Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This question is inspired by the construction of the time evolution for endomotives as given by Connes and Marcolli in their book http://www.alainconnes.org/docs/bookwebfinal.pdf.

Let $M$ be a monoid (countable and discrete) acting on a locally compact Hausdorff space $X$ and consider the $C^*$-algebra $A$ given by the (semigroup) crossed product $$A = C_0(X) \rtimes M$$ Now let us assume given a state $\mu : A \to \mathbb C$ (for example one could try to use the one induced by the integration map $f \in C_0(X) \mapsto \int_X f $ with respect to the Haar measure on $X$, as in the above reference) such that the associated GNS constrution yields a faithful representation $\pi : A \to \mathcal B (H)$, further let us assume that the map $\mathbb R \to Aut(\pi(A)'')$ induced by Tomita-Takesaki theory on the von Neumann algebra given by the bicommutant $\pi(A)''$ restricts to a map $$\sigma : \mathbb R \to Aut(A)$$ My question is now if there are known examples where $\sigma$ is described explicitly in the literature. (Of course my general formulation allows trivial examples which are not asked for...)

In some sense $\sigma$ should depend only on $M$ and its action on $X$ because this is the only source for noncommutativity. Recall that Tomita-Takesaki theory is only visible in the noncommutative case, in the commutative case the $\mathbb R$-action is trivial.

The only cases I know are essentially all given by so called "Bost-Connes type systems", as explained for example in the nice article http://arxiv.org/pdf/0710.3452v2 by Laca, Larsen and Neshveyev. (The best known example is given by the original Bost-Connes system $C(\hat {\mathbb Z}) \rtimes \mathbb N$.)

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

I'd like to take this opportunity to dispel this wrong beleif:

"Recall that Tomita-Takesaki theory is only visible in the noncommutative case, in the commutative case the $\mathbb R$-action is trivial"

As mentioned in this question, given a C*-algebra and/or von Neumann algebra $A$, the modular flow is a homomorphism $i\mathbb R\to \text{BIM}^\times(A)$, where the latter refers to the 2-group of invertible $A$-$A$-bimodules.

Now let's take $A$ to be commutative, and let $X$ be the (locally compact/measurable) space on which $A$ is the algebra of functions.

The bimodule correcponding to $it\in i\mathbb R$ is the module (since $A$ is abelian, the left and right actions coincide) of $it$-densities on $X$. For any $\kappa\in \mathbb C$, a $\kappa$-density is a section of a particular line bundle that exists on $X$.

To simplify by exposition, I'll cheat and assume that $X$ is an oriented manifold: then it's easier to say what a density is: it's a section of $\Lambda^{top}(TX)$. Let $\mathcal L$ be the total space of the line bundle of densities, and let $\mathcal L_+\subset \mathcal L$ be its positive part. Then $\mathcal L_+$ is a principal $\mathbb (R_+,\cdot)$-bundle on $X$. The bundle of $\kappa$-densities is the associated bundle for the representation $\lambda\mapsto \lambda^\kappa$ of $\mathbb R_+$ on $\mathbb C$, and a $\kappa$-density is a section of that bundle.

Now let's go back to the actual question.
If $G$ is a group that acts on $X$ (or monoid, assuming it acts by covering maps), then it induces an action on the bundle of $\kappa$-densities.

The $\big((\text{functions on } X) \rtimes G\big)$-bimodules that describe the modular flow are then given by $$ \big((it)\text{-densities on } X\big) \rtimes G. $$

[Added later]
Let me add a few words in order to connect the above story to the one that you are familiar with:

First of all, $(\text{densities on } X) \rtimes G$ can be identified with $L^1$ of the algebra, i.e. the predual in the von Neumann algebra setting [I'm ignoring all issues relted to completions]. Indeed, an element $$\bigoplus_{g\in G}\mu_g\in(\text{densities on } X) \rtimes G$$ can be identified with the functional that sends an algebra element $$\bigoplus_{g\in G}f_g\in(\text{functions on } X) \rtimes G$$ to the number $\sum_{g\in G}\displaystyle\int_X f_g(x) d\mu_{g^{-1}}(x)$.

Given a state $\phi = \bigoplus_{g\in G}\phi_g$, the modular flow is given by $\sigma_t^\phi(f) = \phi^{it} f \phi^{-it}$ (see p. 1083 of Yamagami's paper Algebraic aspects in modular theory). For an arbitrary such state $\phi$, the expression $\phi^{it}$ can be quite difficult to compute. But if we take $\phi$ such that $\phi_g=0$ for all non-trivial elements of $G$, then things become suddenly much simpler: "computing" $\phi^{it}$ is now almost tautological. We can now go back to the question of computing $\sigma_t^\phi(f)$ and give a complete answer:

The $g$-component of $\sigma_t^\phi(f)$ is given by $\left(\frac{d(g_\*\phi)}{d\phi}\right)^{it} f_g$,

where $f_g$ is the $g$-component of $f$, where $g_\*\phi$ is the density obtained by letting $g\in G$ act on $\phi$, and where $\frac{d(g_\*\phi)}{d\phi}$ is the Radon-Nikodym derivative.

share|improve this answer
Dear André, thank you very much for your answer! I have to admit that I first of all have to understand your definition of the modular flow! Do you have a reference for your description? I've never seen this before, it looks very interesting (and geometric). –  user5831 Jun 20 '11 at 23:12
No. I don't have a reference. However, I do have a (good) reference for the non-commutative L^p-spaces: Yamagami's paper "algebraic aspects in Modular theory" pages.uoregon.edu/njp/Yam.pdf –  André Henriques Jun 20 '11 at 23:20
Thank you very much for the reference. I'll have a look at it. One thing I do not understand is that in your formulation I don't see any representation of the $C^*$-/von Neumann algebra in hand. For me this is like the crucial thing for doing TT. As far as I understand it is a representation-dependend theory. –  user5831 Jun 20 '11 at 23:26
The earliest reference I know for this kind of interpretation of the TT story is Alan Weinstein's paper on the modular class (I think it's the one in J Geom Phys 1997). –  David Ben-Zvi Jun 21 '11 at 1:12
Dear David, thank you very much for the reference! This is very helpful. –  user5831 Jun 21 '11 at 11:26
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.