We prove that if a uniformly bounded (or equidistantly uniformly bounded) Nemytskij operator maps the space of functions of bounded ${\it\varphi}$-variation with weight function in the sense of Riesz into another space of that type (with the same weight function) and its generator is continuous with respect to the second variable, then this generator is affine in the function variable (traditionally, in the second variable).

The aim of this paper is to discuss the commutativity of a Banach algebra $A$ via its derivations. In particular, we prove that if $A$ is a unital prime Banach algebra and $A$ has a nonzero continuous linear derivation $d:A\rightarrow A$ such that either $d((xy)^{m})-x^{m}y^{m}$ or $d((xy)^{m})-y^{m}x^{m}$ is in the centre of $A$ for an integer $m=m(x,y)$ and sufficiently many $x,y$, then $A$ is commutative. We give examples to illustrate the scope of the main results and show that the hypotheses are not superfluous.

For a discrete abelian cancellative semigroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ with a weight function $\omega $ and associated multiplier semigroup $M_\omega (S)$ consisting of $\omega $-bounded multipliers, the multiplier algebra of the Beurling algebra of $(S,\omega )$ coincides with the Beurling algebra of $M_\omega (S)$ with the induced weight.

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

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

$\ell (X)$

$X$

$\mathscr{P}$

$X$

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

$\upsilon X$

$X$

We investigate which weighted convolution algebras ${ \ell }_{\omega }^{1} (S)$, where $S$ is a semilattice, are AMNM in the sense of Johnson [‘Approximately multiplicative functionals’, J. Lond. Math. Soc. (2) 34(3) (1986), 489–510]. We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all ${ \ell }_{\omega }^{1} (S)$ where $S$ has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein [‘Strong Ditkin algebras without bounded relative units’, Int. J. Math. Math. Sci. 22(2) (1999), 437–443]. We also investigate when $({ \ell }_{\omega }^{1} (S), { \mathbb{M} }_{2} )$ is an AMNM pair in the sense of Johnson [‘Approximately multiplicative maps between Banach algebras’, J. Lond. Math. Soc. (2) 37(2) (1988), 294–316], where ${ \mathbb{M} }_{2} $ denotes the algebra of $2\times 2$ complex matrices. In particular, we obtain the following two contrasting results: (i) for many nontrivial weights on the totally ordered semilattice ${ \mathbb{N} }_{\min } $, the pair $({ \ell }_{\omega }^{1} ({ \mathbb{N} }_{\min } ), { \mathbb{M} }_{2} )$ is not AMNM; (ii) for any semilattice $S$, the pair $({\ell }^{1} (S), { \mathbb{M} }_{2} )$ is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent $2\times 2$ matrices.

Towards an involutive analogue of a result on the semisimplicity of ${\ell }^{1} (S)$ by Hewitt and Zuckerman, we show that, given an abelian $\ast $-semigroup $S$, the commutative convolution Banach $\ast $-algebra ${\ell }^{1} (S)$ is $\ast $-semisimple if and only if Hermitian bounded semicharacters on $S$ separate the points of $S$; and we search for an intrinsic separation property on $S$ equivalent to $\ast $-semisimplicity. Very many natural involutive analogues of Hewitt and Zuckerman’s separation property are shown not to work, thereby exhibiting intricacies involved in analysis on $S$.

The notion of BSE algebras was introduced and first studied by Takahasi and Hatori and later studied by Kaniuth and Ülger. This notion depends strongly on the multiplier algebra $M( \mathcal{A} )$ of a commutative Banach algebra $ \mathcal{A} $. In this paper we first present a characterisation of the multiplier algebra of the direct sum of two commutative semisimple Banach algebras. Then as an application we show that $ \mathcal{A} \oplus \mathcal{B} $ is a BSE algebra if and only if $ \mathcal{A} $ and $ \mathcal{B} $ are BSE. We also prove that if the algebra $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ with $\theta $-Lau product is a BSE algebra and $ \mathcal{B} $ is unital then $ \mathcal{B} $ is a BSE algebra. We present some examples which show that the BSE property of $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ does not imply the BSE property of $ \mathcal{A} $, even in the case where $ \mathcal{B} $ is unital.

For a compact subset K of the boundary of a compact Hausdorff space X, six properties that K may have in relation to the algebra A(X) are considered. It is shown that in relation to the algebra A(Dn), where Dn denotes the n-dimensional polydisc, the property of being totally null is weaker than the other five properties. A general condition is given on the algebra A(X) which implies the existence of a totally null set that is not null, and several conditions are stated for A(X) , each of which is sufficient for a totally null set to be a null set.

We present a family of radical convolution Banach algebras on intervals (0,a] which are of Sobolev type; that is, they are defined in terms of derivatives. Among other properties, it is shown that all epimorphisms and derivations of such algebras are bounded. Also, we give examples of nontrivial concrete derivations.

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

Let L1(ω) be the weighted convolution algebra L1ω(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→L1(ω2), the unique continuous extension to a homomorphism between the corresponding weighted measure algebras on ℝ+ is also continuous with respect to the weak-star topologies on these algebras. In this paper we investigate whether similar results hold for homomorphisms from L1(ω) into other commutative Banach algebras. In particular, we prove that for the disc algebra every nonzero homomorphism extends uniquely to a continuous homomorphism which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras A+v on (including the algebra of absolutely convergent Taylor series on ) we prove that every nonzero homomorphism Φ:L1(ω)→A+v extends uniquely to a continuous homomorphism which is also continuous with respect to the weak-star topologies.

A commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

Let Bn denote the unit ball in ℂn, n≥1. Given an α>0, let ℱα(n) denote the class of functions defined for z∈Bn by integrating the kernel (1−〈z,w〉)−α against a complex Borel measure dμ(w), w∈Bn. The family ℱ0(n) corresponds to the logarithmic kernel log (1/(1−〈z,w〉)). Various properties of the spaces ℱα(n), α≥0, are obtained. In particular, pointwise multiplies for ℱα(n) are investigated.

In this paper, we present a constructive proof of the amenability of C(X), where X is a compact space.

Let K and X be compact plane sets such that . Let P(K) be the uniform closure of polynomials on K, let R(K) be the uniform closure of rational functions on K with no poles in K and let A(K) be the space of continuous functions on K which are analytic on int(K). Define P(X,K),R(X,K) and A(X,K) to be the set of functions in C(X) whose restriction to K belongs to P(K),R(K) and A(K), respectively. Let S0(A) denote the set of peak points for the Banach function algebra A on X. Let S and T be compact subsets of X. We extend the Hartogs–Rosenthal theorem by showing that if the symmetric difference SΔT has planar measure zero, then R(X,S)=R(X,T) . Then we show that the following properties are equivalent:

1. (i) R(X,S)=R(X,T) ;

2. (ii) and ;

3. (iii) R(K)=C(K)for every compact set ;

4. (iv) for every open set U in ℂ ;

5. (v) for every p∈X there exists an open disk Dp with centre p such that

1. (i) A(X,S)=R(X,T) ;

2. (ii) for every closed disk in ℂ ;

3. (iii) for every p∈X there exists an open disk Dp with centre p such that

Let M be a forward-shift-invariant subspace and N a backward-shift-invariant subspace in the Hardy space H2 on the bidisc. We assume that . Using the wandering subspace of M and N, we study the relations between M and N. Moreover we study M and N using several natural operators defined by shift operators on H2.

The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

In this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.

Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

In this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.