All Questions
Tagged with oa.operator-algebras free-probability
18 questions
43
votes
8
answers
3k
views
How to quantify noncommutativity?
If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
- 1,890
25
votes
3
answers
2k
views
Why did Voiculescu develop free probability?
I was recently asked why Voiculescu developed free probability theory. I am not very expert in this and the only answer I was able to provide is the classical one: he was challenging the isomorphism ...
- 6,034
10
votes
4
answers
903
views
Relationship between free probability and deterministic graphs?
Consider the $N\times N$ matrix $$
M = \left(\begin{array} \\
0 & 1 & & 0 \\
1 & \ddots & \ddots & \\
& \ddots & \ddots & 1 \\
0 & & 1 & 0 \\
\end{...
- 1,890
8
votes
1
answer
426
views
Is $L(\mathbb{Z}*\mathbb{Z}_{2})$ a free group factor?
This is a reference request for something that is likely to be well-known to operator algebraists. I will not, therefore, include the technical definition of free product of finite von Neumann ...
- 7,047
8
votes
0
answers
362
views
C* algebras of free semicircular systems
It was shown by Pimsner and Voiculescu in 1982 that the reduced group $C^{*}$-algebras $C^{*}_{r}(\mathbb{F}_{n})$ and $C^{*}_{r}(\mathbb{F}_{m})$ are isomorphic if and only if $n = m$ (here, $\mathbb{...
- 629
6
votes
0
answers
1k
views
Relationship between R-transform and free convolution of random matrices?
I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...
- 1,890
5
votes
1
answer
281
views
The definition of amalgamated free product for general von Neumann algebras
When discussing the amalgamated free product von Neumann algebras, people often assume that the algebras are $\sigma$-finite. I am wandering if there is any literature on the amalgamated free product ...
- 481
4
votes
1
answer
715
views
Classical convolution VS Free Convolution
We denote $\varphi:\mathbb R^2\rightarrow\mathbb R$ the addition of real numbers, and $\varphi_*:M_1(\mathbb R^2)\rightarrow M_1(\mathbb R)$ the induced push-forward map (where $M_1(\Delta)$ stands ...
- 2,135
4
votes
1
answer
250
views
Can we solve the FGF problem by finding an appropriate action?
If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
4
votes
0
answers
227
views
Recursive formula for joint moments in free probability
Suppose $\mathfrak{A}$ is an algebra (over $\mathbb{C}$, let's say), $\phi$ a linear functional on $\mathfrak{A}$, and $A_1, A_2 \subset \mathfrak{A}$ subalgebras which are $\phi$-freely independent. ...
- 275
3
votes
1
answer
145
views
Existence of free operators, independent and with given distributions
Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...
- 421
3
votes
0
answers
203
views
Uniqueness of the reduced free product of unital completely positive maps
For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = *...
- 3,984
2
votes
1
answer
199
views
Uniqueness of free complements
Let $A,B$ be subfactors of a II$_1$ factor $M$ with $A*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a free complement for $A$ ...
- 1,411
2
votes
1
answer
199
views
Brown measure of left shift operator
Let $L$ be the left shift operator on $\ell^2(\mathbb{Z})$ with trace $\tau(T) := \langle T \delta_0, \delta_0 \rangle$.
How can I show that the Brown measure of $L$ is the uniform measure on the ...
- 123
2
votes
1
answer
356
views
The fundamental group of the von Neumann algebra of a free group of infinite rank
It is well-known that that the fundamental group (in the sense of Murray and von Neumann) of the factor $L(F_{\mathbb{N}})$ is
$\mathbb{R} \smallsetminus \{0\}$. I think that by the cutting and ...
- 481
2
votes
0
answers
119
views
Random matrices may be asymptotically free but never free themselves?
It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
- 21
1
vote
1
answer
407
views
The problems of global asymptotic freeness
Let $X_{N}\in\mathcal{M}_{N}\big(L^{\infty-}(\Omega,\mathbb{P})\big)$ be a $N\times N$ random complex matrix such its entries $(x_{ij}, 1\leq i, j\leq N)$ be $i.i.d.$, centred with variance $1$. $X_{...
- 137
0
votes
1
answer
308
views
Berry-Esseen type theorem for Monotonic independence
The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases ...
- 152