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 $(S^{\ast }S{)}^{1\mathrm{/}2}$. After isolating the $T$-regular operators, we show that they form a Dedekind-complete order ideal in the natural lattice of order-bounded $R(T)$-module homomorphisms. On this operator lattice the order operations remain $T$-strongly bounded and the adjoint is an order automorphism. We then compare the two moduli and explain why they may differ.