Some Results in the Hyperinvariant Subspace Problem and Free Probability

Some Results in the Hyperinvariant Subspace Problem and Free Probability. (May 2009) Gabriel H. Tucci Scuadroni, B.A., Universidad de la República, Uruguay; Electrical Engineer Diploma, Universidad de la República, Uruguay Chair of Advisory Committee: Dr. Kenneth Dykema This dissertation consists of three more or less independent projects. In the first project, we find the microstates free entropy dimension of a large class of L∞[0, 1]– circular operators, in the presence of a generator of the diagonal subalgebra. In the second one, for each sequence {cn}n in l1(N), we define an operator A in the hyperfinite II1-factor R. We prove that these operators are quasinilpotent and they generate the whole hyperfinite II1-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra, and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of A are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of A∗A. Finally, let {Tk}k=1 be a family of ∗–free identically distributed operators in a finite von Neumann algebra. In this paper, we prove a multiplicative version of the Free Central Limit Theorem. More precisely, let Bn = T ∗ 1 T ∗ 2 . . . T ∗ nTn . . . T2T1 then Bn is a positive operator and B 1/2n n converges in distribution to an operator Λ. We completely determine the probability distribution ν of Λ from the distribution μ of |T |. This gives us a natural map G : M+ → M+ with μ 7→ G(μ) = ν. We study how this map behaves with respect to additive and multiplicative free convolution.

超不变子空间问题与自由概率中的一些结果（2009年5月） 加布里埃尔·H·图奇·斯夸德罗尼，乌拉圭共和国大学文学学士；乌拉圭共和国大学电气工程师文凭 指导委员会主席：肯尼斯·戴克马博士 本论文包含三个或多或少相互独立的项目。在第一个项目中，在对角子代数的一个生成元存在的情况下，我们找到了一大类\(L_{\infty}[0, 1]\) - 循环算子的微观态自由熵维数。在第二个项目中，对于\(\ell_1(\mathbb{N})\)中的每一个序列\(\{c_n\}_n\)，我们在超有限\(II_1\)型因子\(R\)中定义一个算子\(A\)。我们证明这些算子是拟幂零的，并且它们生成整个超有限\(II_1\)型因子。我们表明它们具有与冯·诺依曼代数相关联的非平凡、闭的不变子空间，并且我们提供了足够的证据表明这些算子对于超不变子空间问题是有趣的。我们还给出了它们的一些性质。特别地，我们表明\(A\)的实部和虚部是等分布的，并且我们找到了一个组合公式以及一种解析方法来计算它们的矩。我们给出了一种计算\(A^{*}A\)的矩的组合方法。最后，设\(\{T_k\}_{k = 1}^{\infty}\)是一个有限冯·诺依曼代数中的一族\(*\) - 自由同分布算子。在本文中，我们证明了自由中心极限定理的一个乘法版本。更确切地说，设\(B_n = T_1^{*}T_2^{*}\cdots T_n^{*}T_n\cdots T_2T_1\)，那么\(B_n\)是一个正算子，并且\(B_n^{1/2n}\)依分布收敛到一个算子\(\Lambda\)。我们从\(\vert T\vert\)的分布\(\mu\)完全确定了\(\Lambda\)的概率分布\(\nu\)。这给我们一个自然映射\(G: M_+ \to M_+\)，其中\(\mu \mapsto G(\mu)=\nu\)。我们研究了这个映射关于加法和乘法自由卷积的行为。