Classification of integro-differential $C^{\ast }$-algebras

Abstract

The integro-differential algebra ${\mathcal{F}}_{N,M}$ is the $C^{\ast }$-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 ${\mathcal{F}}_{N,M}$ in terms of its Bratteli diagram. In particular, we show that ${\mathcal{F}}_{N,M}$ does not depend on $M$ but depends on $N$. At the same time, it is known that differential algebras ${\mathcal{H}}_{N,M}$, generated by the operators 1) and 2), do not depend on both dimensions $N$ and $M$, they are all $\ast$-isomorphic to the universal UHF-algebra. We explicitly compute the Glimm-Bratteli symbols (for ${\mathcal{H}}_{N,M}$ it was already computed earlier) $$\mathfrak{n}({\mathcal{F}}_{N,M})=\prod _{n=1}^{\infty }{\left(\begin{array}{cc}n& 0\\ n-1& 1\end{array}\right)}^{\otimes N}{\left(\begin{array}{c}1\\ 1\end{array}\right)}^{\otimes N},\mathfrak{n}({\mathcal{H}}_{N,M})=\prod _{n=1}^{\infty }n,$$ which characterize completely the corresponding AF-algebras.