Another version of "exotic characterization of a commutative H∗-algebra"

Theorem 1.1. Let A be a semisimple complex Banach algebra with the following properties: (i) for every closed right ideal R in A, there exists a closed left ideal L such that R∩L= {0} and R+ L= A (each a∈ A can be written in the form a= a1 + a2 with a1 ∈ R, a2 ∈ L); (ii) if a,b in A are such that ab = ba= 0, then ‖a+ b‖2 = ‖a‖2 +‖b‖2. Then A is a commutative proper H∗-algebra [1]. It is easy to see that each proper commutative H∗-algebra has properties (i) and (ii), stated in the theorem. A properH∗-algebra is a Banach algebraA, whose underlying Banach space is a Hilbert space, which has an involution x→ x∗ such that (xy,z) = (y,x∗z) = (x,zy∗) for all x, y ∈ A. An idempotent is a member e of A such that e2 = e; e is primitive if it cannot be written as a sum, e = e1 + e2, of two nonzero idempotents e1, e2 such that e1e2 = e2e1 = 0 (e = e1 + e2 implies either e1 = 0 or e2 = 0). A Banach algebra A is semisimple if its radical [2] (Jacobson radical) consists of 0 alone. One of the properties of radical [2, Theorem 16] is the following proposition: if R is a right ideal consisting of nilpotents (x ∈ R implies xn = 0 for some positive integer n), then R is included in the radical. This proposition is relevant to both the present note and [4].