Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian algebra generated by a positive element $h\in A$ (the "altitude function").

We refer to [1] where the structure of quantum topography space is originally introduced.

I believe one can think of $(A,M)$ as something stronger than a noncommutative topological space and weaker than a noncommutative metric space.

In the commutative setting, it is known that every locally compact topological group admits (see the main Theorem on page 217 in [2]) a metric structure.

Latremoliere shows how introduce a metric on $(A,M)$ making $(A,M)$ into a quantum locally compact metric space $(A,M,d)$. {\mathbf What quantum groups admit quantum topographic space structure? }

Well, the claim holds true at lease for locally compact topological groups. Indeed, one can construct approximate identity on the topological group $G$ utilising e.g. Urysohn lemma. We take the algebra $A=C(G)$.

In the noncommutative situation, there are situations [3] when quantum group $L^{\infty}(\mathbb{G})$ and the dual algebra $L^{\infty}(\widehat{\mathbb{G}})$ are factors. It seems possible to find some maximal abelian algebras inside factor quantum groups.

The current formulation might be a bit unclear, so please feel free to comment and I will edit the post. Many thanks for sharing your knowledge.

[1] F. Latremoliere. Quantum locally compact metric spaces. J. Funct. Anal., 264(1):362 – 402, 2013.

[2] R. A. Struble. Metrics in locally compact groups. Compositio Math, 28(217-222):2, 1974.

[3] P. Fima. On locally compact quantum groups whose algebras are factors. J. Funct. Anal., 244(1):78–94, 2007.