[Alain Connes][1]: "**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][2] 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][3], $\rho(a) = u. \vert \rho(a) \vert$ 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][4] 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][5] 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][6]* : 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][7], for more details).

*Advertising*: there will have a [Master Class in Modular Theory][8] by [Serban Stratila][9] and [Masamichi Takesaki][10], at Chennai (India) from 24 Nov. to 04 Dec. 2014.


  [1]: http://en.wikipedia.org/wiki/Alain_Connes
  [2]: http://en.wikipedia.org/wiki/Von_Neumann_algebra
  [3]: http://en.wikipedia.org/wiki/Polar_decomposition
  [4]: http://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors
  [5]: http://en.wikipedia.org/wiki/Direct_integral
  [6]: http://en.wikipedia.org/wiki/Tomita%E2%80%93Takesaki_theory
  [7]: http://en.wikipedia.org/wiki/Tomita%E2%80%93Takesaki_theory#Structure_of_type_III_factors
  [8]: http://www.imsc.res.in/~sunder/smpost.pdf
  [9]: http://www.humboldt-club.infim.ro/public_html/MEMBERS/PAGES/stratila.htm
  [10]: http://en.wikipedia.org/wiki/Masamichi_Takesaki