Rickart \(C^*\)-algebras are \(\sigma\)-normal (Q1924578)

A \(C^*\)-algebra \(A\) is said to be a Rickart \(C^*\)-algebra if the right annihilator of any element equals the principal right ideal generated by some projection in \(A\). It is well-known that the set \(P(A)\) of projections of any Rickart \(C^*\)-algebra is an \(\aleph_0\)-complete lattice. We prove that Rickart \(C^*\)-algebras are \(\sigma\)-normal, meaning that the supremum of any increasing sequence of projections of a Rickart \(C^*\)-algebra \(A\), computed in the lattice \(P(A)\), coincides with the supremum of that sequence in \(A_{sa}\), the set of all selfadjoint elements of \(A\).