Base norm spaces--classical, complex, and noncommutative
David P. Blecher, Damon M. Hay
TL;DR
The paper develops a unified, duality-driven extension of base norm space theory from real to complex and noncommutative settings. By introducing complex base norm spaces and noncommutative base norm spaces, it establishes fundamental dualities with complex Archimedean order unit spaces and dual operator systems, respectively, and shows how these bases encode state-space and matrix-state structures via hyperplanes and bases. The approach leverages the dual Taylor norm for complexification and demonstrates that nc base norm spaces arise as preduals of dual operator systems, enabling a tight link between base norms, operator systems, and quantum convexity notions. Concrete instantiations include the Paulsen system, where nc base norm structure aligns with operator system norms and duals (Ng's dual system), illustrating broad applicability to finite-dimensional operator systems and CPTP maps, with implications for quantum information and generalized probabilistic theories.
Abstract
We generalize the theory of base norm spaces to the complex case, and further to the noncommutative setting relevant to `quantum convexity'. In particular, we establish the duality between complex Archimedean order unit spaces and complex base norm spaces, as well as the corresponding duality between their noncommutative counterparts. Additional topics include an exploration of natural connections with various notions of quantum convexity and regularity of noncommutative convex sets, and an analysis of how these concepts interact with complexification. We also define, as in the classical case, a class that contains and generates the noncommutative base norm spaces, but is defined by fewer axioms. We show how this may be applied to provide new and interesting examples of noncommutative base norm spaces.