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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.

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.

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

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 $\mathcal{b}$, then $\mathcal{A}$ admits a faithful representation $\rho$ on $H$ by left multiplication (the regular representation).
 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 dim. 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).

Connes' Radon-Nikodym theorem: let $\Omega'$ be another vacuum vector, then there is a Radon-Nikodym map $u_{t} \in \mathcal{U} ( \mathcal{M})$ [the group of 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 $\delta$.

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.

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).

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 $\delta$.

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.
" There are NC algebras creating their own intrinsic time " would be more correct, and more precisely:
Theorem: a von Neumann algebra of type $\rm III$ creates its own intrinsic time up to inner automorphism.

Anyway, from a noncommutative (associative) algebra $\mathcal{A}$ (with a countable base $\mathcal{b}$) on the field $\mathbb{C}$ , we can generate a von Neumann algebra as follows: let $H = l^2(\mathcal{b})$ be the Hilbert space generated by $\mathcal{b}$, then $\mathcal{A}$ admits a faithful representation $\rho$ on $H$ by left multiplication (the regular representation).
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, and $\mathcal{M}$ is the von Neumann 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 iff 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 dim. 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 $\mathcal{M}$.

NC Radon-Nikodym theorem: Let $\Omega'$ be another vacuum vector, then there exists a Radon-Nikodym map $u_{t} \in \mathcal{U} ( \mathcal{M})$, define 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 $\Omega$, ie, there exist an intrinsic $\delta : \mathbb{R} \to Out (\mathcal{M})$ with $Out (\mathcal{M}) = Aut(\mathcal{M})/Inn(\mathcal{M})$. What Alain Connes calls the own intrinsic time, is precisely $\delta$.
On type $\rm I$ or $\rm II$ the modular action is inner, and so $\delta$ trivial. It's non-trivial for the type $\rm III$.

Remark: the type $\rm III$ factors exist, moreover, in some sense, most of the factors are type $\rm III$.

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.

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.
"There are NC algebras creating their own intrinsic time" would be more correct, and 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 $\mathcal{b}$, then $\mathcal{A}$ admits a faithful representation $\rho$ on $H$ by left multiplication (the regular representation).
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 dim. 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 vector, then there is a Radon-Nikodym map $u_{t} \in \mathcal{U} ( \mathcal{M})$ [the group of 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.