Research On Some Questions In Effect Algebras

Effect algebras were introduced as a mathematical model of quantum computation by American mathematicians Foulis and Bennett in 1994, which generalized orthomodular lattices. Although the history of this kind of abstract effect algebra is very short, many authors in the fields of mathematics and physics are going in for the study of effect algebras. Some related concepts and methods were well developed in the past few years. Based on existing results, we mainly discuss questions concerning convex σ-effect algebras with sequential product, isomorphisms of Hilbert space effect algebras, Hilbert C*-modules and the related effect algebra. This thesis is divided into three parts.In Chapter 1, we introduce some definitions and basic properties about effect algebra, sequential effect algebra and so on. Subsequently, we give several examples about effect algebras and sequential effect algebras.In Chapter 2, we first study some properties of convex σ-effect algebras with sequential product. Secondly, we introduce a theorem due to Wigner on the symmetry transforms in quantum mechanics. We give several forms of Wigner's theorem and prove the equivalence between them. Applying the fundamental theorem of pro-jective geometry, we give a generalization of Unlhorn's theorem. Finally, applying the generalized Unlhorn's theorem, we give several characterizations of isomorphism between different Hilbert space effect algcbras. We provc that every bijection preserving Jordan triple product between different Hilbert space effect algebras is an isomorphism and every bijection preserving sequential product between different Hilbert space effect algebras is an isomorphism, respectively.In Chapter 3, we first discuss some problems concerning basis in a Hilbert C~*-modules and get some results similar to ones in Hilbert spaces. Then we give several propositions about A-linear operators. Finally, we discuss Hilbert C*-module effect algebras with sequential products and prove several results similar to Hilbert space case.