Let $G$ be a locally compact group. We continue our work [A. Ghaffari: $\Gamma $-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of $\Gamma $-amenability of a locally compact group $G$ defined with respect to a closed subgroup $\Gamma $ of $G\times G$. In this paper, among other things, we introduce and study a closed subspace $A_\Gamma ^p(G)$ of $L^\infty (\Gamma )$ and then characterize the $\Gamma $-amenability of $G$ using $A_\Gamma ^p(G)$. Various necessary and sufficient conditions are found for a locally compact group to possess a $\Gamma $-invariant mean.
In the current work, a new notion of n-weak amenability of Banach algebras using homomorphisms, namely (ϕ, ψ)-n-weak amenability is introduced. Among many other things, some relations between (ϕ, ψ)-n-weak amenability of a Banach algebra A and Mm(A), the Banach algebra of m × m matrices with entries from A, are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra L 1 (G) is (ϕ, ψ)-n-weakly amenable for any bounded homomorphisms ϕ and ψ on L 1 (G).
Let F be a class of entire functions represented by Dirichlet series with complex frequencies ∑ ake hλ k ,zi for which (|λ k |/e)|λ k | k!|ak| is bounded. Then F is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. F is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to F have also been established.
Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z(U)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $[F(u),u]F(u)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F(x)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.
A surjective bounded homomorphism fails to preserve n-weak amenability, in general. We however show that it preserves the property if the involved homomorphism enjoys a right inverse. We examine this fact for certain homomorphisms on several Banach algebras.
Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F(u)[F(u),u]^n=0$ for all $u \in L$, where $n\geq 1$ is a fixed integer. Then one of the following holds: \begin {itemize} \item [(1)] there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$; \item [(2)] $R$ satisfies $s_4$ and $F(x)=ax+xb$ for all $x\in R$, with $a, b\in U$ and $a-b\in C$; \item [(3)] $\mathop {\rm char}(R)=2$ and $R$ satisfies $s_4$. \end {itemize} As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.
Let $\mathcal B_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty $. Let $\mathcal B_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty $. Define $\mathcal A^n_c$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\mathcal B_c$. Similarly with $\mathcal A^n_r$ from $\mathcal B_r$. A type of integral is defined on distributions in $\mathcal A^n_c$ and $\mathcal A^n_r$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb N$, the spaces $\mathcal A^n_c$ and $\mathcal A^n_r$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\mathcal B_c$ and $\mathcal B_r$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\mathcal A_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\mathcal A_r^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.
Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a C*-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a C*-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions., Endre Makai Jr., Jaroslav Zemánek., and Obsahuje seznam literatury
In this paper we extend the notion of $n$-weak amenability of a Banach algebra $\mathcal A$ when $n\in \mathbb{N}$. Technical calculations show that when $\mathcal A$ is Arens regular or an ideal in $\mathcal A^{**}$, then $\mathcal A^*$ is an $\mathcal A^{(2n)}$-module and this idea leads to a number of interesting results on Banach algebras. We then extend the concept of $n$-weak amenability to $n \in \mathbb{Z}$.
Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in C n where the fibre is nontrivial, has to exceed n. This is shown not to be the case.