For a locally compact group G, let A(G) be the Fourier ALGEBRA and let A_M (G) be the completion of this ALGEBRA in its multiplier ALGEBRA. In this paper, we show that A(G) is an abstract Segal ALGEBRA in A_M (G). Also, a necessary and sufficient condition for equality of these two ALGEBRAs is given. Then we prove that A_M (G) is an ideal in its second dual if and only if G is discrete. We show that if G is a discrete group, then A_M (G) is a BSE ALGEBRA if and only if G is M-weakly amenable. As a corollary, it is proven that A_M (F_2 ) is a BSE ALGEBRA while A(F_2 ) is not. Finally, we examine our results for the Lebasque-Fourier ALGEBRA and also give a completely new proof for equality of the character space of A(G) and A_M (G).

In this paper the definition and some properties of semigroupoids are considered. Representations, tight representations, and universal representations of a cancellative semigroupoid are discussed. Also, the C*-ALGEBRA of a semigroupoid is introduced and it is shown that source elements transfer to zero by tight representations.

Let A and B be Banach ALGEBRAs, α , β ∈ Hom(A, B), α ≤ 1 and β ≤ 1. We define an (α , β )-product on A× B which is a strongly splitting extension of A by B. We show that these products form a large class of Banach ALGEBRAs which contains all module extensions and triangular Banach ALGEBRAs. Then we consider spectrum, Arens regularity, amenability and weak amenability of these products.

In this paper, we study the relation between the soft sets and soft Lie ALGEBRAs with the lattice theory. We introduce the concepts of the lattice of soft sets, full soft sets and soft Lie ALGEBRAs and next, we verify some properties of them. We prove that the lattice of the soft sets on a fixed parameter set is isomorphic to the power set of a set with respect to set inclusion. In particular, we characterize the atoms in the lattice of the soft sets and the lattice of full soft Lie ALGEBRAs. After that, we introduce the compact elements in the full soft Lie ALGEBRAs and we present the necessary and sufficient conditions for the compactness and atomicness of the lattice of full soft Lie ALGEBRAs. We show that if a full soft Lie ALGEBRA is a compact element of the lattice of full soft Lie ALGEBRA then the parameter set is finite and the Lie ALGEBRA is finitely generated. In the sequel, we study the relationship between prime and maximal ideals in Lie ALGEBRAs and the prime and maximal elements in the lattice of soft Lie ALGEBRAs.

The present study aims at indicating the existence and uniqueness result of system in extended colombeau ALGEBRA. The Caputo fractional derivative is used for solving the system of ODEs. In addition, Riesz fractional derivative of Colombeau generalized ALGEBRA is considered. The purpose of introducing Riesz fractional derivative is regularizing it in Colombeau sense. We also give a solution to a nonlinear heat equation illustrating the application of the theory.