‎We say that a locally compact group $G$ is $C^*$-unitarizable if its‎ ‎full group $C^*$-algebra $C^*(G)$ satisfies Kadison's similarity‎ ‎problem (SP)‎, ‎i.e. every bounded representation of $C^*(G)$ on a‎ ‎Hilbert space is similar to a *-representation‎. ‎We prove that‎ ‎locally compact and unitarizable groups are $C^*$-unitarizable‎. ‎For‎ ‎discrete groups‎, ‎we prove that $C^*$-unitarizable passes to‎ ‎quotients‎. ‎Moreover‎, ‎a given discrete group is $C^*$-unitarizable‎ ‎whenever we can find a normal and $C^*$-unitarizable subgroup with‎ ‎amenable quotient‎.

Let $E$ be a real $q$-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let $T:E\to E$ be a nonexpansive mapping with $F(T)\neq\emptyset.$ Let $A:E\to E$ be an $\eta$-strongly accretive map which is also $\kappa$-Lipschitzian. Let $f:E\to E$ be a contraction map with coefficient $0\lt \alpha\lt1.$ Let a sequence $\{y_{n}\}$ be defined iteratively by $y_{0}\in E,~~ y_{n+1}=\alpha_n\gamma f(y_n)+(I-\alpha_n\mu A)Ty_n,n\geq0,$ where $\{\alpha_n\},~~\gamma$ and $\mu$ satisfy some appropriate conditions. Then, we prove that $\{y_{n}\}$ converges strongly to the unique solution $x^{*} \in F(T)$ of the variational inequality $\langle(\gamma f-\mu A)x^{*},j(y-x^{*})\rangle\leq0,~\forall~y\in F(T).$ Convergence of the correspondent implicit scheme is also proved without the assumption that $E$ has weakly sequentially continuous duality map. Our results are applicable in $l_{p}$ spaces, $1\lt p \lt \infty$.

We consider the Banach space $X=(\Bbb R^2‎, ‎\|\cdot\|)$ with a normalized‎, ‎absolute norm‎. ‎Our aim in this paper is to calculate the modified Neumann-Jordan constant $C'_{NJ}(X)$ and the Zbăganu constant $C_Z(X)$‎.

‎We give asymptotic classification of the positive solutions of a class of‎ ‎two-dimensional nonlinear Volterra integro-differential equations‎. ‎Also‎, ‎we furnish necessary and sufficient conditions for the existence of such‎ ‎positive solutions‎.

‎The main object of the present paper is to discuss some univalence‎ ‎conditions for a family of integral operators‎. ‎Several other‎ ‎closely-related results are also considered‎.

In this paper‎, ‎we introduce the generalized system nonlinear variational inclusions‎ ‎and prove the existence of its solution in normed spaces‎. ‎We provide‎ ‎examples of applications related to a system nonlinear variational‎ ‎inclusions in the sense of Verma‎, ‎a coupled fixed point problem‎, ‎considered by Bhaskar and Lakshmikantham‎, ‎a coupled coincidence‎ ‎point considered by Lakshmikantham and Ćirić‎. ‎Also‎, ‎we‎ ‎generalized coupled best approximations theorem‎.

Given a finite subset $\sigma$ of the unit disc $\mathbb{D}$ and a holomorphic function $f$ in $\mathbb{D}$ belonging to a class $X$, we are looking for a function $g$ in another class $Y$ which satisfies $g_{\vert\sigma}=f_{\vert\sigma}$ and is of minimal norm in $Y$. More precisely, we consider the interpolation constant $c\left(\sigma,\, X,\, Y\right)=\mbox{sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}\mbox{inf}_{g_{\vert\sigma}=f_{\vert\sigma}}\left\Vert g\right\Vert _{Y}.$ When $Y=H^{\infty}$, our interpolation problem includes those of Nevanlinna-Pick and Carathéodory-Schur. If $X$ is a Hilbert space belonging to the families of weighted Hardy and Bergman spaces, we obtain a sharp upper bound for the constant $c\left(\sigma,\, X,\, H^{\infty}\right)$ in terms of $n=\mbox{card}\,\sigma$ and $r=\mbox{max}{}_{\lambda\in\sigma}\left|\lambda\right|\lt 1$. If $X$ is a general Hardy-Sobolev space or a general weighted Bergman space (not necessarily of Hilbert type), we also establish upper and lower bounds for $c\left(\sigma,\, X,\, H^{\infty}\right)$ but with some gaps between these bounds. This problem of constrained interpolation is partially motivated by applications in matrix analysis and in operator theory.

‎Let $\mathcal{A}$ and $\mathcal{B}$ be two Banach algebras and let‎ ‎$\mathcal{M}$ be a Banach $\mathcal{B}$-bimodule‎. ‎Suppose that‎ ‎$\sigma:\mathcal{A} \rightarrow \mathcal{B}$ is a linear mapping and‎ ‎$d:\mathcal{A} \rightarrow \mathcal{M}$ is a $\sigma$-derivation‎. ‎We‎ ‎prove several results about automatic continuity of‎ ‎$\sigma$-derivations on Banach algebras‎. ‎In addition‎, ‎we define a‎ ‎notion for m-weakly continuous linear mapping and show that‎, ‎under‎ ‎certain conditions‎, ‎$d$ and $\sigma$ are m-weakly continuous‎. ‎Moreover‎, ‎we prove that if $\mathcal{A}$ is commutative and $\sigma‎: ‎\mathcal{A} \rightarrow \mathcal{A}$ is a continuous homomorphism‎ ‎such that $\sigma^{2} = \sigma$ then $\sigma d \sigma (\mathcal{A})‎ ‎\subseteq \sigma(Q(\mathcal{A})) \subseteq rad(\mathcal{A})$‎.

‎We introduce the notion of strongly $h$-convex functions (defined on‎ ‎a normed space) and present some properties and representations of‎ ‎such functions‎. ‎We obtain a characterization of inner product spaces‎ ‎involving the notion of strongly $h$-convex functions‎. ‎Finally‎, ‎a‎ ‎Hermite-Hadamard-type inequality for strongly $h$-convex functions‎ ‎is given‎.

‎Topological algebras with the exponential map are examined and‎ ‎several conditions are obtained for an element in a Gelfand-Mazur‎ ‎algebra with the exponential map to be central modulo the left‎ ‎topological radical‎.

‎Some sufficient conditions for the existence of fixed point of mappings‎ ‎satisfying generalized weak contractive conditions is obtained‎. ‎A fixed‎ ‎point theorem for nonexpansive mappings is also obtained‎. ‎As an application‎, ‎some invariant approximation results are derived in cone metric spaces‎.

‎In the peresent paper‎, ‎by using generalized weighted mean and difference‎ ‎matrix of order $m,$ we introduce the sequence spaces $X(u,v,\Delta^{(m)})$‎, ‎where $X$ is one of the spaces $\ell_{\infty}$‎, ‎$c$ or $c_{0}$‎. ‎Also‎, ‎we‎ ‎determine the $\alpha$-‎, ‎$\beta$‎- ‎and $\gamma$-duals of those spaces and‎ ‎construct their Schauder bases for $X\in\{c,c_{0}\}$‎. ‎Morever‎, ‎we give the‎ ‎characterization of the matrix mappings on the spaces $X(u,v,\Delta^{m})$‎ ‎ for $X\in\{\ell_{\infty},c,c_{0}\}.$ Finally‎, ‎we characterize some‎ ‎classes of compact operators on the spaces $\ell_{\infty}(u,v,\Delta^{m})$ and‎ ‎$c_{0}(u,v,\Delta^{m})$ by using the Hausdorff measure of noncompactness‎.

‎In this note the Poisson structures on Banach manifolds are‎ ‎considered‎. ‎Then a Lichnerowicz-Poisson cohomology is reformulated‎ ‎in the Banach setting and its relation with Banach Lie algebroids‎ ‎cohomology is given in a classical way just as in the finite‎ ‎dimensional case‎.

‎In this paper we investigate conditions under which a holomorphic‎ ‎self-map of the unit disk induces a composition operator $C_\phi$‎ ‎with closed range on the weighted Bloch space $\mathcal{B}_{\log}$.‎ ‎Also‎, ‎we introduce a new class of functions the so called‎ ‎$F_{\log}(p,q,s)$ spaces‎. ‎Necessary and sufficient conditions are‎ ‎given for a composition operator $C_\phi$ to be bounded and compact‎ ‎from ${\mathcal{B}_{\log}}$ to $F_{\log}(p,q,s)$. Moreover‎, ‎necessary and sufficient conditions for $C_{\phi}$ from the‎ ‎Dirichlet space $\mathcal{D}$ to the spaces $F_{\log}(p,q,s)$ to be‎ ‎compact are given in terms of the map $\phi$‎.