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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.
... can anyone justify this claim?

Given a von Neumann algebra $M$, its modular flow 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 of invertible $M$-$M$-bimodules. The bimodule $\mathbf{\Phi}(it)$ is the non-commutative $L^p$-space for the value $\frac 1 p=it$.


Given a ring $R$ of characteristic $p$, its Frobenius 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...

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minor nitpick: typo in title –  Yemon Choi Jun 20 '11 at 9:48
    
Thanks: Typo fixed –  André Henriques Jun 20 '11 at 9:58
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That's an interesting question. Personally, I don't understand/know what "the $\mathbb F_1$ Frobenius" should give (just archimedean factors or the whole completed zeta function?) But in the framework of Endomotives Connes (together with Consani and Marcolli) developed a (co)homological approach (based on cyclic homology) which leads to a spectral realization of zetas in form of a canonical $R$-action on a certain homology group which might be considered as an analogue of Frobenius action on l-adic cohomology. This is explained in alainconnes.org/docs/bookwebfinal.pdf pp. 556. –  user5831 Jun 20 '11 at 10:06
    
This may not be of much help (and perhaps you already know the article better than I do) but in his article for the "IMU: Visions and Perspectives" book, Connes starts with some big-picture overview of how one might start with number-theoretic considerations (class field theory) and be led to thinking about modular flow. See pages 1-6 of alainconnes.org/docs/imufinal.pdf –  Yemon Choi Jun 20 '11 at 21:15
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In what sense does the Frobenius provide a spectral realization of a zeta-function? The eigenvalues of the Frobenius operator (acting on cohomology groups if you like) are the zeros of the $\zeta$-function, is this sufficient? –  Junkie Jun 21 '11 at 3:02
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1 Answer

A low tech (naive?) piece of intuition comes straight from the definition of the modular operator and what happens if one tries to carry it over to finite fields.

The nontrivial automorphism $z\mapsto\overline{z}$ in $Gal(\mathbb{C}/\mathbb{R})$ is encoded in a von Neumann algebra via the existence of a *-operation $(zX)^\ast=\overline{z}X^\ast$. When $M$ is faithfully represented in a Hilbert space $H$ with cyclic and separating vector $\Omega$ we construct $SX\Omega:=X^\ast\Omega$, then $\Delta:=|S|^2$, then $\sigma_{t}(X)=\Delta^{it}X\Delta^{-it}$, so the modular group encodes $z\mapsto\overline{z}$ in some sense.

Let $n\in\mathbb{N}$ and note that the Frobenius automorphism of $F_{p^n}$ generates $Gal(F_{p^n}/F_p)$. Take an associative, unital algebra $R$ over $F_{p^n}$ equipped with a bijection $Q:R\to R$, satisfying $Q(zx)=Fr(z)Q(x)$ for all $z\in F_{p^n}$ and $x\in R$. If $R$ is faithfully represented on an $F_{p^n}$ vector space $V$ with cyclic and separating vector $\Omega$ then we obtain a map $S:V\to V$, $S x\Omega=Q(x)\Omega$, which has the property $$S z\xi=Fr(z)S\xi\quad\hbox{for all}\quad z\in F_{p^n},~\xi\in V.$$ Given such a map $T$ we can extract an $F_{p^n}$-linear map $T^n$ (the analogue of $T\rightarrow |T|^{2i}$ for an antilinear operator). Set $\Delta:=S^n$. As $\Omega$ is cyclic and separating and $Q$ is a bijection, $\Delta$ is invertible and we can form the maps $\sigma_{m}(x)=\Delta^{m} x\Delta^{-m}$ for each $m\in\mathbb{Z}$. Then we find that $\sigma_m$ is an $F_{p^n}$ algebra homomorphism and $\sigma_{m_1}\circ\sigma_{m_2}=\sigma_{m_1+m_2}$. If $R$ is a field over $F_{p^n}$ then we have the canoncial map $Q(x)=x^p$ and $\sigma_m(x)=Fr^{nm}(x)$. This extends easily to the case $R=M_n(K)$ for a field $K$. I'm not sure if a `nice' $Q$ exists for a general division algebra.

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Very nice ☺! Would you guess that your answer was one of Connes' motivations for making his claim? I'm just asking because it feels a bit like an "after the fact" justification... –  André Henriques Jun 21 '11 at 19:53
    
No, I don't think he was thinking anything as concrete. Possibly Connes' motivation was that the modular flow plays a similar role in classifying von Neumann algebras to the tools used in classifying local fields. Yemon's reference explains this quite well. –  Ollie Margetts Jun 22 '11 at 16:53
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