  We prove universal lower bounds for discrepancies (i.e. sizes of spectral
gaps of averaging operators) of measure-preserving actions of a locally compact
group on probability spaces. For example, a locally compact Hausdorff
unimodular group $G$, acting continuously, by measure-preserving
transformations, on a compact atomless probability space $(X,\nu)$, with an
orbit $Gx_0$ of measure zero, contained in the support of $\nu$, and with
compact stabilizer (i.e. $G_{x_0}$ is compact) has the following property: any
finite positive regular Borel measure $\mu$ on $G$ satisfies
$$\|\pi_0(\mu)\|_{2\to 2}\geq \|\lambda_G(\mu)\|_{2\to 2},$$ where $\pi_0$
denotes the Koopman representation of $G$, defined by the given action, and
$\lambda_G$ denotes the left-regular representation of $G$. The lower bounds we
prove generalize the universal lower bounds for the discrepancies of
measure-preserving actions of a discrete group. Many examples show that the
generalization from discrete groups to locally compact groups requires some
additional hypothesis on the action (we detail some examples of actions of
amenable groups with a spectral gap, due to Margulis). Well-known examples and
results of Kazhdan and Zimmer show that the discrepancies of some actions of
Lie groups on homogeneous spaces match exactly the universal lower bounds we
prove.
