Classification of integro-differential $C^*$-algebras

Abstract

The integro-differential algebra $\mathscr{F}_{N,M}$ is the $C^*$-algebra generated by the following operators acting on $L^2([0,1)^N\to\mathbb{C}^M)$: 1) operators of multiplication by bounded matrix-valued functions, 2) finite differential operators, 3) integral operators. We give a complete characterization of $\mathscr{F}_{N,M}$ in terms of its Bratteli diagram. In particular, we show that $\mathscr{F}_{N,M}$ does not depend on $M$ but depends on $N$. At the same time, it is known that differential algebras $\mathscr{H}_{N,M}$, generated by the operators 1) and 2), do not depend on both dimensions $N$ and $M$, they are all $*$-isomorphic to the universal UHF-algebra. We explicitly compute the Glimm-Bratteli symbols (for $\mathscr{H}_{N,M}$ it was already computed earlier) $$ \mathfrak{n}(\mathscr{F}_{N,M})=\prod_{n=1}^{\infty}\begin{pmatrix} n & 0 \\ n-1 & 1 \end{pmatrix}^{\otimes N}\begin{pmatrix}1 \\ 1 \end{pmatrix}^{\otimes N},\ \ \ \ \mathfrak{n}(\mathscr{H}_{N,M})=\prod_{n=1}^{\infty}n, $$ which characterize completely the corresponding AF-algebras.