Domination properties of lattice homomorphisms

Let E and F be Banach lattices, T, K : E → F be such that 0 ≤ T ≤ K and T is either a lattice homomorphism or almost interval-preserving. In this paper we will deduce that (1) If K is AM-compact then T also is AM-compact; (2) If either E′ or F has an order continuous norm and K is compact, then T is compact as well; (3) If K is weakly compact then so is T. Some related results are also obtained.      

Laterally closed lattice homomorphisms

Let A and B be two Archimedean vector lattices and let be a lattice homomorphism. We call that T is laterally closed if T(D) is a maximal orthogonal system in the band generated by T(A) in B, for each maximal orthogonal system D of A. In this paper we prove that any laterally closed lattice homomorphism T of an Archimedean vector lattice A with universal completion Au into a universally complete vector lattice B can be extended to a lattice homomorphism of Au into B, which is an improvement of a result of M. Duhoux and M. Meyer [M. Duhoux and M. Meyer, Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Bruxelles 98 (1984) 3-18], who established it for the order continuous lattice homomorphism case. Moreover, if in addition Au and B are with point separating order duals ^′(Au) and B^′ respectively, then the laterally closedness property becomes a necessary and sufficient condition for any lattice homomorphism to have a similar extension to the whole Au. As an application, we give a new representation theorem for laterally closed d-algebras from which we infer the existence of d-algebra multiplications on the universal completions of d-algebras.     

Extensions of orthosymmetric lattice bimorphisms

Let be an Archimedean vector lattice, let be its Dedekind completion and let be a Dedekind complete vector lattice. If is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism that not just extends but also has to be orthosymmetric. As an application, we prove the following: Let be an Archimedean -algebra. Then the multiplication in can be extended to a multiplication in , the Dedekind completion of , in such a fashion that is again a -algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

     

Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space E there is a complex Banach lattice ${\text{FBL}}_{\mathbb{C}}[E]$ containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice $X_{\mathbb{C}}$, every operator $T:E\to X_{\mathbb{C}}$ admits a unique lattice homomorphic extension $\stackrel{?}{T}:{\text{FBL}}_{\mathbb{C}}[E]\to X_{\mathbb{C}}$ with $6\stackrel{?}{T}6=6T6$. The free complex Banach lattice ${\text{FBL}}_{\mathbb{C}}[E]$ is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces E and F can be given so that ${\text{FBL}}_{\mathbb{C}}[E]$ and ${\text{FBL}}_{\mathbb{C}}[F]$ are lattice isometric. The spectral theory of induced lattice homomorphisms on ${\text{FBL}}_{\mathbb{C}}[E]$ is also explored.     

The complexity of locally injective homomorphisms

A homomorphism f:G→H, from a digraph G to a digraph H, is locally injective if the restriction of f to N⁻(v) is an injective mapping, for each v∈V(G). The problem of deciding whether such an f exists is known as the injective H-colouring problem (INJ-HOM_H). In this paper, we classify the problem INJ-HOM_H as being either a problem in P or a problem that is NP-complete. This is done in the case where H is a reflexive digraph (i.e. H has a loop at every vertex) and in the case where H is an irreflexive tournament. A full classification in the irreflexive case seems hard, and we provide some evidence as to why this may be the case.     

Weakly compact homomorphisms of regular Banach algebras

Let A be a uniformly regular Ditkin algebra. It is shown that every weakly compact homomorphism of A into a Banach algebra is finite-dimensional.     

序同态的NC－连续性

本文在ＬＦ拓扑空间上利用良紧性作为背景引进了序同态的ＮＣ－连续性、分子网和理想的ＮＣ－收敛性及ＮＣ－闭包算子等概念，系统地讨论了这些概念之间的关系以及ＮＣ－连续序同态的特性，得到了若干重要的结果。     

A structure of ring homomorphisms on commutative Banach algebras

We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that:

(i)

A ring homomorphism of a commutative -algebra onto another commutative -algebra with connected infinite Gelfand space is either linear or anti-linear.

(ii)

A ring automorphism of is either linear or anti-linear.

(iii)

, and the disc algebra are neither ring homomorphic images of nor .

     

Compact homomorphisms of URM algebras

We show when a homomorphism from a URM algebra into a uniform algebra or into a regular Banach algebra is weakly compact or compact. We prove that every homomorphism from URM algebras into is compact. Finally, we characterize the spectra of compact endomorphisms of URM algebras defined on a connected compact Hausdorff space .

     

Riesz代数上的d-模(英文)

本文引入了Riesz代数上d-模的概念.利用正算子理论讨论了可交换的Riesz代数上的d-模的二次共轭空间的d-模结构,并且研究了由格同态算子或区间保持算子产生的主理想上的特殊d-模.     

An algorithm for the number of path homomorphisms

A homomorphism of a graph G₁=(V₁,E₁) to a graph G₂=(V₂,E₂) is a mapping from the vertex set V₁ of G₁ to the vertex set V₂ of G₂ which preserves edges. In this paper we provide an algorithm to determine the number of homomorphisms from an arbitrary finite undirected path to another arbitrary finite undirected path.     

Characterizations of Jordan derivations and Jordan homomorphisms

Let 𝒜 be a unital Banach algebra and ? be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ? satisfying δ(ST)?=?δ(S)T?+Sδ(T) for any S, T?∈?𝒜 with ST?=?W, where W is a left or right separating point of ?, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ? which satisfies h(S)h(T)?=?h(ST) for any S,?T?∈?𝒜 with ST?=?W is a Jordan homomorphism if h(W) is a separating point of ?.     

Sequentially continuous homomorphisms on products of topological groups

Sequentiality of products of topological groups is investigated.     

格上点式p.q.度量的乘性

本文在可数多个分子格的乘积上构造了一个乘积点式p·q·度量函数,从而证明了点式p·q·度量分子格的可数积是可点式p·q·度量化的.     

A new proof of D. Popescu's theorem on smoothing of ring homomorphisms

We give a new proof of D. Popescu's theorem which says that if is a regular homomorphism of noetherian rings, then is a filtered inductive limit of smooth finite type -algebras. We strengthen Popescu's theorem in two ways. First, we show that a finite type -algebra , mapping to , has a desingularization which is smooth wherever possible (roughly speaking, above the smooth locus of ). Secondly, we give sufficient conditions for to be a filtered inductive limit of its smooth finite type -subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.

     

The complex of maximal lattice free simplices

The simplicial complexK(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Axb), withA a fixed generic (n + 1) ×n matrix. The topological space associated withK(A) is shown to be homeomorphic to ⁿ , and the space obtained by identifying lattice translates of these simplices is homeorphic to then-torus.Corresponding author.The first author was partially supported by Hungarian NSF grants 1907 and 1909, and also by U.S. NSF grant CCR-9111491. The research of the second author was supported by DMS9103608 and the third author by NSF grant SES9121936.