Positive projections and Jordan structure in operator algebras.

Abstract

Review: Let A be a unital C∗-algebra and P:A→A a positive unital projection, i.e., P≥0, P(1)=1 and P2=P. Denoting the selfadjoint elements in A by Ah, the product a∘b=12(ab+ba) turns Ah into a Jordan algebra. The authors show that P(Ah) is itself a Jordan algebra when provided with the new multiplication (x,y)→P(x∘y). (In fact, they prove that when Ah is replaced by an arbitrary unital JC-algebra, the range becomes a JC-algebra.) The paper may be regarded as an attempt to place in a general setting the work of J. Arazy and Y. Friedman [Mem. Amer. Math. Soc. 13 (1978), no. 200; MR0481219 (82b:47023)] characterizing the ranges of contractive projections in the algebra of compact operators on a separable Hilbert space.Reviewed by Dorte Olesen