I've often heard Alain Connes say that the modular flow of Tomita-Takesaki theory should be thought of as a characteristic zero analog of the Frobenius endomorphism. <br> ... can anyone justify this claim?

>Given a von Neumann algebra $M$, its <b>modular flow</b>
is a canonically defined homomorphism
$$
\mathbf{\Phi}: i\mathbb R\quad\longrightarrow\quad \text{BIM}^\times(M)
$$
that, in the presence of a state (or weight), lifts to a homomorphism $i\mathbb R\to Aut(M)$.
Here, $\text{BIM}^\times(M)$ denotes the [2-group][1] of invertible $M$-$M$-bimodules.
The bimodule $\mathbf{\Phi}(it)$ is the non-commutative $L^p$-space for the value $\frac 1 p=it$.
><hr>
>Given a ring $R$ of characteristic $p$, its <b>Frobenius</b> is a canonically defined homomorphism
$$
\mathbf{F}:\mathbb N\quad\longrightarrow\quad End(R)
$$
such that $\mathbf{F}(1)$ sends $x$ to $x^p$.
More generally, $\mathbf{F}(n)$ sends $x$ to $x^{p^n}$.

So far, the only analogy I can see is that both $\mathbf{F}$ and $\mathbf{\Phi}$ are canonically defined actions...


  [1]: http://en.wikipedia.org/wiki/2-group