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Proceedings of the American Mathematical Society

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Ergodic theorems for Fourier transforms of noncommutative analogues of vector measures


Author: Kari Ylinen
Journal: Proc. Amer. Math. Soc. 98 (1986), 655-662
MSC: Primary 43A30; Secondary 22D40, 60G60
MathSciNet review: 861770
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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\Vert {\omega ({\mu _i})} \right\Vert < \infty $, $ {\lim _i}{\mu _i}(G) = 1$, and $ {\lim _i}\left\Vert {\omega (\mu _i^* * {\delta _s} - \mu _i^*)\xi } \right\Vert = 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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0861770-4
Keywords: (Amenable) locally compact group, group $ {C^*}$-algebra, convergence to right invariance, ergodic theorem, homogeneous random field
Article copyright: © Copyright 1986 American Mathematical Society