Quantum differentiability on quantum tori

The core ingredients of the quantised calculus, introduced by A.Connes, are a separable Hilbert space $H$, a unitary self-adjoint operator $F$ on $H$ and a $C^\ast$-algebra $\mathcal{A}$ represented on $H$ such that for all $a \in \mathcal{A}$ the commutator $[F,a]$ is a compact operator on $H$. Then the quantised differential of $a \in \mathcal{A}$ is defined to be the operator $\mathbf{d} a = i[F,a]$. We provide a full characterisation of quantum differentiability in the sense of Connes on quantum tori $\mathbb{T}_\theta^d$. We also prove a quantum integration formula which differs substantially from the commutative case. A joint work with Ed McDonald and Xiao Xiong.