Completely distributive CSL algebras with no complements in $\mathcal {C}_p$

Abstract:

Anoussis and Katsoulis have obtained a criterion for the space $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ to have a closed complement in $\mathcal C_p$, where $\mathcal L$ is a completely distributive commutative subspace lattice. They show that, for a given $\mathcal L$, the set of $p$ for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice $\mathcal L$ for which $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ has no complement for any $p\not =2$. However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices $\mathcal L$, $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ fails to have a complement for any $p\not =2$.