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.