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

Let A be a complete normed complex algebra, let 7r : A-+ A be a nonzero contractive linear projection, and consider 7r(A) as a complete normed complex algebra under the product (x, y)-+ 7r(xy). We prove the following results. -If A is unital and associative (respectively, alternative) and satisfies the von Neumann inequality, and if 7r satisfies the weak conditional expectation property (WCE) 7r(7r(a)7r(b)) = 7r(7r(a)b) = 7r(a7r(b)) for all a, b is an element of A, then 7r(A) is unital and associative (respectively, alternative) and satisfies the von Neumann inequality. -If A is an Arazy algebra, and if 7r satisfies the weak Jordan conditional expectation (namely the symmetrization of (WCE)), then 7r(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 7r is an elementary operator, then the closed multiplication algebra of 7r(A) satisfies the von Neumann inequality. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

By studying projections of complete normed complex algebras under different conditions, a series of results about the properties of the algebras and satisfying inequalities have been obtained.