Lattice norms on the unitization of a truncated normed Riesz space

Positivity - Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first...     

Extensions of orthosymmetric lattice bimorphisms revisited

This note furnishes an example illustrating the following two facts. On the one hand, there exist Archimedean Riesz spaces and with Dedekind-complete and an orthosymmetric lattice bimorphism with lattice bimorphism extension which is not orthosymmetric, where denotes the Dedekind-completion of . On the other hand, there is an associative -multiplication in the same Archimedean Riesz space which extends to a -multiplication in which is not associative. The existence of such an example provides counterexamples to assertions in Toumi, 2005.

     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.     

Some aspects of Riesz multimorphisms

We study the behavior of Riesz multimorphisms on Archimedean f-algebras with unit element and focus on their different multiplicative aspects.     

A relationship between two almost f-algebra products

Let A be an uniformly complete almost f-algebra. Then is a positively generated ordered vector subspace of A with as a positive cone. If is a positive linear operator, we put the linear operator defined by with for all is the algebra of all order bounded linear operators of A). Let denote the range of and let's define a new product by putting for all . It is easily checked that if then , this shows that if it happens that the product is associative then A is an almost f- algebra with respect to this new product. It turns out that a necessarily and sufficient condition in order that be an associative product is that is a commutative subalgebra of . We find necessarily and sufficient conditions on T in order that is an almost f-algebra (respect.; d-algebra, f-algebra) product. Such conditions are described in terms of the algebraic and order structure of the algebra .?The converse problem is also studied. More precisely, let A be an uniformly complete almost f-algebra and assume that is another almost f-algebra product on A. The aim is to find sufficient conditions in order that there exist such that for all . It will be showed that a sufficient condition is that A is a d-algebra with respect to the initial product. An example is produced which shows that the condition "A is a d-algebra with respect to the initial product" can not be weakened. Received November 8, 1999; accepted in final form February 14, 2000.     

A generalization of a theorem by Bigard–Keimel and Conrad–Diem

Let G be an archimedean ℓ-group and \mathfrakP(G){\mathfrak{P}(G)} denote the set of all polar preserving bounded group endomorphisms of G. Bigard and Keimel in [Bull. Soc. Math. France 97 (1969), 381–398] and, independently, Conrad and Diem in [Illinois J. Math. 15 (1971), 222–240] proved that \mathfrakP(G){\mathfrak{P}(G)} is an archimedean ℓ-group with respect to the pointwise addition and ordering. This classical result is extended in this paper to certain sets of disjointness preserving bounded homomorphisms on archimedean ℓ-groups.     

The order bidual of <Emphasis Type="Italic">f</Emphasis> -algebras revisited

It is shown that the order bidual X ^~~ of an Archimedean semiprime f -algebra X has a unit element for the Arens multiplication if and only if every positive linear functional on X extends to a positive linear functional on the f -algebra Orth (X) of all orthomorphisms on X.