  In the given article the notion of infinite norm decomposition of a
C$^*$-algebra is investigated. The norm decomposition is some generalization of
Peirce decomposition. It is proved that the infinite norm decomposition of any
C$^*$-algebra is a C$^*$-algebra. C$^*$-factors with an infinite and a nonzero
finite projection and simple purely infinite C$^*$-algebras are constructed.
