On lattice ordered algebras
Başlık çevirisi mevcut değil.
- Tez No: 400642
- Danışmanlar: DR. KEITH ROWLANDS
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2001
- Dil: İngilizce
- Üniversite: Aberystwyth Unıversıty
- Enstitü: Prıfysgol Aberystwyth
- Ana Bilim Dalı: Yurtdışı Enstitü
- Bilim Dalı: Matematik Ana Bilim Dalı
- Sayfa Sayısı: Belirtilmemiş.
Özet
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Özet (Çeviri)
In the first chapter we establish extension theorems for additive mappings (p : A+ x B+ \-t G+, where A, B are lattice ordered spaces (^-spaces) and C is an order complete ^-space, to the whole of A x J5, thereby extending well-known results for additive mappings between ^-spaces. Thereafter we are mainly concerned with lattice ordered algebras (^-algebras) and mappings on them. Chapter 2 introduces the classes of ^-algebras we are concerned with, their fundamental properties and the relations between them. In particular, the class of /-algebras first appeared in a paper by Birkhoff and Pierce [16] in 1956, to be followed a decade later by the class of almost /-algebras introduced by Birkhoff [15]. Prior to that, in 1962 Kudlacek [38] introduced the notion of d-algebras, and now more than three decades later we introduce the notion of r-algebras, which appears to be a new class of ^-algebras. In Chapter 3 we study the order bidual of these algebras. In particular, by using the methods introduced by Bernau and Huijsmans in [14], we prove that the order continuous bidual of an Archimedean r-algebra is again an Archimedean r-algebra. We also consider the nilpotent elements of the bidual of an /-algebra. This section owes much to the papers of Huijmans and Pagter [29], and Huijmans [26]. We clarify their arguments wherever possible and note in particular how the notion of an orthomorphism on an ^-space yields (a) characterizations for the nilpotent elements of the bidual, and (b) criteria for the bidual to be semi-prime. In Chapter 4 we introduce a new concept, that of a quasi-orthomorphism on an £-space, which generalizes the notion of an orthomorphism. The orthomorphisms on an Archimedean semi-prime /-algebra A are precisely the multipliers on A. It is therefore natural to seek relationships between quasi-orthomorphisms and quasi-multipliers. The latter were first introduced by Akemann and Pedersen in [1, §4], and the theory of quasi-multipliers on a Banach algebra with a bounded approximate identity has subsequently been developed by McKennon [45], Vasudevan and Goel [58, 59], Kassem and Rowlands [35], and Argun and Rowlands [10]. It turns out that, for an order complete Banach /-algebra A with a norm approximate identity, the quasi-orthomorphisms on A are precisely the quasi-multipliers. We also show that, under certain conditions, the algebra of quasi-orthomorphisms on A is ^-isomorphic to the algebra C(K) of continuous real-valued functions on some compact Hausdorff space K.
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