Wyniki wyszukiwania

1

Real-linear isometries between certain subspaces of continuous functions

100%

Jamshidi A., Sady F.

Open Mathematics

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2013

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tom 11

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nr 11

2034-2043

EN

In this paper we first consider a real-linear isometry T from a certain subspace A of C(X) (endowed with supremum norm) into C(Y) where X and Y are compact Hausdorff spaces and give a result concerning the description of T whenever A is a uniform algebra on X. The result is improved for the case where T(A) is, in addition, a complex subspace of C(Y). We also give a similar description for the case where A is a function space on X and the range of T is a real subspace of C(Y) satisfying a ceratin separating property. Next similar results are obtained for real-linear isometries between spaces of Lipschitz functions on compact metric spaces endowed with a certain complete norm.

2

Maps between Banach function algebras satisfying certain norm conditions

100%

Hosseini M., Sady F.

Open Mathematics

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2013

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tom 11

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nr 6

1020-1033

EN

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\bar A$ and $\bar B$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and a homeomorphism ϕ: M B → M A between the maximal ideal spaces of B and A such that for all f ∈ A, [...] where $\hat \cdot$ denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from $\bar A$ onto $\bar B$ inducing a homeomorphism between $M_{\bar B}$ and $M_{\bar A}$ . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.

3

Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

86%

Averill K., Johnston A., Northrup R., Silversmith R., Luttman A.

Open Mathematics

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2012

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tom 10

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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.

Ograniczanie wyników

3 Open Mathematics

2 Sady F.

1 Averill K.

1 Hosseini M.

1 Jamshidi A.

1 Johnston A.

1 Luttman A.

1 Northrup R.

1 Silversmith R.

2 2013

1 2012

Real-linear isometries between certain subspaces of continuous functions

Maps between Banach function algebras satisfying certain norm conditions

Boundaries of weak peak points in noncommutative algebras of Lipschitz functions