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2010 | 91 | 1 | 411-421
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Weak multiplicative operators on function algebras without units

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For a function algebra A let ∂A be the Shilov boundary, δA the Choquet boundary, p(A) the set of p-points, and |A| = {|f|: f ∈ A}. Let X and Y be locally compact Hausdorff spaces and A ⊂ C(X) and B ⊂ C(Y) be dense subalgebras of function algebras without units, such that X = ∂A, Y = ∂B and p(A) = δA, p(B) = δB. We show that if Φ: |A| → |B| is an increasing bijection which is sup-norm-multiplicative, i.e. ||Φ(|f|)Φ(|g|)|| = ||fg||, f,g ∈ A, then there is a homeomorphism ψ: p(B) → p(A) with respect to which Φ is a ψ-composition operator on p(B), i.e. (Φ(|f|))(y) = |f(ψ(y))|, f ∈ A, y ∈ p(B). We show also that if A ⊂ C(X) and B ⊂ C(Y) are dense subalgebras of function algebras without units, such that X = ∂A, Y = ∂B and p(A) = δA, p(B) = δB, and T: A → B is a sup-norm-multiplicative surjection, namely, ||Tf Tg|| = ||fg||, f,g ∈ A, then T is a ψ-composition operator in modulus on p(B) for a homeomorphism ψ: p(B)→ p(A), i.e. |(Tf)(y)| = |f(ψ(y))|, f ∈ A, y ∈ p(B). In particular, T is multiplicative in modulus on p(B), i.e. |T(fg)| = |Tf Tg|, f,g ∈ A. We prove also that if A ⊂ C(X) is a dense subalgebra of a function algebra without unit, such that X = ∂A and p(A) = δA, and if T: A → B is a weakly peripherally-multiplicative surjection onto a function algebra B without unit, i.e. $σ_π(Tf Tg) ∩ σ_π(fg) ≠ ∅$, f,g ∈ A, and preserves the peripheral spectra of algebra elements, i.e. $σ_π(Tf) = σ_π(f)$, f ∈ A, then T is a bijective ψ-composition operator on p(B), i.e. (Tf)(y) = f(ψ(y)), f ∈ A, y ∈ p(B), for a homeomorphism ψ: p(B) → p(A). In this case A is necessarily a function algebra and T is an algebra isomorphism. As a consequence, a multiplicative operator T from a dense subalgebra A ⊂ C(X) of a function algebra B without unit, such that X = ∂A and p(A) = δA, onto a function algebra without unit B is a sup-norm isometric algebra isomorphism if and only if T is weakly peripherally-multiplicative and preserves the peripheral spectra of algebra elements. The results extend to function algebras without units a series of previous results for algebra isomorphisms.
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  • The University of Montana, Missoula, Missoula, MT 59812, USA
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bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-24
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