Semigroup Of Non-commutative Harmonic Analysis And Some C ~ * - Algebras

The paper consists of two parts..We study noncommutitive harmonic analysis on semigroups in first part. We mainly consider completely monotone functions from semigroup to operator algebras. We prove the equivalent conditions of completely monotone function, we also discuss the relationship of positive definite function, semi-positive definite function and completely monotone function when the function take values in a commutitive von Neumann algebras.A classification of certain separable C*-algebras of real rank zero with trivial Ki-group is given in second part. We construct all the extensions of Cuntz algebras by the compact operators, and compute their K-theory. The C*-algebras considered are those that can be expressed as inductive limits of matrix algebras, matrix algebras over Cuntz algebras, matrix algebras over extensions of Cuntz algebras, and their hereditary C*-subalgebras. C*-algebras in the class are not necessary simple. They are, in general, neither finite nor purely infinite. However, the class includes all AF-algebras. and all Kirchberg algebras with UCT and trivial K1-group. It is closed under quotients, inductive limits and tensor products with AF-algebras. The invariant (V(A), [1A]) that we used for unital case is the semigroup of Murry-von Neumann equivalence classes of projections in matrices over C*-algebras together with the class of the unit.