Three-space problems in discrete spectra of C*-algebras and dual C*-algebras

We discuss the three-space problem on discreteness for the Jacobson topolog y on the spectrum of a C*-algebra in detail. As an application, it is shown that a C*-algebra A is a dual C*-algebra if and only if a closed ideal I o f A and the quotient A/I are dual C*-algebras and the open central projecti on, in the second dual of A, corresponding to I is a multiplier for A.