Complementation for right ideals in generalized Hilbert algebras

Abstract:

Let $\mathfrak {A}$ be a generalized Hilbert algebra and let $\mathcal {J}$ be a closed right ideal of $\mathfrak {A}$. Let ${\mathcal {J}^ \bot }$ denote the pre-Hilbert space orthogonal complement of $\mathcal {J}$ in $\mathfrak {A}$. The problem investigated in this paper is: for which algebras $\mathfrak {A}$ is it true that $\mathfrak {A} = \mathcal {J} \oplus {\mathcal {J}^ \bot }$ for every closed right ideal $\mathcal {J}$ of $\mathfrak {A}$? In the case that $\mathfrak {A}$ is achieved, a slightly stronger property is characterized and these characterizations are then used to investigate some interesting examples.