Approximate C∗- and W∗-Algebras on Non-Separable Banach Spaces: Structure, Positivity, and Representation Theory
Abstract
This paper develops a comprehensive algebraic framework for extending the classical theory of C∗ - and W∗ -algebras to non-separable Banach spaces. By constructing Approximate C∗ - and W∗-algebras, defined as directed inductive limits of local operator algebras acting on separable subspaces, we obtain a rigorous structure that preserves the essential algebraic, topological, and spectral properties of classical operator algebras while overcoming the limitations imposed by separability. The first part of the article establishes the fundamental definitions of approximate C∗ -algebras, including approximate positivity, involution stability, and norm consistency under directed limits of projections. We then formulate and prove an Approximate Gelfand–Naimark theorem, showing that every approximately self-adjoint algebra can be faithfully represented as an approximate norm-closed-subalgebra of B(X) for some non-separable Banach space X. The second part introduces Approximate W∗ -algebras, defined through projective limits of preduals corresponding to the local separable components. We demonstrate that these algebras retain weak* - compactness and support approximate normal states, extending the classical duality between W∗-algebras and Banach pre-duals to the non-separable setting. An Approximate Bicommutant Theorem is also established, identifying the approximate double commutant with the closure of the algebra under the approximate weak*-topology. The third part develops an Approximate Representation Theory. By generalising the Gelfand– Naimark–Segal (GNS) construction, we show that every approximate positive functional induces a cyclic approximate representation, and that factor decompositions (of types I, II, and III) extend naturally to this setting through local coherence conditions. Several explicit examples—particularly within l∞, L∞, and C(βN)—illustrate the structural behaviour of approximate C∗- and W∗-algebras, their spectra, and their reflexive properties. This framework provides a unified foundation for analysing operator algebras on non-separable Banach spaces, reconciling algebraic and topological aspects in the absence of countable bases. The results obtained not only generalise classical operator algebra theory but also open the door to future developments in noncommutative geometry and infinite-dimensional mathematical physics.