We consider a semidirect product of two locally compact groups S and T, with S Abelian, denoted by SσT. An action of SσT on S is introduced to make S a homogeneous space of SσT. Then we define a unitary representation from SσT into the unitary group of L2(S) which is our main tool for defining the continuous wavelet transform on L2(S). Also the main properties of the transform are discussed. We prove the Plancherel and inversion formulas and reproducing kernel’s formula for this transform. This is finally specialized to the case of the continuous wavelet transform on L2(Rd).
The purpose of this paper is to introduce an algebra of functions on a semitopological semigroup and to study these functions from the point of view of universal semigroup compactification. We show that the corresponding semigroup compactification of this algebra is universal with respect to the property of being a nilpotent group.
We consider the enveloping semigroup of a flow generated by the action of a semitopological semigroup on any of its semigroup compactifications and explore the possibility of its being one of the known semigroup compactifications again. In this way, we introduce the notion of E-algebra, and show that this notion is closely related to the reductivity of the semigroup compactification involved. Moreover, the structure of the universal EᏲ-compactification is also given.2000 Mathematics Subject Classification: 22A20, 43A60.
Abstract. The purpose of this paper is to introduce an algebra of functions on a semitopological semigroup and to study these functions from the point of view of universal semigroup compactification. We show that the corresponding semigroup compactification of this algebra is universal with respect to the property of being a nilpotent group.
Abstract. We know that if S is a subsemigroup of a semitopological semigroup T , and Ᏺ stands for one of the spaces Ꮽᏼ, ᐃᏭᏼ, Ꮽᏼ, Ᏸ or ᏸᏯ, and ( , T Ᏺ ) denotes the canonical Ᏺ-compactification of T , where T has the property that Ᏺ(S) = Ᏺ(T ) |s , then ( |s , (S)) is an Ᏺ-compactification of S. In this paper, we try to show the converse of this problem when T is a locally compact group and S is a closed normal subgroup of T . In this way we construct various semigroup compactifications of T from the same type compactifications of S.
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