  The notions of column and row operator space were extended by A. Lambert from
Hilbert spaces to general Banach spaces. In this paper, we use column and row
spaces over quotients of subspaces of general $L_p$-spaces to equip several
Banach algebras occurring naturally in abstract harmonic analysis with
canonical, yet not obvious operator space structures that turn them into
completely bounded Banach algebras. We use these operator space structures to
gain new insights on those algebras.
