Slowly oscillating functions in semigroup compactifications and convolution algebras

2007

Studia Math.

Abstract. We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if A has a brai (blai), then the right (left) module action of A on A * is Arens regular if and only if A is reflexive. We find that Arens regularity is implied by the factorization of A * or A * * when A is a left or a right ideal in A * * . The Arens regularity and strong irregularity of A are related to those of the module actions of A on the nth dual A (n) of A. Banach algebras A for which Z(A * * ) = A but A Z t (A * * ) are found (here Z(A * * ) and Z t (A * * ) are the topological centres of A * * with respect to the first and second Arens product, respectively).

“…We know that for every locally compact group G, the group algebra L 1 (G) is left strongly Arens irregular (see [13], [9], [15]). So we have the following result.…”

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confidence: 99%

Arens regularity of module actions

Gordji

2007

Studia Math.

“…The algebra A is said to be Arens regular when Z(A * * ) = A * * . We may recall that any C * -algebra is Arens regular, and that the group algebra L 1 (G) of a locally compact group G is strongly Arens irregular, i.e., Z(L 1 (G) * * ) = L 1 (G) (see [27], or [29] and [14] for different proofs). For more details, the reader is directed for example to [15], [5] or [7].…”

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confidence: 99%

Weak amenability of the second dual of a Banach algebra

Gordji

2007

Studia Math.

“…The most important applications are maybe those related to the cardinality of the set of left invariant means when G is amenable and to Arens irregularity. Indeed, a careful reader will quickly notice that most of the arguments giving the number of left invariant means or leading to the topological center of LUC(G) * being the measure algebra M (G) and that of the second Banach dual of the group algebra L 1 (G) being L 1 (G) are based on sets (or nets) of points taken from the LUC-compactification G LU C of G. See, for example, [8,11,14,20], for the number of left invariant means and other related results, and [7,9,15,16,18,19,22], for the results on topological centers.…”

Section: Introductionmentioning

confidence: 99%

On the spectra of C⁎-algebras of slowly oscillating functions on locally compact groups

Alaste

Advances in Mathematics

2015

We present a study of C * -algebras SO(ϕ) of slowly oscillating functions in the direction of filters ϕ on a locally compact topological group G. We show that SO(ϕ) is an m-admissible subalgebra of C(G) if and only if the closure of the filter ϕ in the LU C-compactification G LU C of G is an ideal of G LU C and that the semigroup compactification of G determined by SO(ϕ) always contains right zero elements. Using this, we characterize a new interesting C * -algebra of bounded continuous functions on G. The spectrum of this C * -algebra determines the universal semigroup compactification of G with respect to the property that the semigroup compactification contains a right zero element. We show that the topological center of this universal compactification is G. As an application of the previous results, we show that every closed ideal of G LU C contained in the ideal U (G) of uniform points of G LU C can be decomposed into 2 2 κ(G) closed left ideals of G LU C .