Extensions of σ-C*-algebras

Let A be a sigma-C*-algebra. The bounded part b(A) of A introduced by Konrad Schmudgen in [4] is a C*-algebra for some C*-norm. We shall show that if A is a split extension of a sigma-C*-algebra B by a closed two-sided ideal I then b(A) will be a split extension of the C*-algebra b(B) by the closed two-sided b(I). A number of results concerning the bounded part of a sigma-C*-algebra are established.