nLab
system of imprimitivity

Definition.

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

  • unitary representation ρ:GU(H)\rho: G\to U(H) on a Hilbert space HH

  • a locally compact Hausdorff space XX with continuous left GG-action

  • a regular projection-valued measure P:B(X)EndHP:B(X)\to End H where B(X)B(X) is the Borel σ\sigma-algebra of XX

such that

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

for all gGg\in G and EB(X)E\in B(X).

An approach via *\ast-representations

In the above definition, one can replace the projection-valued measure PP by a *\ast-representation M:C 0(X)HM : C_0(X)\to H of the C *C^\ast-algebra C 0(X)C_0(X) by defining M(f)=fdPM(f) = \int f dP, 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 MM satisfying this property defines a regular projection-valued measure as above.

Extensions

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

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)