C*-like modules and matrix p-operator norms

2025/05/26 by Calin, Alessandra, Cartwright, Ian, Coffman, Luke +5
#46H35 #47L10 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 46H15 #Secondary 15A60

paper · doi:10.48550/arxiv.2505.19471

Abstract

We present a generalization of Hölder duality to algebra-valued pairings via L^p-modules. Hölder duality states that if p ∈ (1, ∞) and p^′ are conjugate exponents, then the dual space of L^p(μ) is isometrically isomorphic to L^p^′(μ). In this work we study certain pairs (Y,X), as generalizations of the pair (L^p^′(μ), L^p(μ)), that have an L^p-operator algebra valued pairing Y × X → A. When the A-valued version of Hölder duality still holds, we say that (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 × d matrices. We provide counterexamples when A ⊂ M_d^p(ℂ) is not block diagonal.

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