Canonical form of $C^*$-algebra of eikonals related to the metric graph

Abstract

The eikonal algebra $\mathfrak E$ of the metric graph $\Omega$ is an operator $C^*$--algebra defined by the dynamical system which describes the propagation of waves generated by sources supported in the boundary vertices of $\Omega$. This paper describes the canonical block form of the algebra $\mathfrak E$ of an arbitrary compact connected metric graph. Passing to this form is equivalent to constructing a functional model which realizes $\mathfrak E$ as an algebra of continuous matrix-valued functions on its spectrum $\widehat{\mathfrak{E}}$. The results are intended to be used in the inverse problem of reconstruction of the graph by spectral and dynamical boundary data. Bibliography: 28 items.