Representations and positive functionals

We discuss the representation theory of both the locally convex and non-locally convex topological^*-algebras. First we discuss the^*-representation of topological^*-algebras by operators on a Hilbert space. Then we study those topological^*-algebras so that every^*-representation of which on a Hilbert space is necessarily continuous. It is well-known that each^*-representation of aB ^*-algebra on a Hilbert space is continuous. We show that this is true for a large class of^*-algebras more general thanB ^*-algebras, including certain non-locally convex^*-algebras. Finally, we investigate the conditions under which a positive functional on a topological^*-algebra is representable.The research of the first-named author was partially supported by an NSERC grant. This work was done by the second-named author when he was a post-doctoral fellow at McMaster University.     

Weakly almost periodic functionals on certain Banach algebras

For a Lau algebra A, we study the Banach space WAP(A) of all weakly almost periodic functionals on A to obtain some equivalent conditions for the existence of topological left invariant means on a topological left introverted subspace X of A contained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean on X and a common fixed point property.     

Continuity of positive functionals on topological*-algebras

To investigate the continuity of positive functionals on certain topological^*-algebras, we consider general locally convex and non-locally convex topological^*-algebras which need not have identity or continuity of involution. We generalize a number of known results.The research of the first-named author was partially supported by an NSERC grant. The research of the second-named author was carried on when the was a post-doctoral fellow at McMaster University.     

Positive one-parameter semigroups on ordered banach spaces

In this review we describe the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces. The review is in two parts.First we discuss the general structure of ordered Banach spaces and their ordered duals. We examine normality and generation properties of the cones of positive elements with particular emphasis on monotone properties of the norm. The special cases of Banach lattices, order-unit spaces, and base-norm spaces, are also examined.Second we develop the theory of positive strongly continuous semigroups on ordered Banach spaces, and positive weak^*-continuous semigroups on the dual spaces. Initially we derive analogues of the Feller-Miyadera-Phillips and Hille-Yosida theorems on generation of positive semigroups. Subsequently we analyse strict positivity, irreducibility, and spectral properties, in parallel with the Perron-Frobenius theory of positive matrices.     

Representations of Archimedean Riesz spaces by continuous functions

A brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces     

On the variety of Riesz spaces

Finitely generated linearly ordered Riesz spaces are described, leading to a proof that the variety of Riesz spaces is generated as a quasivariety by the Riesz space ? of real numbers. The finitely generated Riesz spaces are also described: they are the subalgebras of real-valued function spaces on root systems of finite height.     

On certain properties of the spaces of order-preserving functionals

In the present paper it is proved that the functor Oτ of τ-smooth order preserving functionals and the functor OR of Radon order preserving functionals, do not change the weight of infinite Tychonoff spaces. It is shown that the density and the weak density of infinite Tychonoff spaces do not increase under these functors. Moreover, if X is a metric space with the second axiom of countability then the spaces Oτ(X) and OR(X) are also metrizable.     

A remark on the representation of vector lattices as spaces of continuous real-valued functions

The well-known Ogasawara-Maeda-Vulikh representation theorem asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space , unique up to a homeomorphism, such that L can be represented isomorphically as an order dense vector sublattice of the universally complete vector lattice C () of all extended-real-valued continuous functions f on for which % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaacmqabaGaeqyYdCNaeyicI4SaeyyQdCLaeyOoaOJaaiiFaiab% gkzaMkabgIcaOiabgM8a3jabgMcaPiaacYhacqGH9aqpcqGHEisPai% aawUhacaGL9baaaaa!4E05!\[\left\{ {\omega \in \Omega :|f(\omega )| = \infty } \right\}\] is nowhere dense. Since the early days of using this representation it has been important to find conditions on L such that consists of bounded functions only.The aim of this short article is to present a simple complete characterization of such vector lattices.     

Some new classes of Banach-Mackey spaces

Some weakenings of property (K) of Antosik for locally convex spaces are introduced: local property (K) and, for spaces with Schauder-type decompositions, two variants of property (K) defined in terms of block- and tail-sequences. It is shown that if a space enjoys any of these new properties, then it is Banach-Mackey. An application to the barrelledness of the spaces of Pettis integrable functions is given, and examples are provided to distinguish between the various K-type properties.     

Boundedness and continuity of additive and convex functionals

Summary It is shown that for any real Baire topological vector spaceX the set classesA(X):={T : for any open and convex setD T, every Jensen-convex functional, defined onD and bounded from above onT, is continuous} andB(X):={T : every additive functional onX, bounded from above onT, is continuous} are equal. This generalizes a result of Marcin E. Kuczma (1970) who has shown the equalityA( ⁿ )=B( ⁿ ) However, the infinite dimensional case requires completely different methods; therefore, even in the caseX = ⁿ we obtain a new (and perhaps simpler) proof than that given by M. E. Kuczma.     

Multipliers on matrix ordered operator spaces and some K-groups

Matrix ordered operator spaces are ‘non-commutative Banach spaces equipped with a non-commutative order’. Examples include C*-algebras as well as their duals. In this article, we define and intrinsically characterize the multiplier algebra for this class of spaces and briefly tackle the problem of extending K-theory to this context.     

Ranges of positive contractive projections in Nakano spaces

We show that in any nontrivial Nakano space X=L_p(·) (Ω, Σ, μ) with essentially bounded random exponent function p(·), the range Y = R(P) of a positive contractive projection P is itself representable as a Nakano space L_pY(·) (Ω_Y Σ_Y, ν_Y), for a certain measurable set Y_Ω⊆Ω (the support of the range), a certain sub-sigma ring Y_Σ⊆Σ (with maximal element Ω_Y) naturally determined by the lattice structure of Y, and a semi-finite measure ν_Y, namely the restriction of some measure Ω on E which is equivalent to μ. Furthermore, we show that the random exponent p_Y(·) associated with such a range can be taken to be the restriction to Ω_Y of the random exponent p(·) (this restriction turns out to be Σ_Y-measurable). As an application of this result, we find Banach lattice isometric characterizations of suitable classes of Nakano spaces. These classes are defined in terms of an important lattice-isometric invariant of Nakano spaces, the essential range of the variable exponent.     

Some fundamental properties for duals of Orlicz spaces

In this paper some basic properties of Orlicz spaces are extended to their dual spaces, and finally, criteria for extreme points in these spaces are given.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

Representation of a class of locally convex (M)-lattices

We prove a representation theorem for Hausdorff locally convex (M)-lattices which are Dedekind σ-complete, and whose topologies are order σ-continuous and monotonically complete. These turn out to be the weighted spaces c₀(T, H), defined in the paper for T ≠ and H ^T₊. We also characterize the dual of c₀(T, H), as the space l¹ (T, H) defined in the last section. The known representation (on c₀(T)) of Banach (M)-lattices with order continuous norm follows as a particular case. We obtain these results by first proving a new general isomorphism theorem, which seems to be of independent interest. Our notion of “monotonic topological completeness” is weaker than the usual completeness and seems to be very convenient in the framework of topological ordered vector spaces.     

On stability of the Cauchy equation on semigroups

Summary We say that Hyers's theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S such that the set {f(x + y) – f(x) – f(y): x, y S} is bounded, there exists an additive functiona:S for which the functionf – a is bounded.Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi. Theorem.Suppose that Hyers's theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S X is a function for which the mapping (x, y) F(x + y) – F(x) – F(y) is bounded, then there exists an additive function A : S X such that F — A is bounded.     

Holomorphy in convergence spaces

We introduce a new approach to infinite dimensional holomorphy. Cast in the setting of closed-embedded linear convergence spaces and based on a categorical definition of derivative, our theory applies beyond the traditional open domains. It reaches certain domains with empty interior (that arise naturally in Fréchet spaces) and gives a fully fledged differential calculus. On open domains our approach provides a new characterization of holomorphic maps. Thus earlier theories become expanded as well as strengthened.NSERC aided     

Hermite functions and weighted spaces of generalized functions

We prove that the Hermite functions are an absolute Schauder basis for many weighted spaces of (ultra)differentiable functions and ultradistributions including the space of Fourier hyperfunctions. The coefficient spaces are also determined. Dedicated to Professor H.-G. Tillmann on the occasion of his 80th birthday