Given a proper coarse structure on a locally compact Hausdorff space $X$, one can construct the Higson compactification for the coarse structure. In the opposite direction given a compactification of $X$, one can construct a coarse structure. We use unitizations of a non-unital C$^*$-algebra $A$ to define a noncommutative coarse structure on $A$. We also set up a framework to abstract coarse maps to this noncommutative setting. The original motivation for this work comes from Physics where quantum phenomenon when probed at large scales give classical results. We show equivalence of the canonical coarse structure on the classical plane $\mathbb{R}^{2n}$ with a certain noncommutative coarse structure on the Moyal plane which models the hypothetical phase space of Quantum physics. If time permits we shall also discuss other examples of noncommutative coarse equivalences. This is a joint work with Prof. Ralf Meyer.

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Last updated: 29 Nov 2023