Let \({\mathscr{C}}_{T}\) be the von Neumann algebra generated by certain compactly supported fermion fields and \({\mathbb{A}}\) be the von Neumann algebra generated by the set of simple adapted processes. We derive the Riesz representation \(L_A^p([0,T];L^q({\mathscr{C}}_T))\simeq(L_A^{p^{\prime}}([0,T];L^{q^{\prime}}({\mathscr{C}}_T)))^*,\) where 1 < p ≤ ∞, 1 < q < ∞ and \(\frac{1}{p}+\frac{1}{p^{\prime}}=1,\,\frac{1}{q}+\frac{1}{q^{\prime}}=1\) . To achieve it, the authors develop an equivalence relation in a more general setting between the above isomorphism and the boundedness of a kind of projection operator. Even in the commutative setting as a special case, their method has certain advantages.
