On discretization of C*-algebras

TL;DR

This paper proves that there is no faithful functor that discretizes C*-algebras into AW*-algebras, including von Neumann algebras, due to fundamental obstructions in their structure.

Contribution

It establishes a negative result showing the impossibility of discretizing C*-algebras into AW*-algebras via faithful functors.

Findings

No faithful functor from C*-algebras to AW*-algebras exists.

The algebra of bounded operators on a separable infinite-dimensional Hilbert space cannot be discretized in this manner.

Obstructions arise from the failure of certain normal factorization properties.

Abstract

The C*-algebra of bounded operators on the separable infinite-dimensional Hilbert space cannot be mapped to a W*-algebra in such a way that each unital commutative C*-subalgebra C(X) factors normally through $\ell^\infty(X)$. Consequently, there is no faithful functor discretizing C*-algebras to AW*-algebras, including von Neumann algebras, in this way.