Tracial Nuclear Dimension Of C~*-algebras

In the year 2008,Toms and Winter proposed a conjecture on the structure theory of C*-algebras,which conjectured that for a infinite dimensional unital simple nuclear C*-algebra A,the following three terms are equivalent to each other:(1)A has finite nuclear dimension;(2)A is Z-stable;(3)A has the strict comparison for positive elements.Many people had made great progress on this conjecture 54,58,51,48,37,22,38],which are deeply revealed the connections between the regularity properties of C*-algebras.This dissertation study the regularity properties of C*-algebras,we give some suffi-cient conditions on when does a C*-algebra is Z-stable.In the first chapter we introduce the background of this dissertation,and introduce some notions and notations that will be used in the following chapters.In the second chapter,by using Cuntz comparison for positive elements,we introduce the concept called Tracial Nuclear Dimension,which is a generalization of the Nuclear Dimension proposed by Winter and Zacharias in their paper[60].In this chapter we show that tracial nuclear dimension carries some useful properties,such as m-comparison,and a result which will be called "tracial covering number" in the third chapter(Theorem 2.4.9).In the third chapter we give the definition of Tracial Covering Number,and show that finite tracial nuclear dimension implies finite tracial covering number.In this chapter we show that with some additional condition,finite tracial covering number(in the unital case)implies tracially Z-absorbing(Theorem3.3.10),and thus reach the strict comparison of positive elements(Corollary3.3.12),moreover,if the C*-algebra is also nuclear,it is even Z-stable(Theorem 3.4.7),and thus finite tracial nuclear dimension and nuclearity implies Z-stability(Theorem 3.4.8).The last part is an appendix,it contains several different definitions of tracial nuclear dimension and some useful properties.