We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on
R
m
...are replaced by non-commutative laws of
m
-tuples. We prove an analog of the Monge–Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu’s non-commutative
L
2
-Wasserstein distance using a new type of convex functions. As a consequence, we show that if (
X
,
Y
) is a pair of optimally coupled
m
-tuples of non-commutative random variables in a tracial
W
∗
-algebra
A
, then
W
∗
(
(
1
-
t
)
X
+
t
Y
)
=
W
∗
(
X
,
Y
)
for all
t
∈
(
0
,
1
)
. Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of
m
-tuples is not separable with respect to the Wasserstein distance for
m
>
1
.
We construct a one parameter family of ICC groups $\{G_t\}_{t > 1}$, with the property that the group factor $L(G_t)$ is isomorphic to the interpolated free group factor $L(\mathbb ...F_t):=L(\mathbb{F}_2)^{1/\sqrt{t-1}}$, for all $t$. Moreover, the groups $G_t$ have fixed cost $t$, are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost.
We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups, ...thus proving that the measure of each atom of any n \times n. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic, and thus the polynomial has a distribution that is real-analytic except at a finite number of points.
We obtain a complete classification of a large class of non-almost-periodic free Araki–Woods factors $\Gamma(\mu, m)''$ up to isomorphism. We do this by showing that free Araki–Woods factors ...$\Gamma(\mu, m)''$ arising from finite symmetric Borel measures $\mu$ on $\mathbb{R}$ whose atomic part $\mu_a$ is nonzero and not concentrated on $\{0\}$ have the joint measure class $\mathcal C(\bigvee_{k \geq 1} \mu^{\ast k})$ as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.
We show that finite-rank perturbations of certain random matrices fit in the framework of infinitesimal (type-B) asymptotic freeness. This can be used to explain the appearance of free harmonic ...analysis (such as subordination functions appearing in additive free convolution) in computations of outlier eigenvalues in spectra of such matrices.
The subject of Operator Algebras is a flourishing broad area of mathematics which has strong ties to many other areas in mathematics including Functional/Harmonic Analysis, Topology, ...(non-commutative) Geometry, Geometric Group Theory, Dynamical Systems, Descriptive Set Theory, Model Theory, Random Matrices and many more. The goal of the Oberwolfach meeting is to give its participants a global view of the subject to maintain and strengthen contacts between researchers from these different directions, making it possible for the most important developments and techniques to be disseminated.
We show that if Γ is a finitely generated finitely presented sofic group with zero first L2-Betti number, then the von Neumann algebra L(Γ) is strongly 1-bounded in the sense of Jung. In particular, ...L(Γ)≆L(Λ) if Λ is any group with free entropy dimension >1, for example a free group. The key technical result is a short proof of an estimate of Jung
We show that the spectral measure of any non-constant non-commutative polynomial evaluated at a non-commutative n-tuple cannot have atoms if the free entropy dimension of that n-tuple is n (see also ...work of Mai, Speicher, and Weber). Under stronger assumptions on the n-tuple, we prove that the spectral measure of any non-constant non-commutative polynomial function is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval's length.
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