functors induced by cauchy extension of c∗-algebras

In this paper, we give three functors p, [⋅]k and f on the category of c∗-algebras. the functor p assigns to each c∗-algebra a a pre-c∗-algebra p(a) with completion [a]k. the functor [⋅]k assigns to each c∗-algebra a the cauchy extension [a]k of a by a non-unital c∗-algebra f(a). some properties of these functors are also given. in particular, we show that the functors [⋅]k and f are exact and the functor p is normal exact.