Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule then $S = A oplus E$, the $l^1$-direct sum of $A$ and $E$ becomes a module extension Banach algebra when equipped with the algebras product $(a,x).(a^prime,x^prime)= (aa^prime, a.x^prime+ x.a^prime)$. In this paper, we investigate $triangle$-amenability for these Banach algebras and we show that for discrete inverse semigroup $S$ with...

The notion of the shell of a Hilbert space operator, which is a useful generalization (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators ...

The notion of the shell of a Hilbert space operator, which is a useful generalization (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators ...

The structure of orthogonal polynomials on R with the weight function |x1 − x2| |x1x2|1e 2 1+x 2 2)/2 is based on the Dunkl operators of type B2. This refers to the full symmetry group of the square, generated by reflections in the lines x1 = 0 and x1− x2 = 0. The weight function is integrable if k0, k1, k0 +k1 > − 1 2 . Dunkl operators can be defined for polynomials taking values in a module o...

We discuss the inner product spaces of the middle homology groups of manifolds of dimensions 2 and 4. We prove that two compact 2-manifolds are homeomorphic if and only if the inner product spaces of their first homology groups are isomorphic. We outline a proof that every inner product space can be realized as the first homology group of some surface. We conclude by proving that two simply con...

In complex product design, product performance driving design is a new and innovative research in the engineering field. For realizing the product performance driving design, an elaborated expert system, Expert Systems for Assisting Mapping from Performance Space to Design Space (ESMPD) is proposed, which have two main modules for mapping from product performance space to interesting design spa...

This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω2q of Cq. The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L(Ω2q). As a bypass, we study the inner product [f, g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables z, z ∈ Cq, in which f(D) is t...

In 1964, Gähler 1 introduced the concept of 2-norm and 2-inner product spaces as generalization of norm and inner product spaces. A systematic presentation of the results related to the theory of 2-inner product spaces can be found in the book in 2, 3 and in list of references in it. Generalization of 2-inner product space for n ≥ 2 was developed by Misiak 4 in 1989. Gunawan and Mashadi 5 in 20...