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