We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of is Arens regular, and give some evidence that this is so if and only if is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebraffi We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of is amenable. We provide an abstract characterization in terms of something like an approximate diagonal, and consider when every ultrapower of a C*-algebra, or a group L1-convolution algebra, is amenable.

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.

We prove that in any compact symmetric space, G/K, there is a dense set of a1,a2∈G such that if μj=mK*δaj*mk is the K-bi-invariant measure supported on KajK, then μ1*μ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1Ka2K, has nonempty interior in G.

Let G be LCA group with an algebraically ordered dual Ĝ. Suppose also that the semigroup P of positive elements in Ĝ is not dense in Ĝ. Subspaces (G) (1 < s < ∞) are defined analogous to the Hardy spaces on the circle group, and the question whether every multiplier from into can be extended to a multiplier from L3(G) into Lq(G) is investigated. If we suppose that s ≠ ∞, the it is shown that such an extension is possible if and only if (s, q) ∈ (1, ∞) × [1, ∞] ∪ {(1, ∞)}. (the negative result for (1, 1) was obtained in a previous paper.)

It is well known that a complex-valued function ø, analytic on some open set Ω, extends to any commutative Banach algebra B so that the action of ø on B commutes with the action of the Gelfand transformation. In this paper, it is shown that if B is a homogeneous convolution Banach algebra over any compact group and if 0 ∈ Ω is a fixed point of ø, then a similar result holds, with the Gelfand transformation replaced by the Fourier-Stieltjes transformation. Care is required, in that discussion of this relation usually requires simultaneous consideration of the extension of ø to B and to certain operator algebras.

A distribution on a Heisenberg type group of homogeneous dimension Q is a biradial kernel of type α if it coincides with a biradial function, homogeneous of degree α — Q, and smooth away from the identity. We prove that a distribution is a biradial kernel of type α, 0 < α < Q, if and only if its Gelfand transform, defined on the Heisenberg fan, extends to a smooth even function on the upper half plane, homogeneous of degree −α/2. A similar result holds for radial kernels on the Heisenberg group.

Let be a homogeneous tree of degree at least three. In this paper we investigate for which values of p and r the (σθ)-Poisson semigroup is Lp – Lr,-bounded, and we sharp estimate for the corresponding operator norms.

In a recent paper In a recent paper the authors proved a multiplier theorem for Hardy spaces Hp (G), 0 < p ≤ 1, defined on a locally compact Vilenkin group G. The assumptions on the multiplier were expressed in terms of the “norms” of certain Herz spaces K (1/p − 1/?r, r, p) with r restricted to 1 ≤ r < ∞ and p < r. In the present paper we show how this restriction on r may be weakened to p ≤ r ∞. Furthermore, we present two modifications of our main theorem and compare these with certain results for multipliers on LP (Rn)-spaces, 1 < p < ∞, due to Seeger and to Cowling, Fendler and Foumier. We also discuss the sharpness of some of our results.

In this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.

Let G be any compact group, connected or disconnected, with dual object Ĝ. We define a family of local Sidon subsets of Ĝ in terms of allowable images of the representations. Using this family we develop a straightforward criterion whereby the existence of infinite Sidon subsets of Ĝ may be decided.

In 1953 P. P. Korovkin proved that if (Tn) is a sequence of positive linear operators defined on the space C of continuous real 2π-periodic functions and limn→rTnf = f uniformly for f = 1, cos and sin. then limn→rTnf = f uniformly for all f∈C. We extend this result to spaces of continuous functions defined on a locally compact abelian group G, with the test family {1, cos, sin} replaced by a set of generators of the character group of G.

Sets of independence are studied for compact abelian hypergroups and they are used, along with Riesz products, to investigate lacunarity questions on the dual object. It is shown that bounded Stechkin sets are always Sidon and that every bounded infinite subset of the dual contains an infinite Sidon set which is also a Λ set. Independent sets are shown to always be Sidon and a necessary condition for Sidonicity is provided. A result of Pisier is used to show that for compact non-abelian groups Sidon and central Λ are equivalent. Several applications are provided, primarily to questions regarding lacunarity on compact groups.

Let G be a locally compact abeian group, (μρ) a net of bounded Radon measures on G. In this paper we consider conditions under which (μρ) is saturated in Lp (G) and apply these results to the Fejér and Picard approximation processes.

The main results of this article are (I) Let B be a homogeneous Banach algebra, A a closed subalgebra of B, and I the largest closed ideal of B contained in A. We assert that for some closed subalgebra J of B. Furthermore, under suitable conditions, we show that A is an R-subalgebra if and only if J is an R-subalgebra. A number of concrete closed subalgebras of a homogeneous Banach algebra therefore are R-subalgebras. For the definition of P(A) and that of an R-subalgebra, see the introduction in Section 1. (II) We give sufficient and necessary conditions for a closed subalgebra of Lp(G), 1 ≦ p ≦ ∞, to be an R-subalgebra.

Let G be a compact, simple, simply connected Lie group. The Lp-norm of a central trigonometric polynomial reduces naturally to a weighted Lp-norm of a trigonometric polynomial on a maximal torus T. The weight is | Δ |2-p, where Δ is the usual Weyl function. If p ≥ 2, we prove that | Δ |2-p satisfies Muckenhoupt's Ap condition if and only if the Lp-norms of the irreducible characters of G are uniformly bounded.

In this paper we study the space of multipliers M(r, s: p, q) from the space of test functions Φrs(G), on a locally compact abelian group G, to amalgams (Lp, lq)(G); the former includes (when r = s = ∞) the space of continuous functions with compact support and the latter are extensions of the Lp(G) spaces. We prove that the space M(∞: p) is equal to the derived space (Lp)0 defined by Figá-Talamanca and give a characterization of the Fourier transform for amalgams in terms of these spaces of multipliers.

Let G be a locally compact Vilenkin group with dual group Γ. We prove Littlewood-Paley type inequalities corresponding to arbitrary coset decompositions of Γ. These inequalities are then applied to obtain new Lp(G) multiplier theorems. The sharpness of some of these results is also discussed.

We extend an uncertainty principle due to Cowling and Price to threadlike nilpotent Lie groups. This uncertainty principle is a generalization of a classical result due to Hardy. We are thus extending earlier work on Rn and Heisenberg groups.

We show that generalized Gaussian estimates for selfadjoint semigroups (e-tA)t ∈ R+ on L2 imply Lp boundedness of Riesz means and other regularizations of the Schrödinger group (eitA)t ∈ R. This generalizes results restricted to semigroups with a heat kernel, which are due to Sjöstrand, Alexopoulos and more recently Carron, Coulhon and Ouhabaz. This generalization is crucial for elliptic operators A that are of higher order or have singular lower order terms since, in general, their semigroups fail to have a heat kernel.

Let A be a commutative Banach algebra with identity of norm 1, X a Banach A-module and G a locally compact abeian group with Haar measure. Then the multipliers from an A -valued function algebra into an X-valued function space is studied. We characterize the multiplier spaces as the following isometrically isomorphic relations under some appropriate conditions: