Multiplicative maps on ideals of operators which are local automorphisms

TL;DR

The paper proves that a multiplicative map on the algebra of bounded operators on a separable Hilbert space, which can be locally approximated by automorphisms, must itself be an automorphism, revealing a reflexivity property.

Contribution

It establishes that local approximability by automorphisms implies the map is an automorphism, extending reflexivity results to multiplicative maps without linearity or continuity assumptions.

Findings

Multiplicative maps approximable by automorphisms are automorphisms.

The result applies to the algebra B(H) of bounded operators on a separable Hilbert space.

No linearity or continuity is required for the map.

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 $\phi:B(H)\to B(H)$ 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).