Continuity of Homomorphisms into Certain Commutative Banach Algebras

Dales, H. G.; McClure, J. P.

doi:10.1112/plms/s3-26.1.69

Cited by 14 publications

(18 citation statements)

References 11 publications

(14 reference statements)

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“…Let We note that in this example A , and hence An , is a strongly decomposable Banach algebra of class 8 with finite dimensional radical and so has unique complete norm topology by Theorem 5-'* of [4]. In comparison to this we note the following result the proof of which is We remark in passing on the similarity between the above expression for |•| and the inequalities for quasianalytic functions on A .…”

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confidence: 78%

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Commutative Banach algebras with non-unique complete norm topology

Loy

1974

Bull. Austral. Math. Soc.

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It has recently teen shown that discontinuous functional calculi exist for certain commutative Banach algebras. Such an algebra thus possesses two distinct calculi so that there exist analytic functions whose action on the algebra is not uniquely determined.In this note a method is given for constructing commutative Banach algebras which admit two inequivalent complete norm topologies and the result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined.The purpose of this paper is to give a simple method for constructing commutative Banach algebras which have non-unique complete norm topology.One of the first examples of such an algebra was that of Feldman (see [J], an example originally constructed to show the failure of the Wedderburn Theorem. This algebra is £" @ C as a vector space with the usual product in lp and trivial multiplication by the second summand, and has a norm in which V is dense. Thus in a sense I has been completed by the adjunction of a radical and the resulting algebra has the non-uniquenessproperty. This is the idea exploited herein, and although the norm of the Feldman example is not obtainable by our approach other inequivalent norms on Z 2 © C may be constructed. Indeed this is essentially done in [I].

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confidence: 78%

“…Banach algebra with finite dimensional radical M , A is an algebra of class 8 (see [4]) and so the isomorphism Q^x is continuous by Theorem 5. 1 * o f [ 4 ] , and so D is continuous.…”

Section: If D : a •+ M Is Such A Derivation Then A @ M Is A Strongly mentioning

confidence: 95%

Commutative Banach algebras with non-unique complete norm topology

Loy

1974

Bull. Austral. Math. Soc.

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“…Since we are differentiating b^z) at z = A fewer times than the multiplicity of A as a root of a^z) = 0, {<2 0 , ...,<2 m _i} is a higher point derivation at (0, A) (see [4] for details). Now, we form a new higher point derivation at (0, A) as follows.…”

Section: The Main Theoremmentioning

confidence: 99%

“…The motivation for this paper comes primarily from a recent result of Dales and McClure (see [4;Theorem 3.1]). They proved that if B is a Banach algebra, all of whose higher point derivations are continuous, then for any Arens-Hoffman extension Aa and any homomorphism v: B -> Aa, v is necessarily continuous.…”

Section: Introductionmentioning

confidence: 99%

Homomorphisms of Banach Algebras into Arens-Hoffman Extensions of a Semi-Simple Algebra

Lindberg

1976

Journal of the London Mathematical Society

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“…The question of uniqueness of norm for algebras with finite-dimensional radical has been discussed in [11], [12], and also implicitly in [4] (see also [2]). The most thorough treatment of this problem has been, however, in [3].…”

Section: Introductionmentioning

confidence: 99%

Automatic continuity for Banach algebras with finite-dimensional radical

Pham

2006

J. Aust. Math. Soc.

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The paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.2000 Mathematics subject classification: primary 46H40.

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Continuity of Homomorphisms into Certain Commutative Banach Algebras

Cited by 14 publications

References 11 publications

Commutative Banach algebras with non-unique complete norm topology

Commutative Banach algebras with non-unique complete norm topology

Homomorphisms of Banach Algebras into Arens-Hoffman Extensions of a Semi-Simple Algebra

Automatic continuity for Banach algebras with finite-dimensional radical

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