Standard homomorphisms and regulated weights on weighted convolution algebras

Ghahramani, F.; McClure, J. P.; Grabiner, Sandy

doi:10.1016/0022-1236(90)90145-b

Cited by 10 publications

(28 citation statements)

References 17 publications

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“…In our previous studies of the standard homomorphism problem [8] and [7], for p = 1, we were able to prove that the weak*-continuous semigroup {μ t } was strongly continuous by coming up with a condition on the weight ω(t) which guaranteed that whenever a sequence {λ n } converged weak* to λ in M(α ), then λ n * / -> λ * / in norm in L ι {ω) for appropriate /. It also turned out [7,Theorem (4.1)] that weak convergence of λ n * / in L ι (ω) implied norm convergence.…”

Section: Compactness and Norm Convergencementioning

confidence: 99%

“…In this paper we show that the IP analogue of a number of questions we have studied ( [10], [8], [11], [7]) involving homomorphisms and semigroups on weighted L 1 spaces on R + = [0, oo) all have positive answers when 1 < p < oo. If ω(t) > 0 is a Borel function on R + which is locally bounded and locally bounded away from 0 and if 1 < p < oo, we let IP(ω) be the Banach space of (equivalence classes of) measurable functions on R + with fω in with the inherited norm ii/ii = II/IUP = \\fω\\ P = (jΓ ι/(tMt)r dt) ) ι l P We are particularly interested in the case that L ι (ω) is a Banach algebra and all IP(ώ) are L 1 (α )-modules under the usual convolution multiplication / * g(x) = / o x f(x -t)g(t) dt.…”

mentioning

confidence: 99%

“…(a) The semigroup a special class of weights [8,Theorem (3.4), p. 284], the regulated weights of Bade and Dales [1]. In §2, we will compare various types of convergence in LP(ω) and show that a class of semigroups, which will include all μ t = φ(δt), are strongly continuous when p > 1.…”

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confidence: 99%

“…§3 will be devoted to proving that all φ are standard for all p > 1. In §4, we observe that, for sequences which do not come from semigroups μ t = φ(δt), the best convergence results require that the weights be regulated, just as when p = 1 [7].…”

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confidence: 99%

“…Types of convergence. Many of the parts of the definition of standard homomorphisms involve comparing various convergence properties for a bounded sequence or net {λ n } in M(α ), just as in the case p = 1 (see [10], [7]). For 1 < p < oo, the dual space of i/(α;) is L q {l/ω), where q is the conjugate exponent to p, under the usual duality (…”

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confidence: 99%

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TheL^ptheory of standard homomorphisms

Ghahramani¹,

Grabiner²

1995

Pacific J. Math.

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Section: Compactness and Norm Convergencementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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TheL^ptheory of standard homomorphisms

Ghahramani¹,

Grabiner²

1995

Pacific J. Math.

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Equivalent weights and standard homomorphisms for convolution algebras on ℝ⁺

Grabiner

2009

Proceedings of the Edinburgh Mathematical Society

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We take a second look at two basic topics in the study of weighted convolution algebras L1(ω) on ℝ+. An early result showed that one could replace the weight $\omega$ with a very well-behaved weight without changing the space L1(ω) as long as L1(ω) was an algebraffi We prove the analogous result for measure algebras when M(ω) is an algebraffi This allows us to preserve not only the norm topology but also the relative weak* topology on L1(ω). A homomorphism between weighted convolution algebras is said to be standard if it preserves generators of dense principal ideals. The original proofs of standardness and its variants are all based on finding the generator of a particular strongly continuous convolution semigroup. In this paper we give much simpler direct proofs of these results. We also improve the statement and proof of the theorem, giving useful properties equivalent to the standardness of a homomorphism.

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Homomorphisms of the algebra of locally integrable functions on the half line

Grabiner

2006

J. Aust. Math. Soc.

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Let 0 be a continuous nonzero homomorphism of the convolution algebra L 1 ' OC (IR + ) and also the unique extension of this homomorphism to A/| OC (1R + ). We show that the map is continuous in the weak* and strong operator topologies on M^, considered as the dual space of C C (R + ) and as the multiplier algebra of L\ x . Analogous results are proved for homomorphisms from(w\), i) to some L'(a>2), and, for each sufficiently large L 1 (coj),

i) to L x (,a>i). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L\ x . We also prove results on convergent nets, continuous semigroups, and bounded sets in M loc that we need in our study of homomorphisms.2000 Mathematics subject classification: primary 43A22, 46J40, 46J45, 43A20,43A10.

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Standard homomorphisms and regulated weights on weighted convolution algebras

Cited by 10 publications

References 17 publications

TheL^ptheory of standard homomorphisms

TheL^ptheory of standard homomorphisms

Equivalent weights and standard homomorphisms for convolution algebras on ℝ⁺

Homomorphisms of the algebra of locally integrable functions on the half line

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Standard homomorphisms and regulated weights on weighted convolution algebras

Cited by 10 publications

References 17 publications

TheLptheory of standard homomorphisms

TheLptheory of standard homomorphisms

Equivalent weights and standard homomorphisms for convolution algebras on ℝ+

Homomorphisms of the algebra of locally integrable functions on the half line

Contact Info

Product

Resources

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TheL^ptheory of standard homomorphisms

TheL^ptheory of standard homomorphisms

Equivalent weights and standard homomorphisms for convolution algebras on ℝ⁺