  A. Gournay defined a notion of $l^{p}$-dimension for subspaces of the
l^{q}-left-regular representation of an amenable discrete group. We give an
alternative definition that works for sofic groups and a different notion for
groups satisfying the Connes embedding conjecture, and for more general
representations on Banach spaces. We extend certain results due to Gournay, as
well as discuss l^{p}-Betti numbers of Free groups.
