Vol. 38, No. 3, 1971

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Lie homomorphisms of operator algebras

C. Robert Miers

Vol. 38 (1971), No. 3, 717–735

A mapping ϕ : M N between -algebras M,N which is -linear, and which preserves the Lie bracket [X,Y ] = XY Y X of elements X,Y in M is called a Lie -homomorphism or just a Lie homomorphism. The main result of this paper states that if ϕ : A B is a uniformly continuous Lie -homomorphism of the C-algebra A onto the C-algebra B then there exists a central projection D in the weak closure of B such that modulo a center-valued -linear map which annihilates brackets, is a -homomorphism and (I D)ϕ is the negative of a anti-homomorphism.

Mathematical Subject Classification 2000
Primary: 46L10
Secondary: 46L05
Received: 30 December 1969
Revised: 30 April 1970
Published: 1 September 1971
C. Robert Miers