Banach Algebra Of Functions Research Articles

Book review: “Banach Function Algebras, Arens Regularity, and BSE Norms” by Harold Garth Dales and Ali Ülger

To the memory of Garth Dales.

The BSE property for some vector-valued Banach function algebras

The BSE property for some vector-valued Banach function algebras

Projective freeness and stable rank of algebras of complex-valued BV functions

AbstractThe paper investigates the algebraic properties of weakly inverse-closed complex Banach function algebras generated by functions of bounded variation on a finite interval. It is proved that such algebras have Bass stable rank 1 and are projective-free if they do not contain nontrivial idempotents. These properties are derived from a new result on the vanishing of the second Čech cohomology group of the polynomially convex hull of a continuum of a finite linear measure described by the classical H. Alexander theorem.

Maximal abelian subalgebras of Banach algebras

Let A be a commutative, unital Banach algebra. We consider the number of different non-commutative, unital Banach algebras C such that A is a maximal abelian subalgebra in C. For example, we shall prove that, in the case where A is an infinite-dimensional, unital Banach function algebra, A is a maximal abelian subalgebra in infinitely-many closed subalgebras of B ( A ) such that no two distinct subalgebras are isomorphic; the same result holds for certain examples A that are local algebras. We shall also give examples of uniform algebras of the form C ( K ) , where K is a compact space, with the property that there exists a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras C such that each C contains B ( C ( K ) ) as a closed subalgebra and is such that C ( K ) is a maximal abelian subalgebra in C.

A Characterization of Coherence of the Algebra of Bounded Uniformly Continuous Functions on a Metric Space and the Spectrum of General Self-Adjoint Banach Function Algebras

A Characterization of Coherence of the Algebra of Bounded Uniformly Continuous Functions on a Metric Space and the Spectrum of General Self-Adjoint Banach Function Algebras

Isomorphism Problem in a Special Class of Banach Function Algebras and its Application

Given a weight function τ, we introduce a new class of Banach function algebras with respect to τ, denoted by C_0b(X, τ ). We provide a complete solution to the isomorphism problem in this class. We further characterize the BSE-extension and the Inoue-Doss ideal associated with it. As an application of our results, we show the equivalence of the four statements: (i) C_0b(X, τ) is of BSE, (ii) C_0b(X, τ) is of BED, (iii) C_0b(X, τ) is Tauberian and (iv) τ is bounded.

On Vector-Valued Banach Function Algebras

We consider vector-valued Banach function algebras on a compact Hausdorff space. Then, we define the subalgebras generated by vector-valued polynomials and rational functions, and determine their maximal ideal spaces and Silov boundaries. We finally make use the results for a certain category of these algebras such as vector-valued Lipschitz algebras, vector-valued Dales–Davie algebras (algebras of vector-valued differentiable functions) and the algebras of vector-valued differentiable Lipschitz functions.

BSE Properties of Some Banach Function Algebras

In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lipα (X, d) and Dales-Davie algebras D(X, M) are BSE-algebras for certain underlying plane sets X. Moreover, we investigate BSE properties of certain subalgebras of Lipα(X, d)suchas LipA (X, α), Lipn (X, α) and Lip(X, M, α). BSE properties of Bloch type spaces $${{\cal B}_\alpha}$$ and Zygmund type spaces Zα are also investigated in different cases of α ∈ ℝ.

Jointly separating maps between vector-valued function spaces

AbstractLetXandYbe compact Hausdorff spaces,Ebe a real or complex Banach space andFbe a real or complex locally convex topological vector space. In this paper we study a pair of linear operatorsS,T:A(X,E) →C(Y,F) from a subspaceA(X,E) ofC(X,E) toC(Y,F), which are jointly separating, in the sense thatTfandSghave disjoint cozeros wheneverfandghave disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pairT:A(X) →C(Y) andS:A(X,E) →C(Y,F) of linear operators, whereA(X) is a regular Banach function algebra onX, such thatf⋅g= 0 impliesTf⋅Sg= 0, for allf∈A(X) andg∈A(X,E). IfTandSare jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism betweenXandYand, furthermore,T−1andS−1are also jointly separating maps.

$p$-Fourier algebras on compact groups

Let G be a compact group. For 1\leq p\leq\infty we introduce a class of Banach function algebras \mathcal A^p(G) on G which are the Fourier algebras in the case p=1 , and for p=2 are certain algebras discovered by Forrest, Samei and Spronk. In the case p\neq 2 we find that \mathcal A^p(G)\cong\mathcal A^p(H) if and only if G and H are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call p -Beurling–Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie G and p > 1 , our techniques of estimation of when certain p -Beurling–Fourier algebras are operator algebras rely more on the fine structure of G , than in the case p=1 . We also study restrictions to subgroups. In the case that G= SU(2), restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well.

Polynomially peripheral range-preserving maps between Banach algebras

Let A and B be two Banach function algebras and p a two variable polynomial $$p(z,w)=zw+az+bw+c$$ , ( $$a,b,c\in {\mathbb {C}}$$ ). We characterize the general form of a surjection $$T: A \longrightarrow B$$ which satisfies $$\mathrm{Ran}_\pi (p(Tf,Tg))\cap \mathrm{Ran}_\pi (p(f,g))\ne \emptyset , (f,g\in A$$ and $$c\ne ab)$$ , where $$\mathrm{Ran}_\pi (f)$$ is the peripheral range of f.

Nonlinear spectral radius preservers between certain non-unital Banach function algebras

Let $\alpha _0\in \mathbb {C} \setminus \{0\}$, $A$ and $B$ be Banach function algebras. Also, let $\rho _1:\Omega _1 \rightarrow A$, $\rho _2:\Omega _2 \rightarrow A$, $\tau _1: \Omega _1 \rightarrow B$ and $\tau _2:\Omega _2 \rightarrow B$ be surjections such that $\|\rho _1(\omega _1)\rho _2(\omega _2)+\alpha _0\|_\infty =\|\tau _1(\omega _1)\tau _2(\omega _2)+\alpha _0\|_\infty $ for all $\omega _1\in \Omega _1, \omega _2\in \Omega _2$, where $\Omega _1$, $\Omega _2$ are two non-empty sets. Motivated by recent investigations on such maps between unital Banach function algebras, in this paper we characterize these maps for certain non-unital Banach function algebras including pointed Lipschitz algebras and abstract Segal algebras of the Talamanca-Herz algebras when the underlying groups are first countable. Moreover, sufficient conditions are given to guarantee such maps induce weighted composition operators.

Quasianalyticity in certain Banach function algebras

Let $X$ be a perfect, compact subset of the complex plane. We consider algebras of those functions on $X$ which satisfy a generalised notion of differentiability, which we call $\mathcal{F}$-differentiability. In particular, we investigate a notion of quasianalyticity under this new notion of differentiability and provide some sufficient conditions for certain algebras to be quasianalytic. We give an application of our results in which we construct an essential, natural uniform algebra $A$ on a locally connected, compact Hausdorff space $X$ such that $A$ admits no non-trivial Jensen measures yet is not regular. This construction improves an example of the first author (2001).

φ-contractibility of some classes of Banach function algebras

In this paper, we study ?-contractibility of natural Banach function algebras on a compact Hausdorff space. As a consequence, we characterize ?-contractibility of the Lipschitz algebra Lip(X,d?), for a compact metric space (X,d). We also characterize ?-contractibility of certain subalgebras of Lipschitz functions including rational Lipschitz algebras, analytic Lipschitz algebras and differentiable Lipschitz algebras.

Orthogonally additive polynomials on Banach function algebras

For a Banach function algebra A, we consider the problem of representing a continuous d-homogeneous polynomial P:A→X, where X is an arbitrary Banach space, that satisfies the property P(f+g)=P(f)+P(g) whenever f,g∈A are such that supp(f)∩supp(g)=∅. We show that such a polynomial can be represented as P(f)=T(fd)(f∈A) for some continuous linear map T:A→X for a variety of Banach function algebras such as the algebra of continuous functions C0(Ω) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lipα(K) for any compact metric space K and α∈]0,1[, the Figà–Talamanca–Herz algebra Ap(G) for some locally compact groups G and p∈]1,+∞[, the algebras AC([a,b]) and BVC([a,b]) of absolutely continuous functions and of continuous functions of bounded variation on the interval [a,b]. In the case where A=Cn([a,b]), P can be represented as P(f)=∑T(n1,…,nd)(f(n1)⋯f(nd)), where the sum is taken over (n1,…,nd)∈Zd with 0≤n1≤…≤nd≤n, for appropriate continuous linear maps T(n1,…,nd):Cn−nd([a,b])→X.

Disjointness-preserving linear maps on Banach function algebras associated with a locally compact group

We introduce a certain property of commutative Banach algebras which we call property OB. We prove that every bounded disjointness-preserving linear map from a commutative Banach algebra with the aforesaid property to any semisimple, commutative Banach algebra is a weighted composition map. Further, it is shown that a variety of important Banach algebras in harmonic analysis have the property OB.

The chain rule for $\mathcal{F}$-differentiation

Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by considering $\mathcal{F}$-differentiable functions on $X$. In this paper, we investigate composition, the chain rule, and the quotient rule for this notion of differentiability. We give an example where the chain rule fails, and give a number of sufficient conditions for the chain rule to hold. Where the chain rule holds, we observe that the Fa\'a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of $\mathcal{F}$-differentiable functions.

Regularity points and Jensen measures for $R(X)$

We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in an earlier paper on non-regularity for Banach function algebras. We show that, even for the natural uniform algebras $R(X)$ (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets $X$ such that $R(X)$ is not regular, but such that $R(X)$ has no non-trivial Jensen measures. The original construction appears in the first author's previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set $X$ with the property that the set of points of discontinuity for $R(X)$ has positive area.

Wiener weighted algebra of functions of infinitely many variables

In this article we consider a weighted Wiener type Banach algebra of functions of infinitely many variables. The main result is a description of the spectrum of this algebra.

On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions

AbstractWe recently introduced a weighted Banach algebra of functions that are holomorphic on the unit disc D, continuous up to the boundary, and of the class C(n) at all points where the function G does not vanish. Here, G refers to a function of the disc algebra without zeros on D. Then we proved that all closed ideals in with at most countable hull are standard. In this paper, on the assumption that G is an outer function in C(n) having infinite roots in and countable zero set h0(G), we show that all the closed ideals I with hull containing h0(G) are standard.