Baruch College’s Ultrametric Completes von Neumann’s Incomplete Tensor Products

Baruch College researchers have revisited a mathematical construction initiated by John von Neumann nearly a century ago, defining a metric for infinite tensor products of Hilbert spaces. The newly defined pseudo-ultrametric, denoted d, is the convergence exponent of the series formed from any pair of representatives, not through standard geometric means. Distinct classes may lie at distance zero, so the researchers pass to the quotient space Γ̃ after establishing the pseudo-ultrametric is not complete, to achieve completeness. The resulting pair (Γ̃, d) is a complete ultrametric space. According to the researchers, a product unitary satisfying the condition infimum ‖x‖=1 |⟨x,Ux⟩−1| > 0 displaces every class to the maximal distance 1.…