nLab canonical Hilbert space of half-densities




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

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

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

Related lecture notes include

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.