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.

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

Last revised on April 2, 2013 at 00:01:52. See the history of this page for a list of all contributions to it.