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 $P:B(X)\to End H$ where $B(X)$ is the Borel $\sigma$-algebra of $X$

such that

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

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

An approach via $\ast$-representations

In the above definition, one can replace the projection-valued measure $P$ by a $\ast$-representation $M : C_0(X)\to H$ of the $C^\ast$-algebra $C_0(X)$ by defining $M(f) = \int f dP$, then

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

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.

References

• Gerald B. Folland, Sec. 6.4 in: A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press (1995) [pdf, gBooks]