# nLab system of imprimitivity

## Definition.

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

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

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

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

such that

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

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

#### An approach via $*$-representations

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

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