On Some Problems On Effect Algebras

This thesis mainly studies a separable effects on the Hilbert space H and sequential products of a convex effect algebra and effect algebra on a Hilbcrt C~*-module.This thesis consists of four chapters.In chapter 1,we give the notions of effect algebras and sequential effect algcbras.Then we give the propertics of them.We also givc a method by which constructs a new effect algebra from a series of effect algcbras.At last, we introduce the definitions of maps between effect algebras and give some propertics and relations of them.In chapter 2,we mainly study the effects on a Hilbert space H.We charactcrize the structure of a bijcctivc map preserving Jordan triplc products on effect algebras of two different Hilbert space and prove that such a mapφhas the formφ(A)= UAU~*,where U is unitary or anti unitary operator.And also prove that a bijective map which prescrvcs scqucntial products in an isomorphism. In this chapter,we introduce thc notion of sequentially independent of n effects on Hilbert space H,and show that n effects on Hilbert space H are sequentially independent if and only if they arc pcrmutable.In chapter 3,we study some problems on convex effect algebras,we prove multiplication of scalars preserves the order on convex effect algebra. Using the property of Archimcdcan on convexσ-effect algebra,we prove that scqucntial products on a convexσ-effect algebra satisfy connect law of scalar multiplication.Finally,we show that thc restriction of cach bijcctivc sequential products of a convcxσ-effect algebra to the sharp elements is also a preserving binary and bijective map.In chapter 4,we introduce the notion of the effect algebra acting on Hilbert C~*-module,and study the scquential products onεA(H)and get some similar results to the effect algebraε(H)'s on Hilbwrt space H.