The Dauns-Hofmann theorem

The Gelfand-Naimark theorem identifies a commutative unital C*-algebra A with C(Spec A). This leads to a natural conjecture that each noncommutative C*-algebra A corresponds to an algebra of operator-valued functions on Prim A. This was the original motivation for the development of C*-bundle theory. The results of this program are not completely satisfactory. Noncommutative generalizations of the Gelfand-Naimark theorem were proved only for rather narrow classes of C*-algebras. On the other hand, some progress has been made, and the Dauns-Hofmann theorem is a good illustration. The theorem states that each C*-algebra is a module over the algebra of continuous functions on the primitive ideal space. In our talk, we discuss the structure of Prim A and prove the Dauns-Hofmann theorem