On Functional Representation of Commutative Locally $A$-Convex Algebras

Arhippainen, Jorma

doi:10.1216/rmjm/1021477242

Cited by 6 publications

(8 citation statements)

References 24 publications

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“…See for example [3], [4], [9] or [10]. Correspondingly function algebras with the weighted topology have been considered in [1], [5], [12]- [15] or [18].…”

Section: Introductionmentioning

confidence: 99%

Generalization of the B *‐algebra (C₀(X),∥ ∥_∞)

Arhippainen

Kauppi

2008

Mathematische Nachrichten

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“…See for example [3], [4], [9] or [10]. Correspondingly function algebras with the weighted topology have been considered in [1], [5], [12]- [15] or [18].…”

Section: Introductionmentioning

confidence: 99%

Generalization of the B *‐algebra (C₀(X),∥ ∥_∞)

Arhippainen

Kauppi

2008

Mathematische Nachrichten

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“…For the use of these types of weights see [2,5,9]. Now (Â, · ˆ) is an A-convex algebra and the Gelfand mapping x →x is a continuous algebra homomorphism from (A, · ) onto (Â, · ˆ).…”

Section: From (4) It Follows That V · (τ )|X(τ )| X For All X ∈ a Andmentioning

confidence: 99%

“…Also if the norm · is irregular this standard representation is not preferable (since we do not know whether the weight function v · is bounded or not) and the use of weighted supremum-norm can give better description. For further information about functional representation of different kind of A-convex algebras see [1][2][3]5,9]. Then · 0 is an A-convex norm on A with m( · 0 ) = ∞ (this implies that · 0 is not complete) and r( · 0 ) = ∞ and · 1 is an m-convex complete norm on A with m( · 1 ) = 1 and r( · 1 ) = ∞.…”

Section: Lemma 1 Suppose That a Has A Unit And M( · ) Is Finite And mentioning

confidence: 99%

“…Instead of supremum norm we will providê A with some weighted supremum norm and show that there are cases in which these weights are not necessarily bounded and that this weighted representation is more closely connected to the topological structure of the algebra (A, · ). We follow here the ideas which were first represented in [5] and later extended and sharpened in [1,2,9].…”

Section: Introductionmentioning

confidence: 98%

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On functional representation of normed algebras

Arhippainen

2007

Journal of Mathematical Analysis and Applications

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Let A be a commutative algebra over complex numbers with a space norm · making the multiplication on A separately continuous. We will study the Gelfand representation of this type of normed algebra. In particular, we look at the cases where the standard Gelfand representation (i.e., the use of supremum-norm on the Gelfand transform algebraÂ) gives different properties from the original algebra (A, · ). We show that there are even Banach algebras for which this type of difficulty may happen. We will provideÂ with some weighted supremum-norm and by using these weights we can avoid the difficulties mentioned above. For the definition of these weights we adopt the ideas of Cochran represented in [A.C. Cochran, Representation of A-convex algebras, Proc. Amer. Math. Soc. 30 (1973) 473-479].

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“…We say that T (P) is weakly regular if for each λ ∈ Λ there is a constant m λ > 0 such that p λ (x) m λ p λ (x) for all x ∈ A. Note that if A has a unit element (denoted by e), then (A, T (P)) is weakly regular (we can take m λ = p λ (e) for each λ ∈ Λ, see [4]). …”

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confidence: 99%