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

TL;DR

This paper characterizes the structure of integro-differential $C^*$-algebras generated by multiplication, differential, and integral operators, revealing their dependence on dimension and providing explicit invariants.

Contribution

It provides a complete classification of the integro-differential $C^*$-algebras using Bratteli diagrams and computes their Glimm-Bratteli symbols, highlighting differences from purely differential algebras.

Findings

$ ext{F}_{N,M}$ depends on $N$ but not on $M$

Explicit computation of Bratteli diagrams for $ ext{F}_{N,M}$

Comparison with differential algebras $ ext{H}_{N,M}$ showing universal behavior

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.