Ergodic theorems for Fourier transforms of noncommutative analogues of vector measures

Abstract:

Let $G$ be a locally compact group and $E$ a complex Banach space. Let $\varphi :G \to E$ be a function which is the Fourier transform of a weakly compact operator $\Phi :{C^*}(G) \to E$ in the sense that ${\Phi ^{**}}(\omega (s)) = \phi (s)$, $s \in G$, where $\omega :G \to {W^*}(G) \subset L({H_\omega })$ corresponds to the universal representation of ${C^ * }(G)$. It is proved that ${\lim _i}\smallint \phi d{\mu _i} = {\Phi ^{**}}({p_\omega })$, where ${p_\omega }$ is the projection onto the space of the common fixed points of all $\omega (s)$, $s \in G$, and ${({\mu _i})_{i \in \mathcal {I}}}$ is an arbitrary net in the measure algebra $M(G)$ satisfying ${\sup _{i \in \mathcal {I}}}\left \| {\omega ({\mu _i})} \right \| < \infty$, ${\lim _i}{\mu _i}(G) = 1$, and ${\lim _i}\left \| {\omega (\mu _i^* * {\delta _s} - \mu _i^*)\xi } \right \| = 0$ for all $s \in G$, $\xi \in {H_\omega }$. If $E$ is a Hilbert space and $\phi$ left (resp. right) homogeneous, the second (resp. first) of the last two limit conditions may be omitted. Finally, a connection of such random fields $\phi$ to a measurability condition is established.