Adjoints, Order and Spectral Moduli on $L^2(T)$

We study adjoints of $R(T)$-linear, $T$-strongly bounded operators on the conditional space $L^2(T)$ and the interaction between its Hilbert-type geometry and its Riesz-space order. This produces two natural operator moduli: the Riesz--Kantorovich order modulus and the spectral modulus $(S^*S)^{1/2}