I'm reading a paper which contains the following line:
$$\|\pi(\mu)f\|^{2n}= \bigg(\int_G\langle \pi(g)f,f\rangle d(\mu^\ast\ast\mu)\bigg)^n$$
Here $G$ is a Lie group, $\mu$ is a Borel probability measure on $G$, $\pi$ is a unitary representation of $G$ on $X$, and $f$ is an $L^2$ function on $X$ with zero mean. This is part of a longer chain of equivalences, so I'm not even sure if this is supposed to be a definition or not. So my questions are: what does $\pi(\mu)f$ mean? what is the norm $\|\cdot\|$? and what is $\mu^\ast$?
The paper is Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups available here
The line appears near the bottom of the second page.