Abstract. We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra $B(H)$, $H$ being a separable Hilbert space. Let $\phi\colon B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of $B(H)$ (these automorphisms may, of course, depend on the point) in the operator norm. Then $\phi $ is an automorphism of the algebra $B(H)$.

AMS Subject Classification (1991): 47D50, 47B49, 46L40

Keyword(s): Reflexivity, automorphisms, ideals of operators, Wigner's unitary-antiunitary theorem

Received January 15, 1999, and in revised form March 26, 1999. (Registered under 2713/2009.)