On EP Elements In C~*-algebras And Related Topics

This thesis mainly studies EP elements in C*-algebras and characterizes their related properties.The full thesis is divided into the following four parts:In the first part,we introduce the historical background and development status of MoorePenrose inverse elements and EP elements,as well as the research results of domestic and foreign scholars in matrix algebra,ring theory,C*-algebras and other fields.Then we give the main results of this thesis.The second part is the Preliminaries,introducing the definitions of Moore-Penrose invertible elements,EP elements,hypo-EP elements,and group invertible elements in C*algebras,and providing the lemmas used in the proof process of this thesis.In the third part,we mainly study the properties of Moore-Penrose invertible elements,EP elements,and hypo-EP elements in C*-algebras,and discuss EP elements and hypo-EP elements by using the properties of C*-algebra.By means of the left and right support and polar decomposition theory in the envelope von Neumann algebra of C*-algebras,we study the relationship between Moore-Penrose invertible elements,EP elements and hypo-EP elements in C*-algebras.Firstly,we prove that the Moore-Penrose invertibility and EP invertibility of an element are independent of the C*-algebras in which the element is located,and give some characterization methods of EP elements and hypo-EP elements in C*-algebras.Secondly,according to the definition of y(a)in C*-algebras,combined with the properties of partial isometry and group invertible elements,we show that if there exists a non-zero element a in a unital C*-algebra that is group invertible and satisfies that γ(a)=‖a‖,a*an=ana*,then a is an EP element.Finally,we discuss how to construct invertible elements in terms of Moore-Penrose invertible elements.In the fourth part,we mainly discuss the properties of EP-spectrum in C*-algebras and study how to characterize finite-dimensional C*-algebras in terms of EP elements.Firstly,by means of the definition of invertible elements in Banach algebras,we introduce the definitions of EP-resolvent sets and EP-spectrum in C*-algebra,and generalize the related conclusions of spectra in C*-algebras.Then,using the conclusion that there must be positive elements of infinite spectrum in infinite-dimensional C*-algebras and the properties of Moore-Penrose invertible elements and EP elements,we prove that every element in a C*-algebra is MoorePenrose invertible if and only if this C*-algebra is finite-dimensional;every element in a C*algebra is EP if and only if this C*-algebra is finite-dimensional and commutative,if and only if this C*-algebra is*-isomorphic to Cn.