Let A be a semisimple, unital, and complex Banach algebra. It is well known and easy to prove that A is commutative if and only $e^xe^y=e^{x+y}$ for all $x,y\in A$ . Elaborating on the spectral theory of commutativity developed by Aupetit, Zemánek, and Zemánek and Pták, we derive, in this paper, commutativity results via a spectral comparison of $e^xe^y$ and $e^{x+y}$ .

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$, every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$. We show that all $C^{\ast }$-algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$-algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$-Drazin invertible elements of a $C^{\ast }$-algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

Let ${\mathcal{A}}$ be a complex unital Banach algebra, let $a$ be an element in it and let $0<\unicode[STIX]{x1D716}<1$. In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the $\unicode[STIX]{x1D716}$-level set of a condition spectrum at a given $(\unicode[STIX]{x1D716},a)$ plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices.

In this paper we define B-Fredholm elements in a Banach algebra A modulo an ideal J of A. When a trace function is given on the ideal J, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operator T acting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], where T0 is a Drazin inverse of T modulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

We give a characterization of ${{C}^{*}}$-normed algebras, among certain involutive normed ones. This is done through the existence of enough specific positive functionals. The same question is also examined in some non normed (topological) algebras.

This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C *-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.

Let $\chi $ be a locally compact metric space and let $\mathcal{A}$ be any of the Lipschitz algebras $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, or $\text{lip}_{\alpha }^{0}\,\chi $. In this paper, we show, as a consequence of rather more general results on Banach algebras, that $\mathcal{A}$ is $C$-character amenable if and only if $\chi $ is uniformly discrete.

We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\triangleright \right)$. We prove that the reduced quantum group ${{C}^{*}}$ -algebra ${{C}_{0}}\left( \mathbb{G} \right)$ can be recovered from the convolution $\triangleright $ by showing that the right $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$-module $\left\langle K\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,T\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is equal to ${{C}_{0}}\left( \mathbb{G} \right)$. On the other hand, we show that the left $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$-module $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,K\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is isomorphic to the reduced crossed product ${{C}_{0}}\left( \widehat{\mathbb{G}} \right){{\,}_{r}}\,\ltimes \,{{C}_{0}}\left( \mathbb{G} \right)$, and hence is a much larger ${{C}^{*}}$ -subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$.

We establish a natural isomorphism between the completely bounded right multiplier algebras of ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright \right)$, and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright $ on $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. We prove that if $\mathbb{G}$ is semiregular, then the space $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,B\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ of right $\mathbb{G}$-continuous operators on ${{L}_{2}}\left( \mathbb{G} \right)$, which was introduced by Bekka (1990) for ${{L}_{\infty }}\left( G \right)$, is a unital ${{C}^{*}}$-subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. In the representation framework formulated by Neufang–Ruan–Spronk (2008) and Junge–Neufang–Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. We also characterize some commutation relations of completely bounded multipliers of $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright \right)$ over $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$.

We show that for a normal locally-$\mathscr{P}$ space $X$ (where $\mathscr{P}$ is a topological property subject to some mild requirements) the subset ${C}_{\mathscr{P}} (X)$ of ${C}_{b} (X)$ consisting of those elements whose support has a neighborhood with $\mathscr{P}$, is a subalgebra of ${C}_{b} (X)$ isometrically isomorphic to ${C}_{c} (Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone–Čech compactification $\beta X$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, ${C}_{\mathscr{P}} (X)$ coincides with the set of those elements of ${C}_{b} (X)$ whose support has $\mathscr{P}$, it moreover becomes a Banach algebra, and simultaneously, $Y$ satisfies ${C}_{c} (Y)= {C}_{0} (Y)$. This includes the cases when $\mathscr{P}$ is the Lindelöf property and $X$ is either a locally compact paracompact space or a locally-$\mathscr{P}$ metrizable space. In either of the latter cases, if $X$ is non-$\mathscr{P}$, then $Y$ is nonnormal and ${C}_{\mathscr{P}} (X)$ fits properly between ${C}_{0} (X)$ and ${C}_{b} (X)$; even more, we can fit a chain of ideals of certain length between ${C}_{0} (X)$ and ${C}_{b} (X)$. The known construction of $Y$ enables us to derive a few further properties of either ${C}_{\mathscr{P}} (X)$ or $Y$. Specifically, when $\mathscr{P}$ is the Lindelöf property and $X$ is a locally-$\mathscr{P}$ metrizable space, we show that

$$\begin{eqnarray*}\dim C_{\mathscr{P}}(X)= \ell \mathop{(X)}\nolimits ^{{\aleph }_{0} } ,\end{eqnarray*}$$

$\ell (X)$

$X$

$\mathscr{P}$

$X$

$$\begin{eqnarray*}Y= {\mathrm{int} }_{\beta X} \upsilon X\end{eqnarray*}$$

$\upsilon X$

$X$

For a locally compact group $ \mathcal{G} $, we introduce and study a class of locally convex topologies $\tau $ on the measure algebra $M( \mathcal{G} )$ of $ \mathcal{G} $. In particular, we show that the strong dual of $(M( \mathcal{G} ), \tau )$ can be identified with a closed subspace of the Banach space $M\mathop{( \mathcal{G} )}\nolimits ^{\ast } $; we also investigate some properties of the locally convex space $(M( \mathcal{G} ), \tau )$.

For a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

Let Σ be a set and σ be a positive function on Σ. We introduce and study a locally convex topology β1(Σ,σ) on the space ℓ1(Σ,σ) such that the strong dual of (ℓ1(Σ,σ),β1(Σ,σ)) can be identified with the Banach space . We also show that, except for the case where Σ is finite, there are infinitely many such locally convex topologies on ℓ1(Σ,σ). Finally, we investigate some other properties of the locally convex space (ℓ1(Σ,σ),β1(Σ,σ)) , and as an application, we answer partially a question raised by A. I. Singh [‘L∞0(G)* as the second dual of the group algebra L1 (G) with a locally convex topology’, Michigan Math. J.46 (1999), 143–150].

We show that if the general real method ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$.

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of is Arens regular, and give some evidence that this is so if and only if is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebraffi We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of is amenable. We provide an abstract characterization in terms of something like an approximate diagonal, and consider when every ultrapower of a C*-algebra, or a group L1-convolution algebra, is amenable.

We deal with the dual Banach algebras for a locally compact group G. We investigate compact left multipliers on , and prove that the existence of a compact left multiplier on is equivalent to compactness of G. We also describe some classes of left completely continuous elements in .

The Arens products are the standard way of extending the product from a Banach algebra to its bidual ″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

Let $(a_n)_{n\ge 0}$ be a sequence of complex numbers, and, for $n\,{\ge}\,0$, let \[ b_n=\sum_{k=0}^n{n\choose k}a_k\quad\hbox{and}\quad c_n=\sum_{k=0}^n{n\choose k}(-1)^{n-k}a_k. \] A number of results are proved relating the growth of the sequences $(b_n)$ and $(c_n)$ to that of $(a_n)$. For example, given $p\,{\ge}\,0$, if $b_n\,{=}\,O(n^p)$ and $c_n\,{=}\,O(e^{\epsilon\sqrt{n}})$ for all $\epsilon\gt0$, then $a_n\,{=}\,0$ for all $n\gt p$. Also, given $0\lt\rho\lt1$, then $b_n,c_n\,{=}\,O(e^{\epsilon n^\rho})$ for all $\epsilon\gt0$ if and only if $n^{1/\rho-1}|a_n|^{1/n}\to 0$. It is further shown that, given $\beta\gt1$, if $b_n,c_n\,{=}\,O(\beta^n)$, then $a_n\,{=}\,O(\alpha^n)$, where $\alpha\,{=}\,\sqrt{\beta^2-1}$, thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredients of the proofs are a Phragmén–Lindelöf theorem for entire functions of exponential type zero, and an estimate for the expected value of $e^{\phi(X)}$, where $X$ is a Poisson random variable.