Operator Algebras on Non-Separable Banach Spaces and Applications in Mathematical Physics

Abstract

Juan Alberto Molina Garcia*

This paper develops a structural and functional framework for operator algebras acting on nonseparable Banach spaces (NSBS). While classical operator algebras—such as C∗ - and W∗ -algebras—are traditionally constructed on separable Hilbert spaces, many physical and mathematical contexts require non-separable or even transfinite structures: quantum field theories with infinitely many degrees of freedom, infinite tensor product systems, and algebras associated with non-measurable state spaces.

We extend the classical operator-algebraic formalism to NSBS by introducing approximate operator algebras, defined through directed nets of weakly compact projections and local separable subspaces. This approach restores the analytic machinery of functional calculus, spectra, and representations, while preserving topological and dual properties within locally separable components. The paper establishes several new results concerning approximate ideals, bicommutants, spectral continuity, and weak operator topologies in NSBS. Furthermore, we analyse the correspondence between approximate representations of C∗ -algebras on NSBS and physical observables in quantum mechanics and field theory.

From a physical perspective, the proposed framework provides a rigorous mathematical description of systems with non-countable degrees of freedom, extending von Neumann’s theory of operator algebras beyond separability. Applications include the representation of infinite spin systems, algebras of observables in non-separable Hilbert–Banach settings, and generalised state spaces in quantum statistical mechanics.

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