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We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures.

For fieldF of characteristic notp containing a primitivepth root of unity, we determine the Galois module structure of the group ofpth-power classes ofK for all cyclic extensionsK/F of degreep.

Recently, Ivan Marin informed us of an error in the proof of Proposition 5.4 in the paper [2] entitled Brauer algebras of simply laced type. He also mentioned this observation in his preprint [3]. Here we give an improved argument. The result is exactly the same as for our incorrect argument and so other results using the proposition remain valid. We are grateful to Ivan Marin for pointing out the error.

Given a positive contraction,P, on C(X) we define the conservative and dissipative parts ofP and establish some properties which are analogous to known ones from measure theory (see [3]). We also prove a ratio limit theorem for certain processes.

Let $$\mathcal{R}$$ be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra $$\mathcal{N}$$ so that for every 1 < p < 2, the Haagerup L p -space associated with $$\mathcal{R}$$ embeds isomorphically into $$\mathcal{N}_ * $$ . We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if $$\mathcal{M}$$ is a semi-finite von Neumann algebra then every reflexive subspace of $$\mathcal{M}_ * $$ embeds isomorphically into L r ( $$\mathcal{M}$$ ) for some r > 1.

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. The finite groups for which ν(G) ≤ 2 were determined by Dedekind and by Schmidt in the early times of group theory. On the other hand, if G is a finite p-group, La Haye and Rhemtulla have proved that either ν(G) ≤ 1 or ν(G) ≥ p. In this note, we determine all finite p-groups satisfying ν(G) = p for p > 2.

We prove localization for random perturbations of periodic divergence form operators of the form ∇ · aω · ∇ near the band edges. Here aω is a matrix function which results from an Anderson type perturbation of a periodic matrix function.