Ergodic and mixing properties of measures on locally compact groups

On mixing and ergodicity in locally compact motion groups

“…On the other hand, when G is non-commutative, the Fourier transform µ(U ) of a probability measure µ is an operator on a Hilbert space, so it is not immediately clear what the appropriate generalizations of these conditions are, to begin with. A natural choice is to try to use the norms of the operators µ(U ) to give conditions for mixing and ergodicity (see [15], [16]), but as it turns out, µ(U ) does not characterize mixing nor ergodicity. Our proof of (B) uses the spectral radius formula (1.1), through which the spectral radius ̺ µ(U ) emerges naturally.…”

“…Finally, we mention two more papers that are related. In [16] Kaniuth considers more general groups G, namely locally compact Hausdorff groups of polynomial growth and with a symmetric group-algebra L 1 (G), but only central probability measures µ ∈ M (G) on such groups; for such measures he gives the necessary and sufficient conditions µ(U ) < 1 and µ(U ) = I, for all non-trivial irreducible U , for µ to be mixing and ergodic by convolutions respectively. Also related, although more loosely, is the paper by Jones, Rosenblatt and Tempelman [15], which, however, has a wider scope.…”