  Buchholz and Grundling (Comm. Math. Phys., 272, 699--750, 2007) introduced a
C$^\ast$-algebra called the resolvent algebra as a canonical quantisation of a
symplectic vector space, and demonstrated that this algebra has several
desirable features. We define an analogue of their resolvent algebra on the
cotangent bundle $T^*\mathbb{T}^n$ of an $n$-torus by first generalizing the
classical analogue of the resolvent algebra defined by the first author of this
paper in earlier work (J. Funct. Anal., 277, 2815--2838, 2019), and
subsequently applying Weyl quantisation. We prove that this quantisation is
almost strict in the sense of Rieffel and show that our resolvent algebra
shares many features with the original resolvent algebra. We demonstrate that
both our classical and quantised algebras are closed under the time evolutions
corresponding to large classes of potentials. Finally, we discuss their
relevance to lattice gauge theory.
