Conditional expectations in complete normed complex algebras satisfying the von Neumann inequality

Abstract

Let A be a complete normed complex algebra, let π:A→A be a nonzero contractive linear projection, and consider π(A) as a complete normed complex algebra under the product (x,y)→π(xy). We prove the following results.-If A is unital and associative (respectively, alternative) and satisfies the von Neumann inequality, and if π satisfies the weak conditional expectation property(WCE)π(π(a)π(b))=π(π(a)b)=π(aπ(b)) for alla,b∈A, then π(A) is unital and associative (respectively, alternative) and satisfies the von Neumann inequality.-If A is an Arazy algebra, and if π satisfies the weak Jordan conditional expectation (namely the symmetrization of (WCE)), then π(A) is an Arazy algebra. (We note that, in the possibly non power-associative setting, Arazy algebras are the adequate substitutes of complete normed unital power-associative complex algebras satisfying the von Neumann inequality.)-If the closed multiplication algebra of A satisfies the von Neumann inequality, and if π is an elementary operator, then the closed multiplication algebra of π(A) satisfies the von Neumann inequality.