Banach-Stone Theorem Research Articles

Weighted composition operators and the Gleason?Kahane?Zelazko theorem

The main purpose of this paper is to investigate extensions of the Banach–Stone theorem and of the Holsztynski theorem to some locally multiplicatively convex associative algebras. The proofs are based on a generalization of a theorem, due to A. Gleason, J.-P. Kahane and W. Zelazko, characterizing continuous characters of a unital associative Banach algebra among all its linear forms.

Continuous functions with compact support

The main aim of this paper is to investigate a subring of the ring of continuous functions on a topological space X with values in a linearly ordered field F equipped with its order topology, namely the ring of continuous functions with compact support. Unless X is compact, these rings are commutative rings without unity. However, unlike many other commutative rings without unity, these rings turn out to have some nice properties, essentially in determining the property of X being locally compact non-compact or the property of X being nowhere locally compact. Also, one can associate with these rings a topological space resembling the structure space of a commutative ring with unity, such that the classical Banach Stone Theorem can be generalized to the case when the range field is that of the reals.

Constructing Banaschewski compactification without Dedekind completeness axiom

The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero‐dimensional Hausdorff topological space as a structure space of a ring of ordered field‐valued continuous functions on the space, and thereby exhibit the independence of the construction from any completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone′s theorem, the Banach‐Stone theorem, and the Gelfand‐Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all zero‐dimensional but not strongly zero‐dimensional Hausdorff topological spaces into three classes that are labeled by inequalities between three compactifications of X, namely, the Stone‐Čech compactification βX, the Banaschewski compactification β0X, and the structure space 𝔐X,F of the lattice‐ordered commutative ring ℭ(X, F) of all continuous functions on X taking values in the ordered field F, equipped with its order topology. Some open problems are also stated.

Some Variations on the Banach-Stone Theorem

Some Variations on the Banach-Stone Theorem

Extreme Point Methods and Banach-Stone Theorems

AbstractAn operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of C0( Q, X) into C0(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.

Homomorphisms on Function Lattices

In this paper we study real lattice homomorphisms on a unital vector lattice \({{\cal L}} \subset {\cal C}(X)\), where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice \({{\cal L}} = {\rm Lip} (X)\) of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case \({{\cal L}} = {\rm Lip}^{\ast} (X)\) of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces.

A Lattice-valued Banach-Stone Theorem

Let X and Y be compact Hausdorff spaces, and let E be a Banach lattice. In this short note, we show that if there exists a Riesz isomorphismΦ:C(X, E) →C(Y, R) such that Φ(f) has no zeros if f has none, then X is homeomorphic to Y and E is Riesz isomorphic to R.

AN ALGEBRAIC APPROACH TO THE BANACH-STONE THEOREM FOR SEPARATING LINEAR BIJECTIONS

Let $X$ be a compact Hausdorff space and $C(X)$ the space of continuous functions defined on $X$. There are three versions of the Banach-Stone theorem. They assert that the Banach space geometry, the ring structure, and the lattice structure of $C(X)$ determine the topological structure of $X$, respectively. In particular, the lattice version states that every disjointness preserving linear bijection $T$ from $C(X)$ onto $C(Y)$ is a weighted composition operator $Tf=h\cdot f\circ\varphi$ which provides a homeomorphism $\varphi$ from $Y$ onto $X$. In this note, we manage to use basically algebraic arguments to give this lattice version a short new proof. In this way, all three versions of the Banach-Stone theorem are unified in an algebraic framework such that different isomorphisms preserve different ideal structures of $C(X)$.

ℕ-compactness and weighted composition maps

In this paper we study some conditions on (not necessarily continuous) linear maps T T between spaces of real- or complex-valued continuous functions C ( X ) C (X) and C ( Y ) C (Y) which allow us to describe them as weighted composition maps. This description depends strongly on the topology in X X ; namely, it can be given when X X is N \mathbb {N} -compact, but cannot in general if some kind of connectedness on X X is assumed. Finally we also give an infimum-preserving version of the Banach-Stone theorem. The results are also proved for spaces of bounded continuous functions when K \mathbb {K} is a field endowed with a nonarchimedean valuation and it is not locally compact.

Isometries of spaces of unbounded continuous functions

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

A Banach-Stone Theorem for Uniformly Continuous Functions

In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X.

A new proof of the Banach-stone theorem

Banach's original proof of the Banach-Stone theorem for compact metric spaces uses peak functions, that is, continuous functions which assume their norm in just one point. We show by using nonstandard methods that the peak point approach also works for compact Hausdorff spaces. The peak functions are replaced by internal functions whose standard part is supported in one monad.

A Cartan theorem for Banach algebras

Let A A be a semisimple Banach algebra, and let Ω A \Omega _A be its spectral unit ball. We show that every holomorphic map G : Ω A → Ω A G\colon \Omega _A\to \Omega _A satisfying G ( 0 ) = 0 G(0)=0 and G ′ ( 0 ) = I G’(0)=I fixes those elements of Ω A \Omega _A which belong to the centre of A A , but not necessarily any others. Using this, we deduce that the automorphisms of Ω A \Omega _A all leave the centre invariant. As a further application, we give a new proof of Nagasawa’s generalization of the Banach-Stone theorem.

A Banach-Stone theorem for a new class of Banach spaces

A Banach-Stone theorem for a new class of Banach spaces

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH −1 are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex.

The cancellation law for compact Hausdorff spaces and vector-valued Banach-Stone theorems

The cancellation law for compact Hausdorff spaces and vector-valued Banach-Stone theorems

Algebraic Characterization of Compact Abelian Groups

The well-known Banach-Stone theorem gives a nice algebraic characterization of compact Hausdorff topological spaces. One might wonder if the topological spaces are also endowed with a (compatible) group structure whether one can obtain a similar characterization for them to be the same, i.e., to be topologically isomorphic. The purpose of this note is to give one such characterization which doesn't seem to have made its way into the literature yet. We begin by stating the result of Banach [1, p. 170] and Stone [7, p. 469] in a somewhat different form than the traditional version.

Affine geometric proofs of the Banach Stone theorems of Kadison and Kaup

This approach was abandoned by Kadison because “the sparseness of knowledge concerning the pure states of an operator algebra makes this procedure seem difficult” [14, p. 326]. Instead he gives an intrinsic proof, depending mainly on spectral theory and the geometry of the underlying Hilbert spaces on which A and B act. Kadison points out that ρ preserves the quantum mechanical structure of the C∗-algebras, i.e., the linear structure and the power structure of self-adjoint elements. It follows, and this is significant for the viewpoint expressed in this paper, that T , given by (1.1), preserves powers of the form aa∗a, aa∗aa∗a, . . . , and hence, by polarization, that T preserves the triple product ab∗c+ cb∗a.

Isometries of a Space of Continuous Functions Determined by an Involution

Let C(X, σ) be the space of all σ-conjugate continuous complex functions, where σ is an involution on X. Surjective isometries of such spaces are investigated and a generalization of the Banach-Stone theorem is proved.

Some affine geometric aspects of operator algebras

The predual of a von Neumann algebra is shown to be a neutral strongly facially symmetric space, thereby suggesting an affine geometric approach to operator algebras and their non-associative analogues. Geometric proofs are obtained for the polar decompositions of normal functionals in ordered and non-ordered settings. A fundamental problem in the operator algebraic approach to quantum mechanics is to determine those algebraic structures in Banach spaces which are characterized by a set of geometrical axioms defining the quantum mechanical measuring process. This problem was solved in the context of ordered Banach spaces by Alfsen, Hanche-Olsen, and Shultz ([2], [1]) and led to a characterization of the state spaces of /2?*-algebras and C*-algebras. The main thrust of the present authors' recent research has been to find those algebraic structures induced on (unordered) Banach spaces in which such quantum mechanical axioms are satisfied. This project, which was initiated in [14] and [15] using the affine geometry of the dual unit ball, is used here to give a geometric proof of the Tomita-Sakai-Effros polar decomposition of a normal functional on a von Neumann algebra. Thus, the purpose of this partially expository paper is to show the richness and power of the affine geometric structure of the dual space of an operator algebra, by working in a purely geometric model. Indeed, since this geometry can be described in terms of the underlying real structure, it can be used to obtain new results in the real structure of operator algebras and in the structure of real operator algebras. For example, by using this approach, Dang ([5]) has shown that a raz/-linear isometry of a C*-algebra is the sum of a linear and a conjugate linear isometry, and hence is multiplicative, thereby obtaining a real analogue of Kadison's non-commutative extension of the BanachStone Theorem. The category of strongly facially symmetric (SFS) spaces (simply called facially symmetric spaces in [14] and [15]) has been shown to