicative, hence an algebra anti-isomorphism.

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Publié le : 2009-01-01

Zbl 1179.47035

EUDML-ID : urn:eudml:doc:286573