Symmetry (Oct 2021)
Classification of Integrodifferential <i>C</i><sup>∗</sup>-Algebras
Abstract
The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli diagram for AF algebras in the same way as was previously performed by J. Glimm for more simple uniformly hyperfinite (UHF) algebras. We apply this symbol to characterize integrodifferential algebras. The integrodifferential algebra FN,M is the C∗-algebra generated by the following operators acting on L2([0,1)N→CM): (1) operators of multiplication by bounded matrix-valued functions, (2) finite-difference operators, and (3) integral operators. Most of the operators and their approximations studying in physics belong to these algebras. We give a complete characterization of FN,M. In particular, we show that FN,M does not depend on M, but depends on N. At the same time, it is known that differential algebras HN,M, generated by the operators (1) and (2) only, 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 HN,M, it was already computed earlier) which completely characterize the corresponding AF algebras. This symbol n is an infinite product of matrices with nonnegative integer entries. Roughly speaking, all the symmetries appearing in the approximation of complex infinite-dimensional integrodifferential and differential algebras by finite-dimensional ones are coded by a product of integer matrices.
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