Ergodic theorems for Fourier transforms of noncommutative analogues of vector measures

Author:

Ylinen Kari

Abstract

Let G G be a locally compact group and E E a complex Banach space. Let φ : G E \varphi :G \to E be a function which is the Fourier transform of a weakly compact operator Φ : C ( G ) E \Phi :{C^*}(G) \to E in the sense that Φ ( ω ( s ) ) = ϕ ( s ) {\Phi ^{**}}(\omega (s)) = \phi (s) , s G s \in G , where ω : G W ( G ) L ( H ω ) \omega :G \to {W^*}(G) \subset L({H_\omega }) corresponds to the universal representation of C ( G ) {C^ * }(G) . It is proved that lim i ϕ d μ i = Φ ( p ω ) {\lim _i}\smallint \phi d{\mu _i} = {\Phi ^{**}}({p_\omega }) , where p ω {p_\omega } is the projection onto the space of the common fixed points of all ω ( s ) \omega (s) , s G s \in G , and ( μ i ) i I {({\mu _i})_{i \in \mathcal {I}}} is an arbitrary net in the measure algebra M ( G ) M(G) satisfying sup i I ω ( μ i ) > {\sup _{i \in \mathcal {I}}}\left \| {\omega ({\mu _i})} \right \| > \infty , lim i μ i ( G ) = 1 {\lim _i}{\mu _i}(G) = 1 , and lim i ω ( μ i δ s μ i ) ξ = 0 {\lim _i}\left \| {\omega (\mu _i^* * {\delta _s} - \mu _i^*)\xi } \right \| = 0 for all s G s \in G , ξ H ω \xi \in {H_\omega } . If E E is a Hilbert space and ϕ \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 ϕ \phi to a measurability condition is established.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference21 articles.

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