nLab
system of imprimitivity
Definition.
Given a locally compact topological group $G$ , a system of imprimitivity on $G$ consists of a

unitary representation $\rho :G\to U(H)$ on a Hilbert space $H$

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

a regular projection-valued measure $P:B(X)\to \mathrm{End}H$ where $B(X)$ is the Borel $\sigma $ -algebra of $X$

such that

$$\rho (g)P(E)\rho (g{)}^{-1}=P(\mathrm{gE})$$ `\rho(g)P(E)\rho(g)^{-1} = P(gE)`

for all $g\in G$ and $E\in B(X)$ .

An approach via $*$ -representations
In the above definition, one can replace the projection-valued measure $P$ by a $*$ -representation $M:{C}_{0}(X)\to H$ of the ${C}^{*}$ -algebra ${C}_{0}(X)$ by defining $M(f)=\int f\mathrm{dP}$ , then

$$\rho (g)M(f)\rho ({g}^{-1})=M({L}_{g}f),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{g}(f)(x):=f({g}^{-1}x).$$ `\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)