Factorization in finite-codimensional ideals of group algebras

Willis, George A.

doi:10.1112/plms/82.3.676

Cited by 7 publications

(6 citation statements)

References 26 publications

(55 reference statements)

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“…J µ is evidently a left ideal in the group algebra L 1 (G), whose annihilator in L ∞ (G) is precisely the space H µ of the bounded µ-harmonic functions. As pointed out by Willis [42,43], ideals of this form appear naturally not only in the theory of µ-harmonic functions but also in the study of amenability and certain factorization questions in group algebras. The quotient L 1 (G)/J µ turns out to be an abstract L 1 -space whose pointwise realization is the boundary needed to represent the µ-harmonic functions by means of a Poisson formula.…”

Section: The Preannihilator Of H µπmentioning

confidence: 99%

The Choquet–Deny Equation in a Banach Space

Jaworski

Neufang

2007

Can. j. math.

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Abstract. Let G be a locally compact group and π a representation of G by weakly * continuous isometries acting in a dual Banach space E. Given a probability measure µ on G, we study the Choquet-Deny equation π(µ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a "Poisson formula" on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law µ. The relation between the space of solutions of the Choquet-Deny equation in E and the space of bounded harmonic functions can be understood in terms of a construction resembling the W * -crossed product and coinciding precisely with the crossed product in the special case of the Choquet-Deny equation in the space E = B(L 2 (G)) of bounded linear operators on L 2 (G). Other general properties of the Choquet-Deny equation in a Banach space are also discussed.

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Section: The Preannihilator Of H µπmentioning

confidence: 99%

The Choquet–Deny Equation in a Banach Space

Jaworski

Neufang

2007

Can. j. math.

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“…In the case where G is amenable, the ideal I has a BAI, and so factors. It is proved in [37] that I always factors weakly, even when it does not have a BAI. For example, let F 2 be the free group on two generators.…”

Section: Trivially (Iv)⇒(v)mentioning

confidence: 99%

Factorization in commutative Banach algebras

2021

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We consider when a (non-unital) commutative Banach algebra has a variety of factorization properties; we list the (obvious) implications between these properties, and then investigate whether any of these implications can be reversed in various classes of commutative Banach algebras. We summarize the known counter-examples to these possible reverse implications, and add further counter-examples. Some results are used to show the existence of a large family of prime ideals in each non-zero, commutative, radical Banach algebra with a dense set of products.

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“…These ideals, which have also been studied by G. Willis in a Banach algebra context (see e.g. [41]), for us correspond to the noncommutative variant of peak sets from the theory of function algebras (see [9,Proposition 6.7] for the correspondence). Here and in the following we write 1 for the identity in A 1 if A is a nonunital algebra.…”

Section: Existence Of R-idealsmentioning

confidence: 99%

Ideals and structure of operator algebras

Almus,

Blecher,

Sharma

2009

Preprint

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Factorization in finite-codimensional ideals of group algebras

Cited by 7 publications

References 26 publications

The Choquet–Deny Equation in a Banach Space

The Choquet–Deny Equation in a Banach Space

Factorization in commutative Banach algebras

Ideals and structure of operator algebras

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