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Register a new userA Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem
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In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof rests heavily on the operator theory on Kaplansky-Hilbert modules, in particular the spectral theorem for Hilbert-Schmidt homomorphisms on such modules. As an application, a generalization of the celebrated Furstenberg-Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.
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The Nevo-Zimmer theorem classifies the possible intermediate $G$-factors $Y$ in $X times G/P to Y to X$, where $G$ is a higher rank semisimple Lie group, $P$ a minimal parabolic and $X$ an irreducible $G$-space with an invariant probability measure. An important corollary is the Stuck-Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set. We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.
The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.
The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutati
The Furstenberg-Sarkozy theorem asserts that the difference set $E-E$ of a subset $E subset mathbb{N}$ with positive upper density intersects the image set of any polynomial $P in mathbb{Z}[n]$ for which $P(0)=0$. Furstenbergs approach relies on a correspondence principle and a polynomial version of the Poincare recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,mathcal{B},mu,T)$ and set $A in mathcal{B}$ with $mu(A) > 0$, one has $c(A):= lim_{N to infty} frac{1}{N} sum_{n=1}^N mu(A cap T^{-P(n)}A) > 0.$ The limit $c(A)$ will have its optimal value of $mu(A)^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $mathbb{Z}/Nmathbb{Z}$. We show that a sequence of modular rings $mathbb{Z}/N_mmathbb{Z}$, $m in mathbb{N},$ is asymptotically totally ergodic if and only if $mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $delta in (0,1]$ and a (not necessarily intersective) polynomial $Q in mathbb{Q}[n]$ such that $Q(mathbb{Z}) subseteq mathbb{Z}$, and write $S = { Q(n) : n in mathbb{Z}/Nmathbb{Z}}$. For any integer $N > 1$ with $mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $mathbb{Z}/Nmathbb{Z}$ such that $|A||B| geq delta N^2$, then $mathbb{Z}/Nmathbb{Z} = A + B + S$.
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $mathbb{Z}G$ has SFC provided at most one copy of the quaternions $mathbb{H}$ occurs in the Wedderburn decomposition of the real group ring $mathbb{R}G$. This generalises the Eichler condition in the case of integral group rings.
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