Thesis (Ph. D.)--State University of New York at Binghamton, Mathematical Sciences Department, 2006. / Includes bibliographical references.

The problem undertaken in this paper is to determine what the algebraic structure of the class of C-sets is, when the notion of sum is to be the "set sum. " While the preliminary work done by Appling took place in the space of additive and bounded real valued functions, the results here are found in the more general setting of a complete lattice ordered group. As a conseque n c e , G . Birkhof f' s book, Lattice Theory, is used as the standard reference for most of the terminology used in the paper. The direction taken is prompted by a paper by W. D. L. Appling, "A Generalization of Absolute Continuity and of an Analogue of the Lebesgue Decomposition Theorem. " Since some of the results obtained provide another approach to a problem originally studied by Nakano, and improved upon by Bernau, reference is made to their work to provide other terminology and examples of alternative approaches to the problem of lateral completion. Thus Chapter I contains a brief history of the notion of C-sets and their relationship to lattice ordered groups, along with a summary of the properties of lattice ordered groups needed for later developments. In addition, several results in the general theory of lattice ordered groups are cited to provide insight into the comparability of the assumptions that will ultimately be made about the groups. Chapter II begins with the axiomatization of the collection of nearest point functions" for the closed A-ideals of the cone of a complete lattice ordered group. The basic results in the chapter establish that the functions defined do indeed characterize the complete A-ideals, and that the maps have a 'nearest point property." The maps are then extended to the entire group and shown to correspond to the "nearest point maps" for a C-set in PAB' Chapter III is devoted to exploring the algebraic structures found in the collection of all closed A-ideal maps, denoted J. J is shown to be a lattice ordered monoid, abelian and complete, containing a maximal group cone P*. It is further shown that the original group cone P is isomorphic to a subset of P*. Chapter IV looks into a rather interesting characterization of P*, one that, in the terminology of Bernau, implies that P* is the cone of the group that is the lateral completion of the original group. A final result is a demonstration that the members of j each have a representation as the sum of an element of P* and an additive element of j.

Thesis (M.S.)--University of Texas at El Paso, 2009. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

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Thesis (Ph. D. in Mathematics)--Vanderbilt University, 2003. / Title from PDF title screen. Includes bibliographical references and index.

Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) Let (G,G+) be a quasi-lattice-ordered group with positive cone G+ Laca and Raeburn have shown that the universal C*-algebra C*(G,G+)introduced by Nica is a crossed product BG+ Xɑ G+ by a semigroup of endomorphisms. Subsequent research centered on totally ordered abelian groups. We generalize the results in [2], [3] and [5] to extend it to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G+ we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly a-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) XɑG+ and B(G/H)+ XβG+. This leads to one of our main results that B(G/H)+ XβG+ is realized as an induced C*-algebra IndG-H (B(G/H+ Xt(G/H)+). Then we use this result to show the existence of the following short exact sequence of C*-algebras 0-IH+ XɑG+ → BG+ XɑG+ → IndG-H (B(G/H+ Xt(G/H)+) → 0. This leads to show that the ideal IH+ XɑG+ is generated by {iBG+(1-1u):u∊H+} and therefore contained in the commutator ideal CG of the C*-algebra BG+ XɑG+. Moreover, we use our short exact sequence to study the primitive ideals of the C* algebra BG+ XɑG+ which is isomorphic to the Toeplitz albebra T(G) of G.

Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) Let (G,G+) be a quasi-lattice-ordered group with positive cone G+ Laca and Raeburn have shown that the universal C*-algebra C*(G,G+)introduced by Nica is a crossed product BG+ Xɑ G+ by a semigroup of endomorphisms. Subsequent research centered on totally ordered abelian groups. We generalize the results in [2], [3] and [5] to extend it to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G+ we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly a-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) XɑG+ and B(G/H)+ XβG+. This leads to one of our main results that B(G/H)+ XβG+ is realized as an induced C*-algebra IndG-H (B(G/H+ Xt(G/H)+). Then we use this result to show the existence of the following short exact sequence of C*-algebras 0-IH+ XɑG+ → BG+ XɑG+ → IndG-H (B(G/H+ Xt(G/H)+) → 0. This leads to show that the ideal IH+ XɑG+ is generated by {iBG+(1-1u):u∊H+} and therefore contained in the commutator ideal CG of the C*-algebra BG+ XɑG+. Moreover, we use our short exact sequence to study the primitive ideals of the C* algebra BG+ XɑG+ which is isomorphic to the Toeplitz albebra T(G) of G.