In https://en.wikipedia.org/wiki/Direct_integral I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or Banach algebras (in particular I am interested in the Schatten classes). Why can't I define a direct integral for those spaces?
Cross-posted and answered on MathOverflow: https://mathoverflow.net/questions/422812/why-is-it-difficult-to-define-a-direct-integral-of-banach-spaces-or-banach-algeb