. O A ] 2 4 Se p 20 01 When Aut ( A ) and Homeo ( Prim ( A ) ) are homeomorphic , where A is a C ∗ - algebra

When Aut(A) and Homeo(Prim(A)) are homeomorphic, where A is a C *-algebra Abstract In this paper, we discuss when Aut(A) and Homeo(Prim(A)) are homeomorphic, where A is a C *-algebra. 1 Preliminaries Let A be C *-algebra. Then the collection Aut(A) of ⋆-automorphisms of A is a group under composition. We give Aut(A) the point-norm topology, that is, α n → α if and only if α n (a) → α(a) for all a ∈ A. Then Aut(A) is a topological group. Let (X, X) and (Y, Y) be topological spaces, and let φ : X → Y be a map. Then φ is called continuous (from (X, X) to (Y, Y)) if the preimage φ −1 (V) of V under φ belongs to X for each V ∈ Y. A bijection between topological spaces is called a homeomorphism if it is continuous and has a continuous inverse.