C*-like modules and matrix p-operator norms

Abstract We present a generalization of Hölder duality to algebra-valued pairings via $$L^p$$ L p -modules. Hölder duality states that if $$p \in (1, \infty )$$ p ∈ ( 1 , ∞ ) and $$p'$$ p ′ are conjugate exponents, then the dual space of $$L^p(\mu )$$ L p ( μ ) is isometrically isomorphic to $$L^{p'}(\mu )$$ L p ′ ( μ ) . In this work, we study certain pairs $$(\textsf{Y},\textsf{X})$$ ( Y , X ) , as generalizations of the pair $$(L^{p'}(\mu ), L^p(\mu ))$$ ( L p ′ ( μ ) , L p ( μ ) ) , that have an $$L^p$$ L p -operator algebra-valued pairing $$\textsf{Y}\times \textsf{X}\rightarrow A$$ Y × X → A . When the A -valued version of Hölder duality still holds, we say that $$(\textsf{Y}, \textsf{X})$$ ( Y , X ) is C*-like. We show that finite and countable direct sums of the C*-like module ( A , A ) are still C*-like when A is any block diagonal subalgebra of $$d \times d$$ d × d matrices. We provide counterexamples when $$A \subset M_d^p(\mathbb {C})$$ A ⊂ M d