Free Banach Spaces Research Articles

Representation Theorems for Operators on Free Banach Spaces of Countable Type

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.

ALGEBRAIC STRUCTURES IN THE SPACES OFLIPSCHITZ FUNCTIONS

The algebra of Lipschitz functions of the metric space and their scalar homomorphisms are studied in this paper, using free Banach spaces.

C-Algebras on some Free-Banach Spaces

The main goal of this work is to study the Gelfand spaces of some commutative Banach algebras with unit within the space of bounded linear operators. We will also show, under special condition, that this algebra is isometrically isomorphic to some space of continuous functions defined over a compact. Such isometries preserve idempotent elements. This fact will allow us to define the respective associated measure which is known as spectral measure. Let us also notice that this measure is obtained by restriction of the reciprocal of the Gelfand transform to the set of characteristic functions of clopen subsets of the spectrum of above algebra. We will finish this work showing that each element of such algebras described above can be represented as an integral of some continuous function, where the integral has been defined through the spectral measure.

A survey on Lipschitz-free Banach Spaces

This article is a survey of Lipschitz\dywiz free Banach spaces and recent progress in the understanding of their structure. The results we present have been obtained in the last fifteen years (and quite often in the last five years). We give a self\dywiz contained presentation of the basic properties of Lipschitz\dywiz free Banach spaces and investigate some specific topics: non-linear transfer of asymptotic smoothness, approximation properties, norm\dywiz attainment. Section 5 consists mainly of unpublished results. A list of open problems with comentary is provided.

Free Banach spaces and the approximation properties

We characterize the metric spaces whose free space has the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property.

Characterization of compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field

Let \documentclass[12pt]{minimal}\begin{document}$\mathcal {C}$\end{document}C be the complex Levi-Civita field and let E be a free Banach space over \documentclass[12pt]{minimal}\begin{document}$\mathcal {C}$\end{document}C of countable type. Then E is isometrically isomorphic to \documentclass[12pt]{minimal}\begin{document}$c_{0}\left( \mathbb {N},\mathcal {C},s\right)\break :=\left\lbrace (x_{n})_{n\in \mathbb {N}}:x_{n}\in \mathcal {C};\lim _{n\rightarrow \infty }|x_{n} |s(n)=0\right\rbrace$\end{document}c0N,C,s:=(xn)n∈N:xn∈C;limn→∞|xn|s(n)=0, where \documentclass[12pt]{minimal}\begin{document}$s:\mathbb {N}\rightarrow \left( 0,\infty \right) .$\end{document}s:N→0,∞. If the range of s is contained in \documentclass[12pt]{minimal}\begin{document}$\left|\mathcal {C}\setminus \left\lbrace 0\right\rbrace \right|,$\end{document}C∖0, it is enough to study \documentclass[12pt]{minimal}\begin{document}$c_{0}\left( \mathbb {N},\mathcal {C} \right)$\end{document}c0N,C, which will be denoted by \documentclass[12pt]{minimal}\begin{document}$c_{0}(\mathcal {C})$\end{document}c0(C) or, simply, c0. In this paper, we define a natural inner product on c0, which induces the sup-norm of c0. Of course, c0 is not orthomodular, so we characterize those closed subspaces of c0 with an orthonormal complement with respect to this inner product; that is, those closed subspaces M of c0 such that c0 = M ⊕ M⊥. Such a subspace, together with its orthonormal complement, defines a special kind of projection, the so-called normal projection. We present a characterization of such normal projections as well as a characterization of another kind of operators, the compact operators on c0.

C0‐semigroups of linear operators on some ultrametric Banach spaces

C0‐semigroups of linear operators play a crucial role in the solvability of evolution equations in the classical context. This paper is concerned with a brief conceptualization of C0‐semigroups on (ultrametric) free Banach spaces . In contrast with the classical setting, the parameter of a given C0‐semigroup belongs to a clopen ball Ωr of the ground field . As an illustration, we will discuss the solvability of some homogeneous p‐adic differential equations.

The Lipschitz free Banach spaces of $C(K)$-spaces

The aim of this note is to prove that if K is any infinite metric compact space, then the Lipschitz free spaces of C(K) and c 0 are isomorphic. This gives an example of non-Lipschitz-homeomorphic Banach spaces whose free Lipschitz spaces are isomorphic. We also derive some results about Lipschitz homogeneity for Banach spaces, from the results of G. Godefroy and N. J. Kalton on Lipschitz free Banach spaces.

Free Banach-Lie algebras, couniversal Banach-Lie groups, and more

The construction of free Banach-Lie algebra over a normed space enables us to build a connected separable Banach-Lie group of which any other connected separable Banach-Lie group is a quotient. New proofs are given to the result on representabil ity of any Banach-Lie algebra as a quotient of an enlargable Banach-Lie algebra (due to van Est and Swierczkowski) and to the result on representabil ity of any topological group as a quotient of a group with no small subgroups (due to successive efforts of Morris and Thompson, the author, and Sipacheva and Uspenskiϊ). 1. Introduction. Over the last 50 years a number of constructions of universal arrows (see, e.g., [Go]) to the categories of topological algebraic systems have been studied. Important contributions are those by Markov [M], Graev [Gr], and Arhangel'skiϊ [A2] on free topological groups, Mal'cev [Me] on free topological algebras, Arens and Eells [AE], Raϊkov [R], and Uspenskiϊ [U] on free Banach spaces and free locally convex spaces. By virtue of these constructions a first ever example of a non-normal Hausdorff topological group was obtained [M], and the representability of any topological group as a quotient group of a zero-dimensional group was proved [Al]. Here we apply the concept of a free complete normed Lie algebra to theory of topological and Lie groups. Our construction is an extension of the well-known construction of Arens-Eells [AE] to the case of normed Lie algebras. Our main result is that there exists a couniversal separable connected Banach-Lie group, that is, such a separable connected Banach-Lie group that any other such Banach-Lie group is its quotient Lie group. This follows from observation that any free Banach-Lie algebra is enlargable, that is, comes from an approriate Banach-Lie group. Also we give entirely new and rather transparent proofs of two earlier known results. Cohomological technique has enabled van Est and independently Swierczkowski [S2] to prove that any Banach-Lie algebra is a quotient algebra of an enlargable Banach-Lie algebra. Here we deduce the result from enlargability of free Banach-Lie algebras. In his book [Ka] Kaplansky asked whether a quotient group of a

Free Banach spaces and representations of topological groups

Free Banach spaces and representations of topological groups