Tomita-Takesaki theory for a simple class of crossed products

Question

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 $C(\hat {\mathbb Z}) \rtimes \mathbb N$.)

Answer 1

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.

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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 $$ \big((it)\text{-densities on } X\big) \rtimes G. $$

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[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.