We construct a class of subspace lattices \({\mathcal {L}}\) on a separable infinite dimensional Hilbert space \(\mathcal {K}\) . Let \({{\,\textrm{Alg}\,}}{\mathcal {L}}\) be the corresponding subspace lattice algebras. We show that every isometric automorphism of \({{\,\textrm{Alg}\,}}{\mathcal {L}}\) is spatial. We also show that \({{\,\textrm{Alg}\,}}{\mathcal {L}}\) are decomposable, and an operator in \({{\,\textrm{Alg}\,}}{\mathcal {L}}\) is single if and only if it is rank 1 under certain conditions.
