Bannon’s Math Research Blog

January 19, 2012

Rokhlin towers for hyperfinite II_1 factors…

When looking at cutting and stacking, there is a noncommutative Rokhlin tower theorem of Connes that can be found here: http://www.alainconnes.org/docs/automorphismes.pdf

This may be useful for Neshveyev-Stormer.

January 13, 2012

Hammer and Chisel versus Soaking

Another thing to try is this: take a problem and try to hit it very hard for two weeks straight. At the end of the two weeks write up what you have. Then file it away. This is what Liming suggested. If the problems are chosen well, experience will overlap strongly and eventually you will have the right facts to solve some nice problems.

This is, sort of, hammer and chisel. Somehow the prospect of payoff may help focus the mind better than the longterm development. If the problems are hard, and you give yourself a deadline, the effect may be the same, though. You will collect things in one place, get stuck, and then move along to another problem. (The new problem will probably be related to the old ones, strongly, and it is as likely that you will find the right fact for another earlier problem while working on the new one as if you were just staring at the one problem forever.)

The important thing is to try really really hard to solve the problem and carefully write things up and then move along.

In fact, the blog may be the best place to collect failed attempts, because the blog is easily searchable. Not sure about this until the numbers of projects/problems tried gets high. It may be better to write drafts and link to them on the webpage.

January 8, 2012

A Hilbert Algebra?

Let $\delta:N \rightarrow H$ be a closable, real, densely defined derivation into an $N-N$ bimodule $H$ . Let $D$ denote the domain of $\delta$ . We suggest the following involutive algebra structure on the orthogonal (Hilbert space) direct sum $D \oplus H$ . Let $(x,\xi)(y,\eta):=(xy,x \eta+\xi y)$ denote the product, and $(x,\xi)^{\#}:=(x^{*},J\xi)$ the involution, where $J$ is assumed to be an antiunitary involution on $H$ that replaces the bimodule structure with its opposite: $J(x \xi y)=y^{*}J\xi x^{*}$ .

From Wikipedia, we have the following definition of Hilbert algebra:

A Hilbert algebra is an algebra $\mathcal{A}$ with involution $\#$ and an inner product $\langle,\rangle$ such that
1) $\langle a,b \rangle= \langle b^{\#},a^{\#} \rangle$ for $a, b \in \mathcal{A}$ ;
2) left multiplication by a fixed a in $\mathcal{A}$ is a bounded operator;
3) $\#$ is the adjoint, in other words $\langle xy,z\rangle = \langle y, x^{\#}z \rangle$ ;
4) the linear span of all products $xy$ is dense in $\mathcal{A}$ .

Let’s explicitly see where our algebra fails these axioms. The involution $J$ is antiunitary on $H$ , and the usual involution $x\Omega \mapsto x^{*}\Omega$ on $L^{2}(M)$ is antiunitary also, so (1) holds. For (2), we need to worry mostly about the “second entry”…perhaps we’ll need to restrict our attention to “bounded vectors”. Sample computation: $\|x \eta+\xi y \| \leq \|x\|\|\eta\|+\|\xi\|\|y\|$ …but the second term is troublesome because the norm on $y$ is the wrong sort. This should be the two-norm. Perhaps if we require $\xi$ to be a bounded vector, then these operators would be bounded, perhaps, and we’d have (2). The condition (4) is easy to satisfy, since $(1,0)$ is the unit element of the algebra we have that the set of products is the entire domain (we’ve taken no closures, yet).

It is easy to see that (3) is the weak point here, in the second coordinate. Let’s record the failure here, though:

$\langle (x,\xi)(y,\eta),(z,\zeta)\rangle=\langle (xy,x \eta+\xi y),(z,\zeta)\rangle$

but $\langle x\eta+\xi y,\zeta\rangle=\langle x\eta,\zeta\rangle+\langle \xi y,\zeta\rangle$

and this is $\langle \eta,x^{*}\zeta\rangle+\langle y, \xi^{*}\zeta\rangle$ . Now the first coordinates are orthogonal (we’re freewheeling here, now) and since $\xi^{*}=J\xi$ can we get this? Looking at this last bit, things don’t quite parse…and we’re left with the Radon-Nikodym business again…so the adjoints don’t work out. Unless, for some mysterious reason, we can get $\xi^{*}\zeta$ to be a vector in $L^{2}(M)$ ! Perhaps this is not so mysterious, as a bounded vector is viewed as a map from $L^{2}(M)$ into $H$ , and its adjoint a map the other way. The question is, what’s the domain of the adjoint? (Among the million other questions like…can we use the orthogonality like this?)

The above will not work, because try as we might, $z$ never comes into play in the second coordinate…

April 28, 2011

Abstract Transverse Measure Theory and the Absence of Cartan Subalgebras

In Connes’s book there is a description of abstract transverse measure theory. (Some functoriality is important in this.) I wonder if one can develop a *noncommutative* abstract transverse measure theory that will allow consideration of the canonical measure associated to the space of leaves of a foliation to make sense even when there is not even a Cartan subalgebra? If so, this may give a hint as to what the canonical semigroup for a von Neumann algebra should be.

December 5, 2008

Radial MASAs in Free Burnside Group Factors

It is easy to see that the von Neumann algebra of an infinite free Burnside group doesn’t posess any MASAs generated by group elements. Since we know that these things are $II_{1}$ -factors, there certainly are MASAs. The idea is to construct one explicitly in the simplest way possible. One idea is to consider an analogue of the Laplacian (radial) MASA in a free group factor. More specifically, we should try to prove that $L_{a}+L_{a^{-1}}+L_{b}+L_{b^{-1}}$ generates a MASA, where $a$ and $b$ are standard generators of an infinite free Burnside group $B(2,p)$ .

December 1, 2008

Faa di Bruno and Free Probability

It is possible to glean many combinatorial identities using Faa di Bruno’s formula for the coefficients of higher derivatives of a composite function, for example, see David Vella’s paper. The resulting identities involve partitions of integers. I imagine that it should be possible to get a connection with free probability theory in which a noncommutative Faa di Bruno formula gives identities involving only noncrossing partitions.

Also, here, in a paper of Brouder, Fabretti and Krattenthaler, a noncommutative co-commutative Hopf algebra is constructed whose abelianization gives precisely the Faa di Bruno Hopf algebra, from which one can recover the classical Faa di Bruno formula. Basically, one considers one-variable generating functions of an infinite sequence of algebraically free coefficients. Also, in this case an explicit formula for the antipode is given in which Catalan numbers appear, and is interpreted in terms of trees and algebra free products. Perhaps this could be a useful starting point for answering the following

It is possible to glean many combinatorial identities using Faa di Bruno’s formula for the coefficients of higher derivatives of a composite function. The resulting identities involve partitions of integers. It should be possible to get a connection with noncommutative probability theory in which a noncommutative Faa di Bruno formula gives identities involving noncrossing partitions. A place to look would be David Vella’s paper and here, a paper of Brouder, Fabretti and Krattenthaler. In the latter paper, a noncommutative co-commutative Hopf algebra is constructed whose abelianization gives precisely the Faa di Bruno Hopf algebra, from which one can recover the classical Faa di Bruno formula. Basically, one considers one-variable generating functions of an infinite sequence of algebraically free coefficients. Also, in this case an explicit formula for the antipode is given in which Catalan numbers appear, and is interpreted in terms of trees and algebra free products… one should also consider noncrossing partitions! There seems to be room here for interpretation of the results in terms of free cumulants in free probability. How can we generalize David Vella’s results to the noncommutative setting and get interesting identities for combinatorial number theory? In David’s paper, one obtains formulas for the coefficients of the Taylor series of a composition of two functions from the coefficients of the Taylor series composed. It seems that he is working in the Hopf algebra in the commutative setting (if the Hopf algebra essentially consists of coefficients by duality). The combinatorial formulas for noncommuting coefficients should give analogous results. The project becomes: read the paper of Brouder, Fabretti and Krattenthaler, understand how to view Vella’s coefficient relations from the Hopf algebra point of view, and then generalize his machine and churn out some nice new identities.

November 18, 2008

The von Neumann Algebra of an Equivalence Relation and the Crossed Product

In Rings of Operators IV, Murray and von Neumann introduced the group-measure space construction, which constructs a von Neumann algebra from starting data $(G,X,\alpha)$ , where $G$ is a countable discrete group, $X=(X,\mathcal{B},\mu)$ is a measure space, and $\alpha$ is a free action of $G$ on $X$ . Since the action is clear in this context, we denote $\alpha(g)(x)$ by $g \cdot x$ throughout. In modern terms, the resulting von Neumann algebra is the crossed product $L^{\infty}(X)\rtimes G$ . It turns out that the (normal *-) isomorphism class of this von Neumann algebra depends only on the orbit equivalence relation $R_{G}(X)=\{(x,g \cdot x)|x\in X, g\in G\}\subseteq X\times X$ , in the sense that if we instead begin with $(G_{1}, X_{1},\alpha_{1})$ and $R_{G_{1}}(X_{1})\cong R_{G}(X)$ modulo null sets then the von Neumann algebras are also isomorphic.

A Borel space $X$ is standard if there is a Borel isomorphism (mod null sets) of $X$ onto [0,1]. Standard Borel spaces are quite common. For example, the Borel space obtained from the topology of any complete separable metric space is standard. Feldman and Moore showed that if $X$ is a standard Borel space, then any countable equivalence relation $R\subseteq X\times X$ that is also a Borel subset of $X\times X$ arises as the orbit equivalence relation of the action of some discrete group on $X$ . Note, though, that this action need not be free. In fact, a result of Furman states that there exists a countable standard equivalence relation that is not the orbit equivalence relation of any free action. However, when the action is free, the associated crossed product von Neumann algebra can be constructed from the equivalence relation alone, without any reference to the group. We suggest below how this may be done.

The crossed product $L^{\infty}(X)\rtimes G$ is the weak operator closure of the *-subalgebra of $B(L^{2}(X)\otimes l^{2}(G))$ generated by

$\{M_{f}\otimes 1: f\in L^{\infty}(X)\}\cup\{\sigma_g\otimes\lambda_g: g \in G\}$

where $\sigma$ here denotes the representation of $G$ on $L^{2}(G)$ extending the action on $L^{\infty}(X)$ defined by $(\sigma_{g}(f))(x)=f(g^{-1} \cdot x)$ , and $\lambda$ is the left regular unitary representation of $G$ on $l^{2}(G)$ . When working with the crossed product, it is convenient to consider the dense *-subalgebra of “polynomials” $\sum_{g}a_{g}u_{g}$ with $a_{g}\in L^{\infty}(X)$ and $u_{g}=\sigma_g\otimes\lambda_g$ with the multiplication rule

$a_{g}u_{g}b_{h}u_{h}=a_{g}\sigma_{g}(b_{h})u_{gh}$

on monomials.

Given an equivalence relation $\mathcal{R}=\mathcal{R}_{G}$ arising from a group action, consider the map

$(x,g\cdot x) \mapsto (x,g)$

from $\mathcal{R}$ into $X \times G$ , and the measure $m$ on $\mathcal{R}$ given by the “pullback” of the product measure on $X \times G$ . We may define a formal matrix product of $a,b \in L^{2}(\mathcal{R},m)$ by

$(ab)(x,k\cdot x)=\sum_{g\in G}a(x,g\cdot x)b(g \cdot x,k\cdot x),$

and define the von Neumann algebra $L(\mathcal{R})$ to be those $T\in B(L^{2}(\mathcal{R},m))$ for which there exists $a\in L^{2}(\mathcal{R},m)$ such that for all $b\in L^{2}(\mathcal{R},m)$ , $ab\in L^{2}(\mathcal{R},m)$ and $Tb=ab$ . In this setting, finitely supported functions give a dense *-subalgebra that, if the action is free, can be identified with the dense subalgebra of the crossed product by the map

$\phi:(a:(x, g^{-1}\cdot x)\mapsto a(x,g^{-1}\cdot x))\mapsto \sum_{g\in G}a_{g}u_{g}$ ,

where $a_{g}(x)=a(x,g^{-1}\cdot x)$ .

We note that the multiplication is preserved:

$\phi(a)\phi(b)=$

$(\sum_{g\in G}a_{g}u_{g})(\sum_{h\in G}b_{h}u_{h})$

$=\sum_{g,h\in G}a_{g}\sigma_{g}(b_{h})u_{gh}$

$=\sum_{k\in G}(\sum_{g\in G}a_{g}\sigma_{g}(b_{g^{-1}k}))u_{k}$ ,

and

$(\sum_{g\in G}a_{g}\sigma_{g}(b_{g^{-1}k})) (x)$

$=\sum_{g\in G}a_{g}(x)\sigma_{g}(b_{g^{-1}k})(x)$

$=\sum_{g\in G}a_{g}(x)b_{g^{-1}k}(g^{-1}\cdot x)$

$=\sum_{g\in G}a(x,g^{-1}\cdot x)b(g^{-1}\cdot x, k^{-1}g\cdot g^{-1}\cdot x)$

$=(ab)(x,k\cdot x)$ ,

therefore

$\phi(a)\phi(b)=\sum_{k\in G}(\sum_{g\in G}a_{g}\sigma_{g}(b_{g^{-1}k}))u_{k}=\phi(ab).$

Notice that under this identification, the “Cartan subalgebra” $L^{\infty}(X)$ in the crossed product becomes the algebra of “diagonal matrices” in $L(\mathcal{R})$ . See Sorin Popa’s survey from his ICM address for more information. If the action is not free the above identification does not work, since $L^{\infty}(X)$ is not maximal abelian in the crossed product, and is not identified with the diagonal algebra, which is maximal abelian in $L(\mathcal{R})$ . In fact it is true that if a von Neumann algebra $M$ possesses a Cartan subalgebra, that is, a maximal abelian *-subalgebra $A$ whose normalizing unitary elements generate all of $M$ , then $M$ can be realized as a generalized (cocycle) crossed product $L^{\infty}(X)\rtimes_{s} G$ , where $A$ is identified with $L^{\infty}(X)$ under the isomorphism.

November 17, 2008

Physical Motivation for Ergodic Theory

In statistical mechanics one aims to derive the macroscopic properties of a system, e.g. temperature and pressure, as probabilistic consequences of dynamical laws on many atoms, or particles. We obtain the macroscopic from the microscopic. Given $n$ particles in our ordinary 3 dimensional space, each particle has a position given by $(x_{i}, y_{i}, z_{i})$ and a momentum given by $(p_{x_{i}},p_{y_{i}},p_{z_{i}})$ . In general, we may consider a dynamical system with $n$ degrees of freedom described by the motion of a point $(q_{1}, q_{2}, ..., q_{n}, p_{1}, p_{2},...,p_{n})$ in $2n-$ dimensional phase space, where $q_{1}, q_{2}, ..., q_{n}$ are generalized positions and $p_{1}, p_{2},...,p_{n}$ are generalized momenta. Each point $P$ in this phase space corresponds to a physical state of the system, and changes in state correspond to the motion of this point in phase space. One useful physical description of the dynamical laws is Hamiltonian dynamics, i.e. assume there is a function $H(p,q,t)$ such that $\frac{\partial p_{i}}{\partial t}=-\frac{\partial H}{\partial q_{i}}$ and $\frac{\partial q_{i}}{\partial t}=\frac{\partial H}{\partial p_{i}}$ for $i \in \{1,2,...,n\}$ . In this picture, $H(p,q,t)$ is the total energy of the system, and conservation of energy requires that motions of $P$ are confined to level “energy surfaces” $E$ of constant energy $c$ , i.e. $E=\{(q,p,t)| H(p,q,t)=c\}$ . Given such an energy surface $E$ , the limit of the volume element between $E$ and neighboring energy surfaces $E+dE$ defines a measure on $E$ that is, by a theorem of Liouville, invariant under the motion of the point $P$ in phase space.

It is a physically reasonable assumption (due to Boltzmann) that if the energy surface $E$ has finite measure, and the motion of $P$ does not form a closed curve, then $P$ should move almost everywhere over the energy surface. Mathematically, this means that any (almost everywhere) invariant subset of $E$ must have full measure or have measure zero. This is the ergodic hypothesis. This hypothesis heuristically justifies the agreement of the time average of a dynamical quantity and its “phase average” (the integral with respect to the above invariant measure).

Here is an intuitive motivation for this justification. Consider $X=[0,1]$ , and $\tau:X \rightarrow X$ a transformation mapping $X$ into itself. Given a point $x\in X$ , consider the orbit of $x$ under $\tau$ . Now, for any subset $A\subseteq X$ , we can consider how often the orbit of $x$ is in $A$ . Let $\chi_{A}$ denote the characteristic function of $A$ and define $T\chi_{A}(x)=\chi_{A}(\tau(x))$ . Then $\frac{1}{n}\sum_{i=0}^{n-1}T^{i}\chi_{A}(x)$ is the average number of times the first $n$ points of the orbit of $x$ are in $A$ , but it is also reminiscent of a Riemann sum which, as $n\rightarrow \infty$ , should approximate the measure of $A$ , provided that the orbit of $\tau$ sufficiently “fills the space”. Here we could have replaced the characteristic function by any measurable function $f$ , and then we would have said that the “time average” $lim_{n\rightarrow \infty} \frac{1}{n} \sum_{i=0}^{n-1}T^{i}f(x)$ of the transformation $T$ applied to the measurable function should agree with the “space average” $\int_{0}^{1}f dx$ , provided that $T$ was an ergodic transformation. With work, this line of investigation yields the mean (von Neumann) ergodic theorem, and the pointwise (Birkhoff) ergodic theorem. For a brisk overview of these see ergodic theory.