Uniqueness of the predual beyond compact metric spaces

Determine whether the predual of the Banach space obtained as the completion of W for the norm ||·||_W is unique in the general setting of Theorem 4 (with V a normed vector space, W a dense subspace equipped with a norm ||·||_W satisfying ||w||_V ≤ r||w||_W, and F defined as the subspace of W′ whose restrictions to the unit ball B_W are continuous for the V-topology). Specifically, ascertain the uniqueness of this predual for non-commutative C*-algebra contexts beyond compact metric spaces, where uniqueness is known.

Background

The paper develops a general framework (Theorem 4) relating a normed space V, a dense subspace W with a stronger norm, and a predual F of the completion of W defined via continuity on the unit ball B_W for the V-topology. This framework is used to recover classical Kantorovich–Wasserstein dualities and extend them to non-commutative settings.

Within this framework, the author notes that uniqueness of the predual is established for compact metric spaces (citing Weaver), but indicates uncertainty about whether such uniqueness holds more generally, particularly in non-commutative C*-algebra contexts addressed in the paper. Clarifying uniqueness is important for the structural analysis of Lipschitz-type spaces and their duality theory in quantum metric settings.