On the spectral theory for Rickart ordered *-algebras (Q2906727)

The authors introduce and study the concept of a Rickart ordered *-algebra (\(RO^*\)-algebra). A Rickart *-algebra is an algebra \(A\) with an involution such that for any element \(x\) in \(A\) its right annihilator is a right principal ideal generated by a projection. If, moreover, additional conditions are added (every positive element has a square root contained in its bicommutant, and some suprema of finite sums of positive elements are self-adjoint), then it is said to be an \(RO^*\)-algebra. The authors define bounded elements in \(A\) and show that the set of all these elements is an \(RO^*\)-algebra which, in a certain norm, is a \(C^*\)-algebra. They show that if \(A\) is commutative, then the set of its self-adjoint elements is a conditionally \(\sigma\)-complete lattice. It is also shown that a maximal commutative subalgebra of \(A\) is an \(RO^*\)-algebra and this result is used for obtaining the spectral decomposition of a self-adjoint element of \(A\).