A Sheaf of von Neumann Algebras and Its Geometry

TL;DR

This paper explores a noncommutative geometric framework for space-time using von Neumann algebras, linking operator algebras with differential geometry, but faces challenges in unifying general relativity and quantum theory.

Contribution

It constructs a generalized differential geometry of space-time via von Neumann algebras derived from a bundle of Hilbert spaces, highlighting the algebraic approach to geometry.

Findings

Geometry transfers to the algebra M_0 and its derivations.

The geometry of M_0 is probabilistic in nature.

The geometry does not extend to the von Neumann algebra M.

Abstract

It is shown that the differential geometry of space-time, can be expressed in terms of the algebra of operators on a bundle of Hilbert spaces. The price for this is that the algebra of smooth functions on space-time has to be made noncommutative. The generalized differential geometry of space-time is constructed in terms of the algebra A (and its derivations) on a transformation groupoid. Regular representation {\pi} of A in the algebra of bounded operators on a bundle of Hilbert spaces leads to the algebra \pi(A) = M_0 which can be completed to the von Neumann algebra M . The representation \pi establishes the isomorphism between A and M_0 which, in turn, implies the isomorphism between moduli of their derivations. In this way, geometry naturally transfers to the algebra M_0 and its derivations. Although geometry, as defined in terms of M_0, is formally isomorphic to that defined in terms of A, it exhibits a strong probabilistic flavour. However, the geometry of M_0 does not prolong to M . This is clearly a serious stumbling block to fully unify mathematical tools of general relativity and quantum theory.