Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Weak multiplicative operators on function algebras without units

100%

Tonev T.

Banach Center Publications

2010

tom 91

nr 1

411-421

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.

Algebra isomorphisms between standard operator algebras

63%

Tonev T., Luttman A.

Studia Mathematica

2009

tom 191

nr 2

163-170

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.

Ograniczanie wyników

1 Banach Center Publications

1 Studia Mathematica

2 Tonev T.

1 Luttman A.

1 2010

1 2009

Wyniki wyszukiwania

Weak multiplicative operators on function algebras without units

Algebra isomorphisms between standard operator algebras