TITLE:  A definition of non-commutative metrisable space
QUESTION [8 upvotes]: If $X$ is a compact metrisable space, a metric $d$ on $X$ can be take as an element of  $C(X\times X)$ such that
(1)  $ev_x\otimes ev_y (d)=d(x,y)\geq 0$ for all $x,y\in X$ (Non-negativity).
(2) $ev_x\otimes ev_y (d)=0$ iff $x=y$ (Identity of indiscernibles).
(3) $ev_x\otimes ev_y (d)=ev_y\otimes ev_x (d)$ for $x,y\in X$ (Symmetry).
(4) For all $x,y,z\in X$, $ev_x\otimes ev_y (d)\leq ev_x\otimes ev_z (d)+ev_z\otimes ev_y (d)$ (Triangle inequality).
Motivated by this, we can give the following definition of quantum compact metrisable space.
A unital nuclear $C^*$-algebra $A$ is called a quantum compact metrisable space if there exists an element $d\in A\otimes A$ such that
(1) $d\geq 0$(non-negativity).
(2) $\psi\otimes \phi(d)=0$ iff $\psi=\phi$ for $\psi,\phi\in P(A)$ where $P(A)$ is the pure state space of $A$(Identity of indiscernibles).
(3) $\psi\otimes \phi(d)=\phi\otimes \psi(d)$ for $\psi,\phi\in P(A)$(Symmetry).
(4) $\psi\otimes \phi(d)\leq \psi\otimes \varphi(d)+\varphi\otimes \phi(d)$ for all $\psi,\phi$ and $\varphi\in P(A)$(Triangle inequality).
I check that for $M_n(C)$, the $C^*$-algebra of $n$ by $n$ complex matrices and found that if a $d$ satisfies non-negativity, identity of indiscernibles and symmetry, then $d$ does not satisfy triangle inequality. So this means that $M_n(C)$ can only admit a semi-metric in this sense. 
My question: is there any genuine quantum compact metrisable space(i.e. noncommutative $C^*$-algebra $A$ with such a $d\in A\otimes A$ satisfying the above properties)?

REPLY [10 votes]: I would suggest that you look at the work of Rieffel on compact quantum metric spaces.  His point of view is not to directly generalize the metric by understanding it as an element of the tensor square $A \otimes A$ (NB: you have not specified which tensor product you use here), but rather he generalizes the Lipschitz seminorm associated to the metric.
As I understand it from Rieffel, it was known already to Kantorovich that the metric on a compact space $X$ is determined by the Lipschitz seminorm on $C(X)$:
$$
L(f) = \sup_{x \neq y} \{\frac{|f(x)-f(y)|}{d(x,y)}\}
$$
via the identity
$$
d(x,y) = \sup\{ |f(x) - f(y)| : L(f) \le 1\}.
$$
Anyway, this doesn't directly answer your question, but this has been a fruitful line of inquiry, and I suggest you look at Rieffel's papers to see if they have anything useful for you.  I think "Metrics on State Spaces" is a good one to start with.