Let $(M,\tau)$ be a semifinite von Neumann algebra, let $J$ be a trace-preserving Jordan isomorphism, and let $(n_k)_{k\geq 1}$ be a random increasing sequence of integers obtained by selecting each integer $n\geq 1$ independently with probability $n^{-\alpha}$, where $0<\alpha<\frac12$. We show that, almost surely, for every $x\in L^1(M)$, $ \frac1m\sum_{k=1}^m J^{n_k}(x)$ converges bilaterally almost uniformly. This extends LaVictoire's classical random $L^1$ ergodic theorem to the non-commutative setting.
Let $\mathcal N$ be a von Neumann algebra equipped with a normal faithful semifinite trace, and let $\gamma$ be a trace-preserving automorphism of $\mathcal N$. We consider the ergodic averages along the prime numbers \[ A_N(x) := \frac1{|P_N|} \sum_{q\in P_N}\gamma^q(x), \qquad P_N:=\{q\leq N:q\ \text{is prime}\}. \]...
Let $X_1,X_2,\ldots$ be i.i.d. complex-valued random variables with arbitrary Borel probability law $\mu$, and set $P_n(z)=\prod_{j=1}^n(z-X_j)$. For every deterministic sequence $k_n=o(n)$, we prove that the empirical zero measure of the $k_n$-th derivative $P_n^{(k_n)}$ converges weakly to $\mu$ almost surely.
We establish a noncommutative version of a familiar Johnson--Maurey--Schechtman--Tzafriri Theorem, by showing that for any $0<p<2$ and a (not necessarily semifinite) von Neumann algebra $\mathcal{M}$ on a separable Hilbert space, if a symmetric quasi-Banach function space $E(0,1) $ containing the function $t\mapsto t^{...
Jing-Hao Huang, M. Junge, F. Sukochev et al.· 1 citation
Over any infinite field we prove existence of polynomial automorphism with prescribed differentials at points. More precisely, let $\K$ be an infinite field and $a_1,\ldots, a_k;$ $b_1,\ldots , b_k$ be two sequences of different points in~$\K^n$, $n\ge 2$. For any sequence $L_1,\ldots, L_k\in SL(n,\K)$ there exists a p...
Z. Jelonek, Gustavo Menani, Maria Michalska· 0 citations
In this paper, we prove that for every integer $k\geq2$ and every $c>1$ sufficiently close to $1$, there is $\kappa>0$ such that every sufficiently large subset of $\{1,\ldots,N\}$ of density at least $(\log\log N)^{-\kappa}$ contains \[ x,\quad x+\lfloor n^c\rfloor,\quad x+\lfloor n^c\rfloor^2, \quad\ldots,\quad x+\lf...
Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian $\ast$-subalgebras of the complex matrix algebra ${\sf M}_n$ can be placed in finite free position: that is, when their relative position realizes this averaging exactly for every p...
Yuki Ueda· 0 citations
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