Centralizing mappings of operator algebras

Abstract

If A is an algebra, a mapping 4: A + A is called centralizing if [4(x), X] E 2, for all x E A where [x, y] = xy yx and 2, is the centre of A. In this paper we prove theorems for certain centralizing mappings of C*-algebras and von Neumann algebras which are related to theorems of Posner [6], Mayne [4], and Herstein [3] for prime or simple rings. Namely, we show that if d: A + A is a derivation on a C*-algebra A with [p(d)(x), x] E 2, for all x E A, where p(t) is a complex polynomial, then p(d)(x) = 0 for all x E A. Moreover, if A is a von Neumann algebra and p(d)(x) = 0 Vx E A there exists z E 2, such that $(u z) = 0 where d(x) = [a, x], a E A. In the case that #: A -+ A is a centralizing *-automorphism of a von Neumann algebra, then A = A, @ A, where + IA, is the identity on A, , and A, is abelian. Although C*-algebras are semi-prime and have many special algebraic properties they are not, in general, prime. In fact, a von Neumann algebra is prime if and only if it is a factor (i.e. its centre consists of scalar multiples of the identity). The presence of central projections (self-adjoint idempotents) in von Neumann algebras means that phenomena of an “either ... or” nature in prime rings can occur simultaneously but on complementary summands in the von Neumann case. This is the situation with regard to Mayne’s theorem which states that for prime rings a centralizing automorphism is the identity or the ring is commutative.