BED properties of some Banach algebras of vector-valued functions

It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, $\mathbb{F}$ ) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where $\mathbb{F}$ = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, $\mathbb{F}$ ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.

Let A be a commutative unital Banach algebra and let X be a compact space. We study the class of A-valued function algebras on X as subalgebras of C(X,A) with certain properties. We introduce the notion of A-characters of an A-valued function algebra A as homomorphisms from A into A that basically have the same properties as evaluation homomorphisms Ex:f↦f(x), with x∈X. We show that, under certain conditions, there is a one-to-one correspondence between the set of A-characters of A and the set of characters of the function algebra A=A∩C(X) of all scalar-valued functions in A. For the so-called natural A-valued function algebras, such as C(X,A) and Lip(X,A), we show that Ex (x∈X) are the only A-characters. Vector-valued characters are utilized to identify vector-valued spectra.

The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces.
 Let $X$ and $Y$ be complex Banach spaces. Let $\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}$ and $\mathbb{P}=\{P_1, P_2,$ \ldots, $P_n, \ldots \}$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $\|A_n\|_1=\|P_n\|_1=1$ and $\deg A_n=\deg P_n=n,$ $n\in \mathbb{N}.$ We consider the subalgebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ of the Fr\'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $\mathbb{A}$ and $\mathbb{P}$, respectively. It is easy to see that $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ are the Fr\'{e}chet algebras as well.
 In this paper we investigate conditions of isomorphism of the topological algebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{\infty})$ of entire functions of bounded type on $L_{\infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{\infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $\ell_{\infty}.$

Daugavet property of Banach algebras of holomorphic functions and norm-attaining holomorphic functions

In this work, we investigate the properties of the topological algebra of entire functions of bounded type, generated by a countable set of homogeneous polynomials on a complex Banach space.
 Let $X$ be a complex Banach space. We consider a subalgebra $H_{b\mathbb{P}}(X)$ of the Fréchet algebra of entire functions of bounded type $H_b(X),$ generated by a countable set of algebraically independent homogeneous polynomials $\mathbb{P}.$ We show that each term of the Taylor series expansion of entire function, which belongs to the algebra $H_{b\mathbb{P}}(X),$ is an algebraic combination of elements of $\mathbb{P}.$ We generalize the theorem for computing the radius function of a linear functional on the case of arbitrary subalgebra of the algebra $H_b(X)$ on the space $X.$ Every continuous linear multiplicative functional, acting from $H_{b\mathbb{P}}(X)$ to $\mathbb{C}$ is uniquely determined by the sequence of its values on the elements of $\mathbb{P}.$ Consequently, there is a bijection between the spectrum (the set of all continuous linear multiplicative functionals) of the algebra $H_{b\mathbb{P}}(X)$ and some set of sequences of complex numbers. We prove the upper estimate for sequences of this set. Also we show that every function that belongs to the algebra $H_{b\mathbb{P}}(X),$ where $X$ is a closed subspace of the space $\ell_{\infty}$ such that $X$ contains the space $c_{00},$ can be uniquely analytically extended to $\ell_{\infty}$ and algebras $H_{b\mathbb{P}}(X)$ and $H_{b\mathbb{P}}(\ell)$ are isometrically isomorphic. We describe the spectrum of the algebra $H_{b\mathbb{P}}(X)$ in this case for some special form of the set $\mathbb{P}.$
 Results of the paper can be used for investigations of the algebra of symmetric analytic functions on Banach spaces.

In this work we investigate the properties of the topological algebras of entire functions, generated by countable sets ℙ={P1, . . . , Pn, . . .} of homogeneous polynomials on complex Banach spaces. In particular, we consider spectra Mbℙ of such algebras and the structure of the ranges of the spectra under the mapping τ : Mbℙ↦ℂ∞ such that τ(φ)=(φ(P1), φ(P2), . . .) for every φ∈Mbℙ. We also investigate conditions of isomorphism of such algebras. Some applications for algebras of symmetric analytic functions of bounded type are obtained.

This note is a conitribution to the study of the subalgebras of the space C(S) of complex-valued continuous functions on a compact Hausdorff space S. We are interested here in finding conditions under which an algebra has maximal ideals other than the obvious ones corresponding to the points of S. We shall restrict ourselves to the case where S is a circle or an interval and shall give two sets of hypotheses under which other maximal ideals do exist. Both theorems depend on deep results about algebras of functions. The first is really a corollary of the theorem of Mergelyan and others which states that oIn any compact set E in the plane, of plane measure zero, an arbitrary continuous function can be uniformly approximated by rational functions having their poles outside E. We are presenting Theorem 1 mainly because its corollary is a statement about polynomials in several complex variables which seems to be new, and which we think is curious. Theorem 2 depends less obviously on the theorem of Silov asserting the existence of idempotents corresponding to the open-closed subsets of the structure space of a commutative Banach algebra. Our theorem is of Stone-Weierstrass type; it states, under hypotheses, that a given algebra either has many maximal ideals or else contains all continuous functions. Let [ be a closed subalgebra of C(S), and let 9 be the collection of functions in W which vanish at a given point of S. Either M is all of 2f, or 9 is a maximal ideal in W, in which case we say that 9J is associated with the given point. If 9 is a maximal ideal which is not of this form, we say that it is not associated with any point of S. We can now state our first theorem.

Throughout this note Z (resp. R, resp. R+, resp. N9 resp. C) is the set of all integer (resp. real, resp. nonnegative real, resp. nonnegative integer, resp. complex) numbers. Also G is a group, e e G its neutral element: K:G—>R+ a submultiplicat ive function (i.e., K{gh) ^ K(g)K(h) for all g,heG) with K(e) = 1; X a Banach space; the Banach algebra of all linear bounded operators on X and the identity. ^m{R) (m e N, m — oo) being the algebra of all m-times differentiable functions on R with the usual topology and Γ — {ze C; \z — 1}, Wm(Γ) is the algebra of all functions f:Γ-*C such that t->f(eu) belongs to unitary if it is 9fm(Γ)-scalar ([2], [4]).

Chapter 7 - FOUNDATIONS OF THE ALGEBRAIC THEORY OF LOGICAL SYNTAX

Many commutative semisimple Banach algebras B including B = C (X), X compact, and B = L1 (G), G locally compact, have the property that every homomorphism from B into C1[0, 1] is compact. In this paper we consider this property for uniform algebras. Several examples of homomorphisms from somewhat complicated algebras of analytic functions to C1[0, 1] are shown to be compact. This, together with the fact that every homomorphism from the disc algebra and from the algebra H∞ (∆), ∆ = unit disc, to C1[0, 1] is compact, led to the conjecture that perhaps every homomorphism from a uniform algebra into C1[0, 1] is compact. The main result to which we devote the second half of this paper, is to construct a compact Hausdorff space X, a uniformly closed subalgebra [Formula: see text] of C (X), and an arc ϕ: [0, 1] → X such that the transformation T defined by Tf = f ◦ ϕ is a (bounded) homomorphism of [Formula: see text] into C1[0, 1] which is not compact.

Preface

This paper deals with the automatic continuity theory for the convolution algebra of all Bochner integrable functions from a locally compact abelian group G into an arbitrary unital complex Banach algebra A. For non-compact G, it is shown that all epimorphisms and all derivations on this vector-valued group algebra are necessarily continuous while for compact G, such results depend heavily on the automatic continuity properties of the range algebra a.

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Lambda (textbf{1})=1, is multiplicative, that is, Lambda (ab)=Lambda (a)Lambda (b) for all a,bin A. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localization A_{P} is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra Asubseteq {mathbb {F}}^{X} over a subfield {mathbb {F}} of {mathbb {C}}, contains all the bounded functions in {mathbb {F}}^{X}, then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in (0,infty ) satisfy the GKŻ property, while the algebra of compactly supported distributions does not.

this space becomes a real Banach space. Finally, provided with the usual multiplication of functions, this space becomes a real Banach algebra. The function which is identically one on X is the multiplicative identity, and we denote it by e. According to the Banach-Stone theorem [5],2 if X is compact, then the topology of X is reflected in the algebraic and metric structure of C(X) in the sense that if C(X) and C( Y) are equivalent as Banach spaces, then X is homeomorphic to Y. This general structural relation suggests particularization: what are the specific algebraic or metric properties of C(X) which correspond to specific topological features of X? Significant results have been obtained in this direction by Myers, Eilenberg, and others [1; 2; 3; 4]. This paper is drawn along these lines, and is concerned in particular with those spaces of continuous functions which are defined on the topological product of two compact Hausdorff spaces, or on compact Hausdorff spaces which contain such a product.3 The central problems are those of characterization; for the cases of products, and spaces which contain a product subspace, characterizations have been obtained. I have also obtained characterizations of continuous functions spaces over fiber spaces and fiber bundles, but the results are quite technical and not very revealing, and therefore are not of sufficient interest to include here.

The present paper is devoted to 2-local derivations. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-lo\-cal derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and \mbox{2-local} derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every \mbox{2-local} derivation on a ∗-algebra C(Q,Mn(F)) or C(Q,Nn(F)), where Q is a compactum, Mn(F) is the ∗-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, Nn(F) is the ∗-subalgebra of Mn(F) defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.