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2009 | 191 | 2 | 163-170
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Algebra isomorphisms between standard operator algebras

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If X and Y are Banach spaces, then subalgebras 𝔄 ⊂ B(X) and 𝔅 ⊂ 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 ∈ 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: 𝔄 → 𝔅 is called peripherally-multiplicative if it satisfies the equation $σ_{π}(φ(A)∘φ(B)) = σ_{π}(AB)$ for all A,B ∈ 𝔄. We show that any peripherally-multiplicative and surjective map φ: 𝔄 → 𝔅, 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 ∈ 𝔄 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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  • Department of Mathematical Sciences, The University of Montana, Missoula, MT 59812-1032, U.S.A.
  • Division of Mathematics and Computer Science, Box 5815, Clarkson University, Potsdam, NY 13699, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-2-4
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