Compact Operators In C*-algebra Generated By A Weighted Unilateral Shift

In1919,the genius mathematician,or maybe scientist we shall call him, von Neumanninitiated the study of operator algebras.Laterly, he predicted together with Murry that thisalgebra will make a big diference in the theory of unitary representation of groups, quantummathematical formalism, formal integral on operator rings and abstract Ring theory. In1971,Sakai declared that all of this had came to be realized. After Alen Connes constructed hisnoncommutative geometry and topology on the foundation, an increasing number of mathe-maticians were attracted into this fields.As for every C algebra is consisted with operators, which is not a trivial fact, the con-nection between operator and the algebra has always been a interesting part of the theory ofoperator algebras. This article was composed down this line by asking whether there are anycompact operators in a C algebra generated by a single operator.After consider about someexamples and applied a technique invented by professor Youqing,Jiand associated professorSen,Zhu, we obtained the total characterization.