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Algebra isomorphisms between standard operator algebras

100%
Tonev T., Luttman A.
Studia Mathematica
|
2009
|
tom 191
|
nr 2
163-170
EN
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.
2
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Norm conditions for uniform algebra isomorphisms

100%
Luttman A., Lambert S.
Open Mathematics
|
2008
|
tom 6
|
nr 2
272-280
EN
In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / {0} and T: A → B is a surjective map, not assumed to be linear, satisfying $$ \left\| {T(f)T(g) + \lambda } \right\| = \left\| {fg + \lambda } \right\|\forall f,g \in A, $$ then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism $$ \tilde T:{\rm A} \to Be \oplus \bar B(1 - e) $$ such that $$ T(f) = \kappa \left( {\tilde T(f)e + \gamma \overline {\tilde T(f)} (1 - e)} \right) $$ for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.
3
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Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

81%
Jiménez-Vargas A., Lee K., Luttman A., Villegas-Vallecillos M.
Open Mathematics
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2013
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tom 11
|
nr 7
1197-1211
EN
Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $\mathbb{K}$-valued Lipschitz functions on X - where $\mathbb{K}$ is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = {f(x): |f(x)| = ‖f‖∞} of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb{K}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.
4
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Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

71%
Averill K., Johnston A., Northrup R., Silversmith R., Luttman A.
Open Mathematics
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2012
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tom 10
|
nr 2
646-655
EN
It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, $\mathbb{F}$ ) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where $\mathbb{F}$ = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, $\mathbb{F}$ ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.
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