nLab canonical Hilbert space of half-densities

Idea

Given a manifold $X$, there is a Hilbert space $L^2(X)$ whose elements are square-integrable half-densities on $X$, and whose inner product is given by tensoring two half-densities to an actual density and then integrating that over $X$.

This Hilbert space is often called the canonical Hilbert space of $X$, in contrast with the L-2-spaces $L^2(X,\mu)$ of $\mu$-integrable functions for every choice of measure $\mu$ on $X$.

The notion of canonical Hilbert spaces originates in the context of geometric quantization in (Guillemin-Sternberg). It directly corresponds to the “canonical” construction of groupoid convolution algebras (see there for details) from sections of half-density bundles without choice of a Haar measure.

References

The notion originates in geometric quantization around

• Victor Guillemin, Shlomo Sternberg, Geometric asymptotics Math. Surveys, 14, Amer. Math. Soc., Providence, R.I., 1977; MR 58 # 24404.

Related lecture notes include

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

Applications in operator algebra theory appear in

• Alain Connes, Noncommutative Geometry