A class of Banach spaces

Let G be a separable locally compact unimodular group of type I, (G) over cap be its dual, is a measurable field of, not necessary bounded, operators on (G) over cap such that (p) over cap(pi) is self-adjoint, (p) over cap(pi) >= I for mu-almost all pi is an element of (G) over cap, and A((p) over cap) (G) = {f (x):= integral(G) over cap Tr[(f) over cap(pi)pi(x)(-1)]d mu(pi), (f) over cap is an element of L-1((G) over cap), parallel to f parallel to(p) over cap = integral(G) over cap Tr vertical bar(p) over cap(pi)vertical bar(pi)vertical bar d mu(pi) < infinity). We show that A((p) over cap) (G) is a Banach space endowed with the norm vertical bar vertical bar f vertical bar vertical bar((p) over cap), and we generalize this result to the matricial group G = G(nm), m >= n, of a local field.