nLab
system of imprimitivity

Definition.

Given a locally compact topological group G, a system of imprimitivity on G consists of a

  • unitary representation ρ:GU(H) on a Hilbert space H

  • a locally compact Hausdorff space X with continuous left G-action

  • a regular projection-valued measure P:B(X)EndH where B(X) is the Borel σ-algebra of X

such that

ρ(g)P(E)ρ(g) 1=P(gE)\rho(g)P(E)\rho(g)^{-1} = P(gE)

for all gG and EB(X).

An approach via *-representations

In the above definition, one can replace the projection-valued measure P by a *-representation M:C 0(X)H of the C *-algebra C 0(X) by defining M(f)=fdP, then

ρ(g)M(f)ρ(g 1)=M(L gf),L g(f)(x):=f(g 1x).\rho(g)M(f)\rho(g^{-1}) = M(L_g f), \,\,\,\,L_g(f)(x) := f(g^{-1}x).

On the other hand, any M satisfying this property defines a regular projection-valued measure as above.

Extensions

Remark: A possible extension is to replace X by a measurable space with a measurable left action of G.

Applications

This concept is important in Mackey machinery and in the applications to the study of coherent states and Berezin quantization.

  • sec. 6.4 in: Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995

Created on June 4, 2011 14:58:52 by Zoran Škoda (31.45.147.163)