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I heard two quotes, one from Alain Connes and an other one from Orlov. Alain Connes was talking about noncommutative geometry and he said the following: " a noncommutative algebra creates its own internal time "

In a talk by Orlov about Mirror symmetry, he was asked if he considers the monoidal structure on the derived category of a scheme. Orlov said that "the monoidal structure is not natural in this context and we should not base our theory on this structure " he added "The tensor product is a NATURAL structure to consider in the category of Motives"

I will be happy if someone can put some enlightenment to what Connes and Orlov meant (if the quotations above make sense ) ?

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  • $\begingroup$ There is a very particular way in which those quotes make sense to me, but I'm afraid is not a mathematical way so I can't really answer :) $\endgroup$ – godelian Nov 5 '14 at 23:40
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    $\begingroup$ The first quote is, IIRC, a reference to the modular automorphism group on a Type III factor. At the risk of being Philistine, it is yet another example of people taking Connes's rhetorical flourishes more seriously than, say, the actual mathematics he does in his papers $\endgroup$ – Yemon Choi Nov 6 '14 at 0:06
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    $\begingroup$ This should really be two separate questions, by the way $\endgroup$ – Yemon Choi Nov 6 '14 at 0:06
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    $\begingroup$ For the first quote see en.wikipedia.org/wiki/Tomita%E2%80%93Takesaki_theory. I agree that these should be separate questions. $\endgroup$ – Qiaochu Yuan Nov 6 '14 at 0:26
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    $\begingroup$ Your first question, regarding Connes, is answered here: mathoverflow.net/questions/57820/… $\endgroup$ – Daniel Moskovich Nov 6 '14 at 1:11
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Here is a guess about the remark of Orlov. Suppose that one wants to define a good notion of noncommutative scheme, given that an affine noncommutative scheme is an associative algebra. Trying to define the spectrum of an associative algebra leads to various problems (c.f. this answer), so a different approach is needed. On the other hand there are various invariants of assocative algebras, like algebraic K-theory, which all happen to be Morita invariant: they depend only on derived categories. Further, a theorem of Bondal and Van den Bergh says that the derived category of a commutative scheme is equivalent to the derived category of some dg-algebra. This suggests to define a noncommutative scheme as the derived category of a dg-algebra. This will then include both affine noncommutative schemes and commutative schemes. This is the approach practised by the Moscow school (Bondal, Orlov, Kaledin, Lunts, etc.).

This explains the first remark: it is essential to consider noncommutative spaces only up to Morita equivalence. If we would consider the symmetric monoidal structure in the definition of noncommutative scheme, then any two commutative schemes would be isomorphic in the noncommutative world iff they were isomorphic in the usual commutative sense. See work of Balmer for this. In fact most interesting morphisms in the noncommutative world do not respect the symmetric monoidal structures coming from the commutative world.

Regarding the second remark, suppose that we want a theory of motives for noncommutative schemes. In fact, such theories already exist after Cisinski-Tabuada and Robalo. For example, following Robalo, we can echo the Morel-Voevodsky construction in our setting: take the (infinity-)category of noncommutative schemes, pass to the free cocompletion (infinity-presheaves), impose A^1-homotopy invariance and descent with respect to a noncommutative version of the Nisnevich topology, and then formally stabilize with respect to P^1. This gives a noncommutative version of the stable motivic homotopy category. Now suppose that we were to include symmetric monoidal structures in the picture, i.e. start with the infinity-category of symmetric monoidal dg-categories, and repeat the same construction. I have not checked this, but I think that one would just recover (an enlargement of) the Morel-Voevodsky stable motivic homotopy category of commutative schemes. My guess is that Orlov may have had something along these lines in mind.

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  • $\begingroup$ Two things. First: does Morita invariance imply dependence only on derived categories? A priori it only means dependence only on categories of modules. Cannot different algebras have equivalent derived categories without having equivalent categories of modules? Second: afaik, derived category of a general noncommutative algebra simply does not have any monoidal structure. For that you need something like Hopf algebra structure (maybe up to Morita). $\endgroup$ – მამუკა ჯიბლაძე Nov 8 '14 at 18:36
  • $\begingroup$ I'm not sure I understand your remarks. 1) Here, It is about derived Morita equivalences. 2) The point is that we are considering differential graded algebras and not only algebras. $\endgroup$ – Max Nov 8 '14 at 19:13
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    $\begingroup$ Right, I should have said derived Morita invariance. And Orlov's remark was about the derived category of a commutative scheme. $\endgroup$ – AAK Nov 8 '14 at 19:48
  • $\begingroup$ @Max Seems like 1) has been clarified. As for 2), I don't understand how dg would change anything in the context of my comments; still, my second one is flawed. Tensor structure on modules over a commutative ring does not have to do anything with Hopf structure, and a general commutative ring does not have any. The essence is, for $A$-modules $M$ and $N$, to produce an $A$-module from the $A\otimes A$-module $M\otimes N$. Hopf structure would restrict it along the diagonal $A\to A\otimes A$; while for any commutative $A$ one may extend it along $A\otimes A\to A$. $\endgroup$ – მამუკა ჯიბლაძე Nov 9 '14 at 6:33
  • $\begingroup$ In your first question there was two questions. 1) Morita invariance imply dependence only on derived categories? 2)Cannot different algebras have equivalent derived categories without having equivalent categories of modules $\endgroup$ – Max Nov 9 '14 at 18:16
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Alain Connes: "a noncommutative algebra creates its own intrinsic time".

First of all, as Yemon Choi commented, this quote of Alain Connes is a slogan, not a theorem.
"Most of the NC algebras create their own intrinsic time" would be a bit more correct, more precisely:
Theorem: a von Neumann algebra of type $\rm III$ creates its own intrinsic time up to inner automorphisms.

In the rest of the answer we will see how we can generate a von Neumann algebra from a given NC algebra, we will define all the notions appearing in the above theorem and explain what does it mean.

From a noncommutative (unital associative) algebra $\mathcal{A}$ (with a countable base $\mathcal{b}$) over $\mathbb{C}$ , we can generate a von Neumann algebra as follows: let $H = l^2(\mathcal{b})$ be the Hilbert space generated by $\mathcal{b}$, let $H_0 = \{v \in H \ \vert \ a.v \in H \ \forall a \in \mathcal{A} \}$ (supposed dense in $H$) and $\rho$ the left regular representation of $\mathcal{A}$ on $H_0$. If $\forall a \in \mathcal{A}, \ \rho(a)$ is bounded, then $\mathcal{M} = (\rho(\mathcal{A}) \cup \rho(\mathcal{A})^*)''$ is the von Neumann algebra generated by $\mathcal{A}$ [else, by the polar decomposition, $\rho(a) = u. \vert \rho(a) \vert$ with $u$ a partial isometry (bounded), and $\mathcal{M}$ is the vN algebra generated by these partial isometries]. Note that $a \to a^*$ is the involution and $\mathcal{E}''= (\mathcal{E}')'$, is the bicommutant of $\mathcal{E} \subset B(H)$ the algebra of bounded operators.

A von Neumann algebra $\mathcal{M}$ is a factor if and only if its center is trivial: $\mathcal{M} \cap \mathcal{M}' = \mathbb{C}$.
Every von Neumann algebra $\mathcal{M}$ decomposes as a direct integral of factors (Murray - von Neumann).
There are three types of factors:
a factor is type $\rm I$ if it admits projections with a finite dimensional range;
else it is type $\rm II$ if it admits no projection equivalent to an own subprojection;
else it is type $\rm III$ (and we can prove that all the projections are equivalent).

Modular theory : let $\mathcal{M}\subset B(H)$ be a von Neumann algebra. Let $\Omega \in H$ be a cyclic and separating vector (i.e., $\mathcal{M}.\Omega$ and $\mathcal{M}'.\Omega$ are dense in $H$). Let $S : H \to H$ be the closure of the anti-linear map $a\Omega \to a^{*}\Omega$, it admits a polar decomposition $S = J\Delta^{1/2}$, with $J$ anti-linear unitary and $\Delta$ positive.
$JMJ = \mathcal{M}'$, $\Delta^{it} \mathcal{M}\Delta^{-it} = \mathcal{M}$ and $\sigma_{\Omega}^{t}(a) = \Delta^{it} a \Delta^{-it}$ gives the modular action of $\mathbb{R}$ on $\mathcal{M}$.

Connes' Radon-Nikodym theorem: let $\Omega'$ be another vacuum (i.e. cyclic-separating) vector, then there is a Radon-Nikodym map $u_{t} \in \mathcal{U} ( \mathcal{M})$ [unitary operators in $\mathcal{M}$], defined such that $u_{t+s} = u_{t} \sigma_{t}^{\Omega'} (u_{s})$ and $\sigma_{t}^{\Omega'} (x) = u_{t} \sigma_{t}^{\Omega}(x) u^{\star}_{t}$. Then, modulo $Inn(\mathcal{M})$, $\sigma_{t}^{\Omega} $ is independent of the choice of $\Omega$, i.e., there exist an intrinsic group morphism $\delta : \mathbb{R} \to Out (\mathcal{M}) = Aut(\mathcal{M})/Inn(\mathcal{M})$.
What Alain Connes calls the own intrinsic time, is precisely $\delta$.

For the type $\rm I$ or $\rm II$, the modular action is inner, and so $\delta$ is trivial (i.e. $\ker(\delta) = \mathbb{R}$). It's non-trivial for the type $\rm III$. A factor is type $\rm III_1$ if and only if $\ker(\delta) = \{0 \}$. The type $\rm III_1$ factors exist, moreover, in some sense, most of the factors are $\rm III_1$ (see Structure of type III factors, for more details).

Advertising: there will have a Master Class in Modular Theory by Serban Stratila and Masamichi Takesaki, at Chennai (India) from 24 Nov. to 04 Dec. 2014. The videos of the lectures are available here.

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    $\begingroup$ I have a comment on the "up to inner automorphisms" part of Sebastien's answer. The automorphism $\sigma^t_\Omega:M\to M$ does depend on $\Omega$, and indeed two different states $\Omega$ and $\Omega'$ produce automorphisms $\sigma^t_\Omega$ and $\sigma^t_{\Omega'}$ that are only equal in $Out(M)$. But there's is a better statement than merely saying that $\sigma^t_\Omega$ and $\sigma^t_{\Omega'}$ become equal in $Out(M)$. The better statement is that the bimodules ${}_MM_{\sigma^t_\Omega,M}$ and ${}_MM_{\sigma^t_{\Omega'},M}$ (where one twists the right actions) are canonically isomorphic. $\endgroup$ – André Henriques Nov 10 '14 at 20:40
  • $\begingroup$ @AndréHenriques, the quote becomes: "a NC algebra creates its own intrinsic one-parameter bimodule". $\endgroup$ – Sebastien Palcoux Nov 10 '14 at 21:24
  • $\begingroup$ Well, at occasions (I was in the audience) Connes by inner time may mean not $\delta$ but the $\Omega$ dependent modular flow $\sigma^\Omega$. This is what happens, for example for the non commutative 2-torus in his work about "modular curvature" $\endgroup$ – Nicola Ciccoli Nov 19 '14 at 8:55
  • $\begingroup$ A cyclic and separating vector $\Omega$ is also called a vacuum or also a bicyclic vector. A factor admits a bicyclic vector, and more generally, see this notes of Serban Stratila (prop. page 2). $\endgroup$ – Sebastien Palcoux Nov 24 '14 at 17:41

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