The Extended Cone b-Metric-like Spaces over Banach Algebra and Some Applications

In this paper, we introduce the structure of extended cone b-metric-like spaces over Banach algebra as a generalization of cone b-metric-like spaces over Banach algebra. In this generalized space we define the notion of generalized Lipschitz mappings in the setup of extended cone b-metric-like spaces over Banach algebra and investigated some fixed point results. We also provide examples to illustrate the results presented herein. Finally, as an application of our main result, we examine the existence and uniqueness of solution for a Fredholm integral equation.

An existence theorem for nonlinear functional Volterra integral equations via Petryshyn's fixed point theorem

<abstract><p>Using the method of Petryshyn's fixed point theorem in Banach algebra, we investigate the existence of solutions for functional integral equations, which involves as specific cases many functional integral equations that appear in different branches of non-linear analysis and their applications. Finally, we recall some particular cases and examples to validate the applicability of our study.</p></abstract>

Hyers Stability and Multi-Fuzzy Banach Algebra

In this paper, we define multi-fuzzy Banach algebra and then prove the stability of involution on multi-fuzzy Banach algebra by fixed point method. That is, if f:A→A is an approximately involution on multi-fuzzy Banach algebra A, then there exists an involution H:A→A which is near to f. In addition, under some conditions on f, the algebra A has multi C*-algebra structure with involution H.

Invariance and normality of projections in the dual of Banach algebras

We study the classes of invariant and natural projections in the dual of a Banach algebra $A$. These type of projections are relevant by their connections with the existence problem of bounded approximate identities in closed ideals of Banach algebras. It is known that any invariant projection is a natural projection. In this article we consider the issue of when a natural projection is an invariant projection.

Some Properties of Cone Inner Product Spaces over Banach Algebra

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

Weak module amenability for the second dual of a Banach algebra

In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

On holomorphic reflexivity conditions for complex Lie groups

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$ .

CENTRE OF BANACH ALGEBRA VALUED BEURLING ALGEBRAS

We prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$ -approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$ . As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity.