It is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natu-ral multiplications extending the one on A. But the essence of A ∗,is completely di, erent. By de, ning new products, we investigate some algebraic and spectral properties of odd duals of A. We will show re-lations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A ∗, , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras.

In this paper we introduce the notion of φ-commutativity for a Banach algebra A, where φ is a continuous homomorphism on A and study the concept of φ-weak amenability for φ-commutative Banach algebras. We give an example to show that the class of φ-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We characterize φ-weak amenability of φ-commutative Banach algebras and prove some hereditary properties. Moreover we verify some of the previous available results about commutative weakly amenable Banach algebras, for φ-commutative φ-weakly amenable Banach algebras.

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.

For semisimple, commutative Banach algebras A and B, some Banach algebra properties of the Cartesin product A×B are characterized in terms of A and B. A couple of results are also proved for non-commutative Banach algebras.

Let  $\left\{a_\alpha\right\}_{\alpha\in I}$ be a bounded net in a Banach algebra $A$ and $\varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $\|aa_\alpha-\varphi(a)a_\alpha\|\to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups such  as Segal algebras and $L^1$-algebras are responsive to this concept. It is also shown that $Wap(A)$ has a left invariant $\varphi$-mean if and only if there exists a bounded net $\left\{a_\alpha\right\}_{\alpha\in I}$ in $\left\{a\in A;\ \varphi(a)=1\right\}$ such that $\|aa_\alpha-\varphi(a)a_\alpha\|_{Wap(A)}\to0$ uniformly for all $a$ in weakly compact subsets of $A$. Other results in this direction are also obtained.