On Signed a-Adic Expansions and Weakly Almost Periodic Functions

Ruppert, Wolfgang

doi:10.1112/plms/s3-63.3.620

Cited by 8 publications

(8 citation statements)

References 24 publications

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“…Then Ruppert made the important breakthrough: he found a class of weakly almost periodic functions which could be described in elementary terms [14]. His direct with the exception of the calculations which show that two classes of functions are weakly almost periodic (Theorems 2.3 and 2.8).…”

Section: Introductionmentioning

confidence: 99%

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The set of idempotents in the weakly almost periodic compactification of the integers is not closed

Bordbar

Pym

1999

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

“…Ruppert [14] uses the description 'almost homomorphism'.) It is obvious from the last expression in equations ( * ) that x → c, x is an oid map.…”

Section: Introductionmentioning

confidence: 99%

The set of idempotents in the weakly almost periodic compactification of the integers is not closed

Bordbar

Pym

1999

Trans. Amer. Math. Soc.

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“…Now 26) so that ν * = µ and µ * = ν. Thus b * = Θ(µ) * = Θ(µ * ) = Θ(ν) = c, (5.27) and the involution is weak * -continuous.…”

Section: Examplesmentioning

confidence: 99%

Shift invariant preduals of ℓ 1(ℤ)

Daws

Haydon²,

Schlumprecht

et al. 2012

Isr. J. Math.

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The Banach space 1 (Z) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on 1 (Z) weak * -continuous. This is equivalent to making the natural convolution multiplication on 1 (Z) separately weak * -continuous and so turning 1 (Z) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C 0 (K) (for countable locally compact K) and 1 (Z) is c 0 (Z). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak * -continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c 0 . We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of Z. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c 0 .

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“…Before βN became an object of serious concern, Brown and Moran had already thrown considerable light on the structure of the semigroup of idempotents in wN as an application of some sophisticated techniques in harmonic analysis [6]. Only recently has an elementary approach to their results been found, by Ruppert [24], and his methods have led to some -but not all that many -other advances. A reasonably up-to-date account of our state of knowledge can be found in Ruppert's survey [23].…”

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confidence: 99%

The weakly almost periodic compactification of a direct sum of finite groups

Bordbar

Pym

1998

Math. Proc. Camb. Phil. Soc.

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The success of the methods of [24] and [4] in investigating the structure of the weakly almost periodic compactification wℕ of the semigroup (ℕ, +) of positive integers has prompted us to see how successful they would be in another context. We consider wG, where G=[oplus ]i∈ωGi is the direct sum of a sequence of finite groups with its discrete topology. We discover a large class of weakly almost periodic functions on G, and we use them to prove the existence of a large number of long chains of idempotents in wG. However, the closure of any singly generated subsemigroup of wG contains only one idempotent. We also prove that the set of idempotents in wG is not closed. The minimal idempotent of wG can be written as the sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of βℕ, we find subsemigroups SF of wG which are ‘carried’ by closed subsets F of βω. wG contains the free abelian product of the semigroups SF corresponding to families of disjoint subsets F. Usually SF is a very large semigroup, but for some points p∈βω, Sp can be quite small.

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On Signed a-Adic Expansions and Weakly Almost Periodic Functions

Cited by 8 publications

References 24 publications

The set of idempotents in the weakly almost periodic compactification of the integers is not closed

The set of idempotents in the weakly almost periodic compactification of the integers is not closed

Shift invariant preduals of ℓ 1(ℤ)

The weakly almost periodic compactification of a direct sum of finite groups

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