Axiomatic theory for C*-algebras

Abstract: We present an axiomatic frame (in Prt I of this book) in which many results of the K-theory for C*-algebras are proved. Then we construct an example for this axiomatic theory (in Part II), which generalizes the classical theory for C*-algebras. This last theory starts by associating to each C*-algebra F the C*-algebras of square matrices with entries in F. Every such C*-algebra of square matrices can be obtained as the projective representation of a certain group with respect to a complex valued Schur function (also called normalized factor set or multiplier or two-co-cocycle in the mathematical literature) for this group. The above mentioned generalization consist in replacing this Schur function by an arbitrary Schur function which satisfies some axiomatic conditions. Moreover this Schur function can take its values in a commutative unital C*-algebra E. In this case this K-theory deos not apply to the category of C*-algebras but to the category of E-C*-algebras, which are C*-algebras endowed with a supplementary structure obtained by an exterior multiplication with elements of E (every C*-algebra can be endowed with such a supplementary structure). Up to some definitions and notation Part II is independent of Part I.