Velocity maps in von Neumann algebras

L. J. Bunce and John David Maitland Wright

Vol. 170 (1995), No. 2, 421–427

DOI: 10.2140/pjm.1995.170.421

Abstract

When A is a C^∗-algebra, a function d : A_sa → A_sa is said to be a velocity map if, for each commutative subalgebra B of A_sa, d : B → A_sa is a derivation.

Let A be a norm closed ideal, or quotient, in a von Neumann algebra without Type I₂ part and let P(A) be the set of projections in A. It is shown that if d : P(A) → A is a bounded function such that d(ef) = ed(f) + d(e)f whenever ef = fe, then d extends uniquely to a derivation of A. Hence every velocity map of A_sa bounded on the unit ball extends to a derivation of A.

Mathematical Subject Classification 2000

Primary: 46L57

Secondary: 46L60, 82C10

Milestones

Received: 14 December 1992

Revised: 11 June 1993

Published: 1 October 1995