Izvestiya: Mathematics, 2022, Volume 86, Issue 4, Pages 621–666 DOI: https://doi.org/10.4213/im9179e (Mi im9179)

DOI: https://doi.org/10.4213/im9179e

Funding agency	Grant number
Russian Foundation for Basic Research	20-01-00627A
Volkswagen Foundation
Ministry of Science and Higher Education of the Russian Federation	075-15-2019-1620
Contest «Young Russian Mathematics»
This paper has been written with the support of the Russian Foundation for Basic Research (grant no. 20-01-00627A), the Volkswagen Foundation, the Leonhard Euler Mathematical Institute (agreement no. 075-15-2019-1620), and in part by the Russian Young Mathematicians Contest.

Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 4, Pages 3–50
DOI: https://doi.org/10.4213/im9179

Bibliographic databases:

§ 1. Introduction

1.1. On this paper

There is an approach, called the boundary control method (BC-method) [1], to inverse problems of mathematical physics. The approach has a pronounced interdisciplinary character: it is based on the connections of inverse problems with system theory and control theory and uses asymptotic methods, functional analysis, operator theory, etc. An algebraic version of the BC-method, based on connections with Banach algebras, provided a new solution to the problem of recovering a Riemannian manifold from boundary data [1]–[3]. Our perspective goal is to apply this version to inverse problems on graphs. The present paper is a step in this direction.

The algebraic version is based on the following fundamental fact: a topological space can be characterized in terms of an appropriate algebra. For example, a compact Hausdorff space $\Omega$ is determined (up to homeomorphism) by the algebra $\mathfrak{A}=C(\Omega)$ of continuous functions (Gel’fand, 1943). Moreover, the spectrum of this algebra, i.e., the set $\widehat{\mathfrak{A}}$ of its irreducible representations endowed with a suitable topology, is homeomorphic to the original space: $\widehat{\mathfrak{A}}\cong\Omega$. As a consequence, by taking any copy $\mathfrak{A}'$ of the algebra $\mathfrak{A}$ and finding its spectrum $\widehat{\mathfrak{A}'}\cong\widehat{\mathfrak{A}}\cong\Omega$, we obtain a homeomorphic copy of the space $\Omega$. This enables us to solve the reconstruction problem: a copy $\mathfrak{A}'$ is extracted from the inverse problem data, and its spectrum $\widehat{\mathfrak{A}'}$ provides the solution to the reconstruction problem: a homeomorphic copy of the manifold $\Omega$ is recovered. An important part of the approach consists in finding the algebra $\mathfrak{A}'$ in terms of the known data. For this algebra, we take the eikonal algebra $\mathfrak{E}$ determined by the dynamical system which describes the wave propagation in $\Omega$.

Variants of the BC-method for inverse problems on graphs are proposed in [4]–[6]. The version using the eikonal algebra was suggested in [7] and extended in [8]. Our work develops this approach. Its general direction is to study the relations between the properties of the algebra $\mathfrak{E}$ (block structure, algebraic invariants, representations) and the graph geometry. A promising goal is the recovery of a graph from its boundary data.

1.2. The eikonal algebra

For clarity and without loss of generality, we can regard $\Omega$ as a connected compact graph in $\mathbb{R}^3$ consisting of smooth curves (edges) $\{e_1,\dots,e_l\}=E$, which are joined at interior vertices^1[x]1Any metric graph admits such a realization. $\{v_1,\dots,v_m\}=V$. There are boundary vertices $\{\gamma_1,\dots,\gamma_n\}=\Gamma$, with a single edge outgoing from each of them. The metric (inner distance) in $\Omega$ is induced by the Euclidean metric of $\mathbb{R}^3$.

The edges of the graph are “material”: oscillations (waves) propagate along them, being initiated by point sources (controls), which are placed at the boundary vertices. The waves move from the boundary with unit velocity, gradually filling the graph. The process is described by the dynamical system

It is possible to control waves coming from a part $\Sigma\subset\Gamma$ instead of the whole boundary. In this case, we use controls of the class

For every boundary vertex we have the family of reachable sets $\mathscr{U}^t_\gamma: =\{u^f(\,{\cdot}\,,t)\mid f\in\mathscr{F}^T_\gamma\}$, $0\leqslant t\leqslant T$, and the corresponding projectors $P^t_\gamma$ in $\mathscr{H}$ to $\mathscr{U}^t_{\gamma}$. The operator $E^T_\gamma:=\int_0^Tt\,dP^t_\gamma$ is called the eikonal corresponding to the vertex $\gamma$. The eikonals are self-adjoint operators belonging to the algebra $\mathfrak{B}(\mathscr{H})$ of bounded operators.

Given a $C^*$-algebra $\mathfrak{A}$ and a set $S\subset\mathfrak{A}$, we write $\vee S$ for the $C^*$-algebra generated by this set, i.e., the minimal $C^*$-subalgebra in $\mathfrak{A}$ containing $S$. The eikonal algebra corresponding to a family $\Sigma\subset\Gamma$ of boundary vertices is the operator $C^*$-algebra

1.3. Results and comments

It was established in [7] that the algebra $\mathfrak{E}^T_\Sigma$ has a block structure: it is isomorphic to some subalgebra of $\bigoplus_{j=1}^J C([0,\epsilon_j];\mathbb M^{m_j})$ and differs from the latter by the existence of relations between blocks. For the simplest graphs (three-beam stars). the nature of these relations and their evolution with respect to $T$ were considered in [8].

The main result of this paper is a canonical block form of the eikonal algebra. This form is, firstly, distinguished by the absence of relations between blocks and, secondly, invariant: up to trivial transformations (permutation of blocks, change of parameterization, etc.), it is determined by any copy of the algebra (1.1). This enables us to hope that $\mathfrak{E}^T_\Sigma$ may be useful in inverse problems. whose data determine it up to isometry.

Of course, the effectiveness of the approach can be fully judged only by seeing concrete applications to inverse problems. There are no such applications in this paper but the above results seem to be an important step in this direction.

The eikonal algebra belongs to the class of $C^*$-algebras with finite-dimensional representations of different dimensions [9], [10]. The reduction to a canonical form is equivalent to constructing a functional model which realizes $\mathfrak{E}^T_\Sigma$ as an algebra of continuous matrix-valued functions on its spectrum. It belongs to the class of models described by Vasil’ev [9].

The definition (1.1) reproduces the definition of the corresponding algebras used in [1]–[3] for reconstruction of manifolds. The success of this application motivated our attempt to transfer this approach to problems on graphs. The non-commutativity of the algebra $\mathfrak{E}^T_\Sigma$ is an obstacle. It has already been encountered in the inverse problem of electrodynamics [3], but there it was solved by taking a quotient by the ideal of compact operators and thus reducing the problem to the commutative algebra $C(\Omega; \mathbb R)$. The non-commutativity of $\mathfrak{E}^T_\Sigma$ is irremovable, which makes its study much more difficult.

Inverse problems on graphs are a topical subject. Various formulations and approaches were given by Avdonin, Kurasov, Novachik, A. Mikhailov and V. Mikhailov, Kuchment, Yurko. The BC-method and other approaches were used in [11]–[17] to solve dynamic and spectral inverse problems for various classes of graphs. Yurko and his followers used a spectral approach to inverse problems for differential operators on graphs [18]–[20]. We mention an informative review [21] by Kuchment and Berkolayko on the whole subject of quantum graphs, including inverse problems on them.

The introductory part of the present paper is rather long and essentially repeats the corresponding sections of [7] and [8]. This is unavoidable since the presentation of facts and results related to $\mathfrak{E}^T_\Sigma$ requires a solid preparation. The technical part is rather complicated since we deal with an extremely general object: an arbitrary compact connected metric graph having a boundary. The study of simple examples in [7], [8] with illustrations may be instructive. We also recommend [22], where the reduction of $\mathfrak{E}^T_\Sigma$ to a canonical form is described in detail for a simple graph.

This paper is addressed to specialists in $C^*$-algebras with a taste for applications and/or specialists in the field of mathematical physics who share the idea of the usefulness of abstractions. The eikonal algebra is a complex and many-faceted object worthy of a thorough study. The possible relations between its algebraic invariants and graph geometry are particularly interesting in inverse problems. For example, we mention a conjecture on the correspondence between clusters in the spectrum of $\mathfrak{E}^T_\Sigma$ and interior vertices of $\Omega$. It is very interesting how the presence of cycles in the graph affects the structure of the spectrum.

§ 2. Waves on graphs

2.1. Graphs

Let $\Omega=E\cup W$ be a connected compact graph in $\mathbb{R}^3$ with edges $\{e_1,\dots,e_L\}=E$ and vertices $\{w_1,\dots,w_M\}=W$. The edges are smooth^2[x]2Throughout the paper, smooth means $C^\infty$-smooth. curves whose ends are vertices. It is convenient to think of edges as open without including their ends. A vertex $w$ and an edge $e$ are incident (we write $w\prec e$) if $w$ is an end of $e$. The vertices $\{\gamma_1,\dots,\gamma_N\}=\Gamma$, each of which is incident to a single edge, are boundary vertices, and those in the set $\{v_1,\dots,v_{M-N}\}=V=W\setminus\Gamma$ are interior.

The valence of a vertex $w$ is the number $\mu(w)$ of edges that are incident to $w$. When $\gamma\in \Gamma$, we have $\mu(\gamma)=1$. We also assume that there are no vertices with $\mu(w)=2$. Hence, $\mu(v)\geqslant 3$ for all interior vertices.

The graph is equipped with the metric (internal distance) $\tau$ induced by the Euclidean metric of $\mathbb{R}^3$. Thus $\tau(a,b)$ is the minimum length of piecewise smooth curves lying in $\Omega$ and connecting $a$ and $b$. Given a set $A\subset\Omega$, we denote its metric neighborhood of radius $r$ by

Each edge $e$ is parameterized by the arclength $\tau$ beginning at one of its ends. For a function $y$ on the graph, the sign of its derivative $dy/d\tau$ with respect to arclength depends on the end chosen, but the second derivative $d^2y/d\tau^2$ is independent of this choice. Given a vertex $w$ and an incident edge $e$, we define the derivative in the direction outgoing from $w$,

The metric $\tau$ on a graph determines a (real) Hilbert space $\mathscr{ H}=L_2(\Omega)$ with scalar product

Define the Kirchhoff class

2.2. Waves

The initial-boundary value problem describing the wave propagation on a graph is of the form

By the definition (2.1), the solution $u^f$ satisfies the homogeneous string equation $u_{tt}-u_{\tau \tau}=0$ on each edge $e$. Hence we see that the waves propagate from the boundary into $\Omega$ with unit velocity. It follows that if the control acts from a part $\Sigma\subseteq\Gamma$ of the boundary, i.e., $\operatorname{supp}f\subset\Sigma\times[0,T]$. then

We write $\delta_x$ for the Dirac measure, i.e., the functional on $C(\Omega)$ acting by the rule $\langle\delta_{x},y\rangle=y(x)$. Let $\delta=\delta(t)$ be the Dirac delta function. Our immediate goal is to define and describe the fundamental solution of the problem (2.2)–(2.5) with $T=\infty$, corresponding to the control $f=\delta_\gamma\delta(t)$, which acts instantaneously from a boundary vertex $\gamma$.

To describe the solution $u^{\delta_\gamma \delta}$, it is convenient to use the following formalism of “impulse dynamics”.

0. By an impulse we mean the measure $a\delta_x$; the constant $a\ne 0$ is called its amplitude.

1. Each impulse $a\delta_{x(t)}$ moves along an edge with velocity $1$ in one of the two possible directions, so that $|\dot x(t)|=1$ for $x(t)\in e$.

2 (superposition principle). Impulses move independently of each other. If at the moment $t$ there are several impulses $a_1\delta_{x(t)},\dots,a_p\delta_{x(t)}$ located at the point $x(t)\in\Omega\setminus\Gamma$, they are combined to form the impulse $[\,a_1+\dots + a_p\,]\delta_{x(t)}$.

3 (passing through an interior vertex). Moving along an edge $e$ and passing through an interior vertex $v$, the impulse $a\delta_{x(t)}$ splits into $\mu(v)$ impulses: one reflected and $\mu(v)-1$ transmitted. The reflected impulse moves along $e$ in the opposite direction and has amplitude $(2-\mu(v)/\mu(v))a$. Each transmitted impulse moves along its own edge (incident to $v$) away from $v$ and has amplitude $(2/\mu(v))a$.

Therefore, the total amplitude is

4 (reflection from the boundary). As soon as an impulse $a\delta_{x(t)}$ reaches a vertex $\gamma\,{\in}\,\Gamma$, it instantly inverts its direction and changes its amplitude from $a$ to $-a$.

Adopting these rules, we can describe the solution $u^{\delta_\gamma\delta}$ as follows (recall that $\tau$ is the distance in $\Omega$).

$(*)$ When $0\leqslant t\leqslant\tau(\gamma,V)$, we have $u^{\delta_\gamma\delta}=\delta_{x(t)}$, where $x(t)$ is the point of an edge $e\succ\gamma$ such that $\tau(x(t),\gamma)=t$. Thus, at small times, $u^{\delta_\gamma\delta}$ is a single impulse with unit amplitude entering the graph from the vertex $\gamma$ and moving along $e$ with unit velocity.

$(**)$ Further evolution at times $t>\tau(\gamma,V)$ is determined by the rules 1–4.

It is easy to see that this description is quite deterministic. At each moment of time $t\geqslant 0$, the solution $u^{\delta_\gamma\delta}$ represents a finite set of impulses moving in $\Omega$. From physical point of view, this picture describes, for example, the propagation of sharp signals (voltage spikes) in an electrical net (a graph made up of wires).

Thus, the fundamental solution is a space-time distribution in $\Omega\times\{t\geqslant 0\}$. Its structure is such that the time convolution

From now on, we regard the function $u^f$ defined by the relation (2.7) as a (generalized) solution for controls of the form described above. In the more general case when $f\in L_2(\Gamma\times[0,T])$ is of the form $f=\sum_{\gamma\in\Sigma}f_\gamma$ with $f_\gamma=\delta_\gamma\varphi_\gamma$, we put

2.3. Hydras

Here we introduce a space-time graph which is used to efficiently describe waves.

Fix a boundary vertex $\gamma$. Regarding the fundamental solution as a space-time distribution, we define the set

Zoom inZoom out

Figure 1.A hydra

We define the projections

– If $h\in H_\gamma$ is such that $\pi(h)=x\in\Omega\setminus\Gamma$ and $\rho(h)=t>0$, then we have $u^{\delta_\gamma \delta}(\,{\cdot}\,,t)=a\delta_x(\,{\cdot}\,)$ and define $a(h):=a$.

– If $h\in H_\gamma$ is such that $\pi(h)\in \Gamma$ and $\rho(h)>0$, we put $a(h):=0$.

– If $h\in H_\gamma$ is such that $\pi(h)=\gamma$ and $\rho(h)=0$, we put $a(h):=1$.

We see that the amplitude is a piecewise constant function defined on the whole hydra $H_\gamma$ (Fig. 2). Note that at self-intersection points $p$ we have $a(p)=-4/9+1/3=-1/9$ by rule 2 of impulse evolution. The self-intersection points like $p$ are the vertices of $H_\gamma$ projecting to $\Omega\setminus[V\cup\Gamma]$.

Zoom inZoom out

Figure 2.Amplitude on the hydra

Here is the representation for which the hydra was introduced (see [7]). Writing $h=(x,t)\in H_\gamma$ and $a(h)=a(x,t)$, we see from (2.7) that

These representations are quite efficient: they can be used to calculate the values of waves. However, the forthcoming analysis of the eikonal algebra requires their modification, which will be described now. It uses a partition of the graph $\Omega$ into parts (families) consistent with the structure of the hydra. This partition was described in [7] in full detail with graphic illustrations.

2.4. The partition $\Pi$

In what follows we deal with truncated hydras

We introduce the following general notion. Let $X$ be a set and let $\sim_0$ be a reflexive symmetric (but, generally speaking, not transitive!) binary relation on $X$. Elements connected by $\sim_0$ will be referred to as neighbors. This relation determines an equivalence relation by the following rule. We say that $x$ and $y$ are equivalent (and write $x\sim y$) if $X$ contains a finite set of elements $x_1,\dots,x_n$ such that $x\sim_0 x_1\sim_0 \dots\sim_0 x_n\sim_0 y$.

The equivalence class $[x]$ of an element $x\in X$ with respect to $\sim$ can be described constructively. We define an operation $\operatorname{ext}$ extending any subset $B\subset X$ by the rule

Consider an equivalence of this kind on the hydra. We say that points $h,h'\in H^T_\gamma$ are neighbors ($h\overset{\gamma}\sim_0 h' $) if at least one of the following conditions holds: $\pi(h)=\pi(h')$ or $\rho(h)=\rho(h')$. We write $\stackrel{\gamma}{\sim}$ for the equivalence relation generated by this notion of neighboring. Every equivalence class

With every point $x\in\overline{\Omega^T_\gamma}$ we associate the set

A point $h\in H_\gamma$ on a complete hydra is called a corner point if $\pi(h)\in V\cup\Gamma$ or $h$ is a self-intersection point (like $p$ on Fig. 2). The latter points are the vertices of valence 4 projecting to $\Omega\setminus[V\cup\Gamma]$.

On the truncated hydra $H^T_\gamma$, besides the corner points of the complete hydra, we declare the points in $\rho^{-1}(T)$ to be corner points. The set of all corner points of the truncated hydra is denoted by $\operatorname{Corn}H^T_\gamma$.

The lattice $\mathscr{L}[\operatorname{Corn} H^T_\gamma]$ divides the hydra into a finite number of open space-time intervals. The amplitude $a$ is constant on each interval.

The points in the finite set

Let $\omega=(c,c')\subset \Pi$ be a maximal interval consisting of regular points. Maximality of $\omega$ means that its ends $c$ and $c'$ are critical points, so that it is impossible to extend $\omega$ keeping its interior points regular. It is easy to see that the set

Comparing the definitions (2.13) and (2.14), we arrive at the representation

Let $\omega'\subset\Pi$ be a maximal interval not lying in the family $\Phi$. It determines another family $\Phi'=\Lambda[\omega']$ consisting of cells, and so on. As a result, $\Pi$ is the finite union of disjoint families $\Phi^1,\dots,\Phi^J$, each of which consists of disjoint cells:

In parallel to the determination set (2.13), each $x\in\overline{\Omega^T_\gamma}\setminus\Gamma$ is associated with the set

Consider a family $\Phi=\bigcup_{k=1}^{m_\Phi}\omega_k\subset\Pi$. It is easy to check that the set

We shall use the functions $\tau^i\colon \Phi \to [0,T]$ associated with the partition of the graph into families. They are introduced as follows.^4[x]4This definition differs from that in [7]. The reason for the change will be explained later. For $x\in\Phi$, we set

The property (2.20) enables us to extend the functions $\tau^i$ to critical points. Namely, if $x\in\omega=(c,c')$, $x\to c$, then $\tau^i(c)=t_i$ or $\tau^i(c)=\widetilde t_i$ depending on which of the representations (2.20) holds.

For each family $\Phi\subset\Pi$, we have its own functions $\tau^i$ and we denote them by $\tau^i_\Phi$ if necessary. Moreover, by (2.19) and (2.20), the equality $\tau^i_\Phi(x)=\tau^{i'}_{\Phi'}(x)$ for different $\Phi$ and $\Phi'$ is possible only at critical points $x$.

Our partition of the graph into families is motivated, in particular, by the fact that the waves $u^f$ depend on the controls $f$ locally in the following sense. We see from (2.9) that the values of $u^f(\,{\cdot}\,,T)|_{\Phi}$ are determined by the values of $f|_{\Xi[\Phi]}$. Moreover, the conditions

The modification of (2.9) and (2.10), mentioned at the end of the previous subsection, consists in an efficient representation of waves on determination sets. It uses functions (vectors) on $\Lambda[x]$ which will be described now.

2.5. Amplitude vectors

We still fix a boundary vertex $\gamma\in\Gamma$, a number $T>0$ and the corresponding partition $\Pi$.

Suppose that $A=\{x_1,\dots,x_m\}\subset\overline{\Omega^T_\gamma}$ is a finite $\Lambda$-closed set and

It follows from (2.6) that the value

Indeed, we see from the proof of Lemma 1 that if $T<T_\gamma$, then the set $\rho^{-1}(t)\subset H^T_\gamma$ is non-empty for $0\leqslant t\leqslant T$, which is equivalent to the equality $\Xi[\overline{\Omega^T_\gamma}]=[0,T]$.

We now reconsider the representation (2.9). In terms of amplitude vectors, it can be written in the form

The final step of modification of the original representation (2.9) consists in passing to a more convenient system of amplitude vectors in (2.22).

2.6. The $\beta$-representation of waves

Again, let $A\,{=}\,\Lambda[A]\,{=}\,\{x_1,\dots,x_m\}\,{\subset}\, \overline{\Omega^T_\gamma}$. We introduce the space $\mathbf{l}_2(A)$ of functions (vectors) on $A$ with scalar product

For every $x\in\overline{\Omega^T_\gamma}$ we have its own $\beta$-set over the determination set $\Lambda[x]$. By Lemma 1, the vectors $\alpha^i$ are linearly independent if $T<T_\gamma$. Hence, for these $T$, the vectors $\beta^i$ are all non-zero and $\operatorname{dim}\mathbb{A}[\Lambda[x]]=n\leqslant m$.

In terms of the $\beta$-set over the set $A=\Lambda[x]$, the representation (2.22) takes its final form

2.7. The hydra $H^T_\Sigma$

The concepts and objects introduced above correspond to a single boundary vertex $\gamma$. In what follows, we use the notation $\overset{\gamma}\sim$, $\mathscr L_\gamma$, $\Lambda_\gamma$, etc., to indicate this if necessary.

Given any set $\Sigma\subseteq\Gamma$ of boundary vertices, we define a space-time graph

The projections from $H^T_\Sigma$ to $\overline{\Omega^T_\Sigma}$ and $[0,T]$ are $\pi((x,t)):=x$ and $\rho((x,t)):=t$. By $\pi^{-1}$ and $\rho^{-1}$ we mean the full preimages in $H^T_\Sigma$.

By definition, we write $h\sim_0 h'$ on $H^T_\Sigma$ if $\pi(x)=\pi(x')$ and/or $\rho(x)=\rho(x')$. This relation of neighboring determines an equivalence relation $h\overset{\Sigma}\sim h'$. We denote the equivalence class (lattice) of a point $h\in H^T_\Sigma$ by $\mathscr L_\Sigma[h]$. The operation $H^T_\Sigma\supset B\mapsto \mathscr L_\Sigma[B]$ is a (topological) closure.

The set $\Lambda_\Sigma[x]:=\pi(\mathscr{L}_\Sigma[\pi^{-1}(x)])$ will be called the determination set of $x\in\overline{\Omega^T_\Sigma}$. Note the obvious embedding $\Lambda_\gamma[x]\subset\Lambda_\Sigma[x]$ for $\gamma\in\Sigma$. The operation $\overline{\Omega^T_\Sigma}\supset A\mapsto \Lambda_\Sigma[A]$ is a (topological) closure.

The set of corner points $\operatorname{Corn}H^T_\Sigma$ consists of all corner points of the hydras $H^T_\gamma\subset H^T_\Sigma$ plus the (transversal) intersection points of the edges of different hydras $H^T_\gamma$. The critical points in $\overline{\Omega^T_\Sigma}$ are $\Theta_\Sigma:=\pi(\mathscr L_\Sigma[\operatorname{Corn}H^T_\Sigma])$, the regular points are $\Pi_\Sigma:=\overline{\Omega^T_\Sigma}\setminus\Theta_\Sigma$. There is a partition

Suppose that $\gamma\in\Sigma$, $\Phi\subset\Pi_\Sigma$ and $x\in \Phi$. We easily see that the set $\Lambda_\Sigma[x]\cap {\Omega^T_\gamma}$ is $\Lambda_\gamma$-closed (in $\Omega^T_\gamma$). Hence it has a $\beta$-set of vectors $\beta^1_{\gamma\Phi},\dots, \beta^{n_\Phi}_{\gamma\Phi}$. We define them on the whole $\Lambda_\Sigma[x]$, extending them from $\Lambda_\Sigma[x]\cap \Omega^T_\gamma$ to $\Lambda_\Sigma[x]$ by zero.

Repeating the construction for all $\gamma\in\Sigma$, we obtain the $\beta$-sets

For a given family $\Phi$ and different vertices $\gamma\in\Sigma$, the set $\{\beta^1_{\gamma\Phi},\dots,\beta^{n_{\Phi}}_{\gamma\Phi}\}$ contains the same number of vectors (namely, $n_{\Phi}$) and the functions $\tau^i_\Phi$ are the same. Nevertheless, we assume by definition that

§ 3. Eikonals

3.1. Reachable sets and projectors

Here we consider the problem (2.2)–(2.5) as a dynamical system and equip it with control theory attributes (spaces and operators).

The space of controls $\mathscr{F}^T :=L_2(\Gamma \times [0,T])$ with the inner product

The space $\mathscr{H}=L_2(\Omega)$ is said to be internal; the waves $u^f(\,\cdot\,,t)$ can be regarded as evolutions of its elements in time. For every set $B\subset\Omega$ we define the subspace $\mathscr H\langle B\rangle:=\{y\in\mathscr H\mid \operatorname{supp}y\subset\overline B\}$ of functions localized in $B$.

The set of waves

The locality of the “control-wave” correspondence noted in (2.21) leads to a decomposition into families:

We fix a boundary vertex $\gamma\in\Sigma$. Let $P^T_{\gamma}$ be the projector in $\mathscr{H}$ to the subspace $\mathscr{U}^T_\gamma$. We shall discuss its properties and describe its action.

By (3.1), we have a representation

It was shown in [7] that the projectors $P^T_{\gamma}\langle\Phi\rangle$ can be expressed in terms of the vectors (2.25) and the corresponding functions $\beta^i_{\gamma\Phi}(\,{\cdot}\,)$ as follows:

We see from (3.2) and (3.3) that if the function $y\in\mathscr{H}$ to be projected to $\mathscr{U}^T_\gamma$ is continuous and $y|_{\Lambda_\gamma[x]}\equiv 0$, then $(P^T_\gamma y)|_{\Lambda_\gamma[x]}\equiv 0$. In other words, the values of $P^T_\gamma y$ on $\Lambda_\gamma[x]$ are completely determined by the values of $y$ on $\Lambda_\gamma[x]$. This is the reason why the sets $\Lambda_\gamma[x]$ as well as $\Lambda_\Sigma[x]$ are called determination sets.

Recall that the vectors $\beta^i_{\gamma\Phi}$ are elements of the subspace $\mathbb A[\Lambda_\Sigma[x]]\subset\mathbf{l}_2(\Lambda_\Sigma[x])$. It follows from the above that the projector $P^T_\gamma$ determines in $\mathbf{l}_2(\Lambda_\Sigma[x])$ the operator $p_{\gamma\Phi}[x]$ which is the projector onto $\mathbb A[\Lambda_\gamma[x]]\subset\mathbb A[\Lambda_\Sigma[x]]$. Let $\chi_1,\dots, \chi_{m_\Phi}$ be the (standard) basis in $\mathbf{l}_2(\Lambda_\Sigma[x])$ consisting of the indicators of points of the set $\Lambda_\Sigma[x]$. In this basis, we have $(\beta^i_{\gamma\Phi})_k=\langle\beta^i_{\gamma\Phi},\chi_k\rangle$. By (3.3), the matrix of the projector $p_{\gamma\Phi}[x]$ takes the following form in this basis:

3.2. Eikonals

The family $\{P^s_\gamma\mid 0\leqslant s\leqslant T\}$ of the projectors in $\mathscr{H}$ to the reachable sets $\mathscr{U}^s_\gamma$ determines the eikonal operator (briefly, the eikonal)

It was shown in [7] that we have a representation

The operator $E^T_\gamma$ induces the operator $e_{\gamma\Phi}[x]$ in $\mathbf{l}_2(\Lambda_\Sigma[x])$. By (3.4) and (3.5), its matrix in the basis $\chi_1,\dots, \chi_{m_\Phi}$ is of the form

By what was said above, we easily obtain some general properties of eikonals as operators in $\mathscr{H}$ (see, e.g., [23]).

3.3. Parameterization

We choose a family $\Phi=\bigcup_{k=1}^{m_{\Phi}}\omega_k\subset\Pi_\Sigma$. Let $\omega=(c,c')\subset \Phi$ be one of its cells. Recall that all the cells are of the same length $\epsilon_\Phi=\tau(c,c')$. Given any $x\in\omega$, we write $x=x(r)$ if $\tau(c,x)=r$. Along with $x$, the determination set also turns out to be parameterized: $\Lambda_\Sigma[x(r)]=\{x_k(r)\}_{k=1}^{m_{\Phi}}$. When $r$ varies in the interval $(0,\epsilon_\Phi)$, the points $x_k(r)$ continuously change their position and cover the cells $\omega_k$. Thus, the family $\Phi$ is parameterized.

All elements of the representations (3.3) and (3.5) are also parameterized by $r$: the vectors

Parameterization determines matrix representations of functions and operators on the graph.

Let $\Phi\subset\Pi_\Sigma$ be a parameterized family, $y\in\mathscr H$ a function on the graph, and $x=x(r)\in\Lambda_\Sigma[x(r)]=\{x_k(r)\}_{k=1}^{m_{\Phi}}\subset\Phi$, $0<r<\epsilon_{\Phi}$. We easily see that the map

We now describe the parameterization of spaces and operators corresponding to the whole partition $\Pi_{\Sigma}$. The analog of the decomposition (3.1) takes the form

The parameterization of the whole $\Pi_\Sigma$ is given by the operator $U:=\bigoplus\!\sum\limits_{\Phi\subset\Pi_\Sigma} \! U_\Phi$:

3.4. Shifted eikonals

For technical reasons, the following operators (shifted eikonals) are more convenient for the purpose of describing the algebra generated by eikonals:

§ 4. The algebra of eikonals

4.1. Definitions and general facts

Recall that a $C^*$-algebra $\mathfrak{A}$ is a Banach algebra with an involution $a \to a^*$ satisfying the following identity [24], [25]:

By $\mathbb{M}^n$ we mean the algebra of real $(n\times n)$-matrices regarded as operators in $\mathbb{M}^n$ and endowed with the operator norm. It is irreducible.

We write $C([a,b],\mathbb{M}^n)$ for the algebra of continuous $\mathbb{M}^n$-valued functions with norm $\|c\|=\sup_{a\leqslant t\leqslant b} \|c(t)\|_{\mathbb{M}^n}$. By the same symbol, we denote the operator (sub)algebra in $\mathfrak{B}(L_2([a,b];\mathbb{M}^n))$ whose elements multiply $\mathbb R^n$-valued square-summable functions by functions in $C([a,b],\mathbb{M}^n)$. The correspondence $c\mapsto c\cdot$ establishes an isomorphism between these algebras.

A $C^*$-subalgebra $\mathfrak A\subset \mathbb{M}^n$ is said to be irreducible if $\mathfrak A\cong \mathbb{M}^k$, where $k\leqslant n$. Such an algebra takes a block-diagonal form In a suitable basis of $\mathbb R^n$ and consists of two blocks, one of which is $\mathbb{M}^k$ and the other (if available) is equal to zero.

Here is a summary of known results.^8[x]8There is a mistake in the analogous summary “On matrix algebras” in [8]: statement 3 is incorrect. However, the results of [8] remain valid after appropriate corrections.

A representation of a $C^*$-algebra $\mathfrak{A}$ is a homomorphism $\pi\colon \mathfrak{A}\to\mathfrak{B}(H)$, where $H$ is a Hilbert space. Equivalence $\pi\sim\pi'$ of representations means that $\iota\,\pi(a)=\pi'(a)\iota$, $a\in\mathfrak{A}$, where $\iota\colon H\to H'$ is an isometry of the representation spaces. A representation is irreducible if the operators $\pi(\mathfrak{A})$ have no common non-zero invariant subspace in $H$.

The spectrum of a $C^*$-algebra $\mathfrak{A}$ is the set $\widehat{\mathfrak{A}}$ of the equivalence classes of its irreducible representations. The equivalence class (the point of the spectrum) corresponding to a representation $\pi$ will be denoted by $\widehat\pi$. The spectrum is endowed with the canonical Jacobson topology [24], [25].

An isomorphism $\mathrm{u}\colon\mathfrak{A}\to\mathfrak{B}$ of $C^*$-algebras determines a correspondence between their representations,

We put

The central object of this paper is the eikonal algebra of the graph $\Omega$,

By the functional calculus of self-adjoint operators and since the projectors $P^i_{\gamma \Phi}$ in (3.12) are orthogonal, we have

4.2. Representations and connections between blocks

By (3.2) and (3.5) we have a representation

In the more explicit block-matrix notation for the representation (4.6) we have

Consider the tuples of projectors $\mathbb P_{\Phi^j}:=\{P^i_{\gamma \Phi^j}\mid i=1,\dots,n_{\gamma \Phi^j};\, \gamma\in\Sigma\}$. Using the algebras

We introduce the projectors

The first relation follows directly from the lemma, and the second can easily be derived from the first by using Proposition 4. By (4.13), there are no connections between the blocks $[U\mathfrak{E}^T_\Sigma U^{-1}]_j$ under these constraints on the coordinates.

If the coordinates of $\mathbf{r}$ take extreme values, (4.12) does not hold: there can be links between the blocks in the matrix algebra $[U\mathfrak{E}^T_\Sigma U^{-1}](\mathbf{r})$. It is these links that distinguish the left- and right-hand sides in the embedding (4.10): the latter consists of standard algebras and has no such links. We explain this by examples.

It follows from the definition and properties of the functions $\tau^i_{\gamma \Phi^j}$ (see (2.19)) that the equality $\tau_{\gamma \Phi^j}^i(r_j)=\tau_{\gamma\Phi^{j'}}^{i'}(r_{j'})$ is possible only at extreme values of the coordinates, i.e. when $r_j\in\{0,\epsilon_j\}$ and $r_{j'}\in\{0,\epsilon_{j'}\}$. Let the functions $\tau^i_{\gamma\Phi^j}$ be such that $\tau^i_{\gamma\Phi^j}(\epsilon_j)=\tau^{i+1}_{\gamma\Phi^j}(\epsilon_j)=\tau$. Then the $j$th block of the eikonal in (4.8) is of the form

Projectors belonging to distinct blocks may also be linked. Let $\tau^i_{\gamma \Phi^j}$ be such that $\tau^i_{\gamma\Phi^j}(\epsilon_j)=\tau^{i'}_{\gamma\Phi^{j'}}(\epsilon_{j'})=\tau$ for distinct $j$ and $j'$. Then the blocks with numbers $j$ and $j'$ in (4.8) are of the form

4.3. Reducibility

Generally speaking, the algebras $\mathfrak{P}_{\Phi^j}$ defined by the families of projectors $\mathbb P_{\Phi^j}$ in (4.9) are reducible. By Proposition 3, we have

The first step is to successively relabel the algebras occurring in (4.14):

This is the point where the subscript $\gamma$ of the functions $\tau^i_\Phi$ becomes necessary (see (2.26)) because the originally equal functions $\tau_{\gamma \Phi^j}^i=\tau_{\gamma'\Phi^j}^i$ for a fixed family $\Phi^j$ can correspond to the projectors $P_{\gamma \Phi^j}^i$ and $P_{\gamma'\Phi^j}^i$ which, after renumbering, belong to different blocks $\mathfrak{P}_l$ and $\mathfrak{P}_{l'}$.

By Corollary 2 and the subsequent comments, connections between the blocks of the algebra $U\mathfrak{E}^T_\Sigma{U^{-1}}$ can occur only on the boundaries of the intervals $[0,\varepsilon_j]$. To describe these connections in detail, we shall use the following formalism.

Fix a vertex $\gamma\in\Sigma$ and consider a triple $(k,l,r_l)$ corresponding to the value $\tau_{\gamma l}^k(r_l)$ of the function $\tau_{\gamma l}^k$. We say that triples $(k,l,r_l)$ and $(k',l',r_{l'})$ are related and write $(k,l,r_l)\leftrightarrow (k',l',r_{l'})$ if $\tau_{\gamma l}^k(r_l)=\tau_{\gamma l'}^{k'}(r_{l'})$. By the properties (2.19) and (2.20), this is possible only for extreme values of the parameters $r_l\in\{0,\varepsilon_l\}$ and $r_{l'}\in\{0,\varepsilon_{l'}\}$; only these values occur in the proposition below. The same properties lead to the following properties of the relation $\leftrightarrow$.

We divide the set of all triples $(k,l,r_l)$ into types $\mathbf 1$, $\mathbf 2$, and $\mathbf 3$ in accordance with Proposition 6.

We have already noticed that the difference between the algebras $U\mathfrak{E}^T_\Sigma U^{-1}$ and $\bigoplus_{l=1}^L \!\! C([0,\varepsilon_l]; \mathfrak{P}_l)$ consists in the possible relations between the blocks $(U\mathfrak{E}^T_\Sigma U^{-1})(\mathbf{r})$ of the matrix algebra in the case when the coordinates $r_l$ take extreme values. To study these relations, we use the boundary algebra

4.4. Algebras generated by one-dimensional projectors

Let $\mathscr{G}$ be a Hilbert space with inner product $\langle\,{\cdot}\,,{\cdot}\,\rangle$, and let $P^1,\dots,P^n$ be one-dimensional projectors: $P^i=\langle\,{\cdot}\,,\beta^i\rangle\beta^i$, $\|\beta^i\|=1$. We put

Write $B=B_1\,{\cup}\,{\cdots}\,{\cup}\, B_q$ as the union of equivalence classes and put $\mathscr A_k:=\operatorname{span}B_k$. It follows from the definition of $\sim$ that $\mathscr A_k\perp \mathscr A_l$ for $k\neq l$. Thus we have a decomposition $\mathscr A=\mathscr A_1\oplus\dots\oplus \mathscr A_q$, which obviously reduces all projectors $P^i$.

Reducibility is obvious; the equality for the dimensions follows since every vector $\beta^i\in\mathscr A_k$ is cyclic in $\mathscr A_k$ for the part $\mathfrak{P}|_{\mathscr A_k}$. The decomposition (4.21) can be obtained by using the procedure (2.11), (2.12).

The following considerations model the situation arising in the study of the eikonal algebra. Namely, the possible relations between the blocks in the representation (4.8) will be discussed in an abstract form.

Let $\mathscr{G}_k$, $k=1,2,3$, be three Hilbert spaces, each endowed with a set $P^1_k,\dots,P^{n_k}_k$ of one-dimensional projectors: $P^i_k=\langle\,{\cdot}\,,\beta^i_k\rangle\beta^i_k$, $\|\beta^i_k\|=1$, where the vectors $\beta^i_k$ belong to the sets

We say that an algebra $\mathfrak{Q}\subset\mathfrak{P}$ separates (or does not connect) the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$ in (4.22) if, along with any element $q_1\oplus q_2\oplus q_3\in\mathfrak{Q}$, it contains the $q_1\oplus O_2\oplus q_3'$ and $O_1\oplus q_2\oplus q_3''$, where $q_3'$, $q_3''$ are some elements of $\mathfrak{P}_3$. Otherwise we say that these blocks are connected. We similarly define a connection (or its lack) for any pair of blocks in (4.22).

Note the following obvious fact. If $\mathfrak{Q}$ admits a system of generators each of which is of the form either $q_1\oplus O_2\oplus q_3'$ or $O_1\oplus q_2\oplus q_3''$, then it separates $\mathfrak{P}_1$ and $\mathfrak{P}_2$.

Let

Using this map, we define the projectors

We now discuss the conditions under which this algebra $\mathfrak{Q}$ separates or connects the blocks $\mathfrak{P}_1$ and $\mathfrak{P}_2$.

Divide the set $\mathbb{P}$ (defined in (4.24)) into the parts $\mathbb{P}_k:=\{\mathcal{P}_k^i\mid i=1,\dots,{n}_k\}$, $k=1,2,3$. For every part we have the matrices

We point out an important circumstance. Under the hypotheses of Theorem 1, the existence of a connection between $\mathfrak{P}_1$ and $\mathfrak{P}_2$ excludes connection between any of these blocks and $\mathfrak{P}_3$. This follows from the italicized remark on the separation of $\mathfrak{Q}_{12}$ and $\mathfrak{Q}_3$ at the beginning of part 4 of the proof.

The following result of Korikov [26] enables us to clarify the structure of the algebra $\mathfrak{Q}_{12}$ in Theorem 1. Suppose that $\mathcal{T}(\mathbb{P}_1)=\mathbb{P}_2$ and accordingly $n_1=n_2=:n$. We enumerate these sets consistently:

It is also easy to show that if at least one equality in Lemma 4 does not hold, then the algebra $\mathfrak{Q}_{12}$ does not connect $\mathfrak{P}_1$ and $\mathfrak{P}_2$. Thus, simultaneous fulfillment of the conditions in Theorem 1 and Lemma 4 guarantees that the blocks are connected and there is an isomorphism $\mathcal{I}$, while lack of at least one of these conditions leads to separation of the corresponding blocks by the algebra $\mathfrak{Q}$.

Consider the more general situation when there are $N$ Hilbert spaces $\mathscr{G}_k$, $k=1,\dots,N$. Each $\mathscr{G}_k$ has its own set of one-dimensional projectors

We endow the total set of generators

We form all possible pairs $\{\mathfrak{P}_k, \mathfrak{P}_{k'}\}$, $k\ne k'$, of the blocks constituting the algebra $\mathfrak{P}$ in (4.31) and choose the pairs connected by the algebra $\mathfrak{Q}$ (in the same sense as $\mathfrak{P}_1$, $\mathfrak{P}_2$ in Theorem 1 and Lemma 4). This choice is unique because, as noted after the proof of the theorem, each $\mathfrak{P}_k$ can be connected with only one $\mathfrak{P}_{k'}$. We renumber the blocks in (4.31), distinguishing the pairs of connected blocks and independent blocks:

The map $\mathcal T$ determines the algebra ${\mathfrak{Q}}$ in terms of the generating projectors $Q^i_k$ in (4.32). The map $\widetilde{\mathcal T}$ similarly determines the projectors $\widetilde Q^i_k$ generating the same algebra. A special feature of the latter projectors is in their form. By construction, we have

We put $\mathbb{Q}_k:=\{\widetilde{\mathcal{Q}}_k^i\mid i=1,\dots,n_k\}$. The following result summarizes our considerations. Recall that the blocks are numbered according to (4.33).

4.5. The canonical form

We now use the results established above to describe the structure of the eikonal algebra. The description will be given by a canonical form (representation).

The grouping of projectors in (4.16) in accordance with the decomposition (4.15) is the partition of the sets $\mathbb P_{\Phi^j}$ into the equivalence classes $\mathbb P_{\Phi^j}^k$ with respect to $\sim$ (they are also the classes $\mathbb P_{l}$). It is motivated by Proposition 7 and prepares our application of Theorem 1.

The role of the algebra $\mathfrak P$ in Theorem 1 (see (4.31)) is played by the algebra

To write the eikonal algebra in the canonical form, we use junction of its blocks connected by the boundary algebra. Let us describe this construction.

Let $\mathfrak{A}=\dot C([0,\varepsilon];\mathfrak{P})$ and $\mathfrak{B}=\dot C([0,\varepsilon'];\mathfrak{P}')$ be standard algebras such that

Given any elements $a\in\mathfrak{A}$ and $b\in\mathfrak{B}$ with $a(\varepsilon)=\mathcal I b(0)$, we define an element $a\sqcup b\in C([0,\varepsilon+\varepsilon'],\mathfrak{P})$ by the rule

Note that the algebras $\mathfrak{A}^{\oplus}\mathfrak{B}$ and $\mathfrak{A}\sqcup\mathfrak{B}$ are isomorphic. To summarize, we say that the algebras $\mathfrak{A}$ and $\mathfrak{B}$ admit a junction through the ends $r=\varepsilon$ and $r'=0$. Note that the algebras $\mathfrak{A}(0)$ and $\mathfrak{B}(\varepsilon')$ do not influence the possibility of junction and its result.

Obvious changes in (4.37) enable us to define the junction $\mathfrak{A}\,\sqcup\,\mathfrak{B}\,{\subset}\, C([0,\varepsilon+ \varepsilon'];\mathfrak{P})$

We endow the algebra $\mathfrak{A}=\dot C([0,\varepsilon],\mathfrak{P})$ with a transposition $t\colon a\mapsto a^t$, $a^t(r): =a(\varepsilon-r)$, $r\in[0,\varepsilon]$, and put $\mathfrak{A}^t:=\{a^t\mid a\in\mathfrak{A}\}$. An isomorphism ${\mathcal M}\colon \mathfrak{P}\to\mathfrak{P}$ induces a transformation $\check{\mathcal M}\colon C([0, \varepsilon],\mathfrak{P})\to C([0,\varepsilon],\mathfrak{P})$, $(\check{\mathcal M}a)(r): =\mathcal M [a(r)]$, $r\in[0,\varepsilon]$. For a standard algebra $\mathfrak{A}$, we put $\check{\mathcal M}\mathfrak{A}:=\{\check{\mathcal M} a\mid a\in\mathfrak{A}\}\subset C([0,\varepsilon],\mathfrak{P})$.

The algebras $\mathfrak{A}^t$ and $\mathcal M\mathfrak{A}$ are also standard. We see that $\check{\mathfrak{A}}\cong{\mathfrak{A}}^t\cong{\mathfrak{A}}$ and

We are now ready to describe the transformation of the eikonal algebra to the canonical form. It is given by the following procedure.

Step 1. Define an isomorphism $\mathbf{U}_0\colon \mathfrak{E}^T_{\Sigma}\to U\mathfrak{E}^T_\Sigma U^{-1}\stackrel{(4.18)}{\subset} \bigoplus_{l=1}^L C([0,\varepsilon_l],\mathfrak{P}_l)$ by specifying it on the generators:

By the properties of the map $\mathcal T$, the existence of a connection between the blocks implies that $n_{\gamma l}=n_{\gamma l'}$ for any $\gamma\in\Sigma$, and the junctions $[\mathbf{U}_0 E^T_{\gamma}]_l\sqcup [\mathbf{U}_0 E^T_{\gamma}]_{l'}$ can be represented in the following way:

To be definite, suppose that $\tau^k_{\gamma l}(r)= t^k_{\gamma l}+r$. Since $\tau^k_{\gamma l}(\varepsilon_{l})=\tau^{k'}_{\gamma l'}(0)$ and since the equality $\tau^k_{\gamma l}(r_{l})=\tau^{k'}_{\gamma l'}(r_{l'})$ is possible only for extreme values of the parameters $r_l$ and $r_{l'}$ (in our case $r_l=\varepsilon_l$, $r_{l'}=0$), it follows that $\tau^{k'}_{\gamma l'}(r)= t^{k'}_{\gamma l'}+r$, with $t^{k'}_{\gamma l'}= t^{k}_{\gamma l}+\varepsilon_l$. This leads to the following equalities:

We have studied the case when the boundary values $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l(\varepsilon_l)$ and $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}(0)$ are related. The cases of other possible connections between the boundary values admitting junction of blocks (see (4.38)), can be studied in a similar way.

Step 2. Suppose that the blocks $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_l$ and $[\mathbf{U}_0 \mathfrak{E}^T_{\Sigma}]_{l'}$ are admissible for junction. We define a map

Successively replacing all pairs of related blocks by their junctions in this way, we arrive at an isomorphism $\mathbf{U}_N\colon \mathfrak{E}^T_{\Sigma}\to\mathbf{U}_N\mathfrak{E}^T_{\Sigma}$ such that there are no blocks of type $\mathfrak{Q}_{k}^I$ in the boundary algebra $\partial(\mathbf{U}_N\mathfrak{E}^T_{\Sigma})$. Thus, the map $\mathbf{U}_N$ transforms the eikonal algebra to an algebra of the same structure as $\mathbf{U}_0\mathfrak{E}^T_{\Sigma}$, but with independent blocks. Note also that several (more than two) blocks of the original representation $\mathbf{U}_0\mathfrak{E}^T_{\Sigma}$ can be joined into a single new block in the process of transformation to the representation $\mathbf{U}_N\mathfrak{E}^T_{\Sigma}$.

Step 3. At this final step, the irreducibility of the algebras $\mathfrak{P}_l$ is used again. For each $\mathfrak{P}_l$ we find a transformation realizing the isomorphism $\mathfrak{P}_l\,{\cong}\,\mathbb M^{\varkappa_l}$. Then we can define an isomorphism $\mathbf{U}\colon \mathfrak{E}^T_{\Sigma}\,{\to}\, \bigoplus_{l=1}^\mathcal L \dot C([0,\zeta_l];\mathbb{M}^{\varkappa_l})$ by specifying it on generators in the following way:

Thus, by successive transformations of the original parametric form (4.6), we obtain a form of the same structure, but with independent blocks which are standard algebras. We state the final result.

We call a representation (form) of this kind canonical. It is not unique, but we can show that any two such representations differ from each other by renumeration of the blocks $[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l$, their transposition $[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l\to[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l^t$, and any isomorphisms $[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l\to\check{\mathcal M}[\mathbf{U}\mathfrak{E}^T_{\Sigma}]_l$. The ambiguity associated with transposition obviously corresponds to the two directions in which the argument of $\widetilde\tau_{\gamma l}^k$ can vary on the interval $[0,\delta_l]$ (two possible parameterizations of the $l$th block).

Here is a comment on Theorem 2. The original parametric form (4.8) of the eikonal algebra corresponded to a partition of the graph $\Omega^T_\Sigma$ into families $\Phi^1,\dots,\Phi^J$. We conjecture that the transition to the canonical form corresponds to a new partition. This is not proven, but it is supported by well-known examples [7], [8].

§ 5. Transformation to the canonical form

5.1. Canonical representation and spectra

The main advantage of the canonical form is that complete information about the spectrum of the algebra $\mathfrak{E}^T_\Sigma$ can easily be extracted from it and a number of its invariants are revealed. We proceed to describe them.

Let us compare the content of Theorem 2 and Proposition 2.

Since the canonical form is non-unique (as mentioned after Theorem 2), we fix the numbering and parameterization of its blocks. We also rewrite this form in a new convenient notation (replacing $\zeta_l$ by $\varepsilon_l$ and $s_{{\gamma l}}$ by $n_{\gamma l}$, and removing $(\,\widetilde{\ }\,)$):

Let

As mentioned in § 4.1, the spectrum of the standard algebra $\dot C([a,b]; \mathbb M^n)$ consists of (the equivalence classes of) irreducible representations corresponding to the interior points of $[a,b]$ as well as (possibly empty) clusters $\{\widehat\pi^1_a,\dots,\widehat\pi^{n_a}_a\}$ and $\{\widehat\pi^1_b,\dots,\widehat\pi^{n_b}_b\}$ adjacent to the ends of $[a,b]$. Equipped with the Jacobson topology, this spectrum is homeomorphic to a space which may naturally be called a closed interval with split ends. It is described by the following construction (see, e.g., [27]).

Consider $n_a$ half-open intervals $[a,b)$ and $n_b$ half-open intervals $(a,b]$ with topology from $\mathbb R$. Identify all their interior points with equal coordinates. The resulting quotient space $\mathscr{S}_{[a,b]}$ consists of the part $S_{(a,b)}$ homeomorphic to $(a,b)$ and two sets of pairwise inseparable points (two clusters) $K_a$ and $K_b$. The clusters consist of $n_a$ and $n_b$ points and correspond to $a$ and $b$ respectively. Each cluster is inseparable from $S_{(a,b)}$. The part $S_{(a,b)}:=\operatorname{int}\mathscr{S}_{[a,b]}$ is the set of interior points, each of which has a neighborhood homeomorphic to an (open) interval of the real axis.

By the first representation (5.1), the spectrum of the algebra $\mathbf U \mathfrak{E}_{\Sigma}^T$ is homeomorphic to a disjoint union of closed intervals

The spectra of the algebras $\mathfrak{E}_{\Sigma}^T$ and $\mathbf U \mathfrak{E}_{\Sigma}^T$ are connected by the homeomorphism $\mathbf U_*\colon \widehat{\mathfrak{E}_{\Sigma}^T}\to\widehat{\mathfrak{E}_{\Sigma}^T}$ (see (4.2)). Hence, $\widehat{\mathfrak{E}_{\Sigma}^T}$ is homeomorphic to the space $\mathscr{S}^T_\Sigma$ and admits a representation

5.2. Coordinates

Recall that the algebras $\mathfrak{E}_{\gamma}^T=\vee\{{{E}^T_{\gamma}}\}\subset\mathfrak{E}_{\Sigma}^T$, which correspond to individual eikonals, are said to be partial. We denote by $\pi$ and $\widehat\pi$ an irreducible representation and its equivalence class.

The correspondence $\varphi({E}^T_{\gamma})\leftrightarrow\varphi$ is an algebra isomorphism between $\mathfrak{E}_{\gamma}^T$ and $C(\sigma_{\mathrm{ac}}( E^T_\gamma))$, which is given by the first equality in (4.5) (see [23]). Each $\mathfrak{E}_{\gamma}^T$ is a commutative subalgebra in $\mathfrak{E}_{\Sigma}^T$. Its spectrum (set of characters) $\widehat{\mathfrak{E}_{\gamma}^T}$ is exhausted by Dirac measures:

If a commutative algebra has a finite number of generators, they constitute coordinates on its spectrum (see [28], Ch. III, Theorem 6). We shall use an adequate analog of this fact for non-commutative $C^*$-algebras whose generators are self-adjoint operators with simple spectrum. Namely, with every $\widehat\pi\in\widehat{\mathfrak{E}_{\Sigma}^T}$ we associate the set $\{\widehat\pi|_{\mathfrak{E}_{\gamma}^T}\mid \gamma\in\Sigma\}$ of its restrictions to partial algebras. Each element of this set is a representation of the form (5.4), already provided with $\gamma$-coordinates $t^1_{\gamma},t^2_{\gamma},\dots$ . The correspondence

The correspondence (5.5) is not injective in general, but its restriction to $\operatorname{int}\widehat{\mathfrak{E}_{\Sigma}^T}$ is injective. Since the set $\widehat{\mathfrak{E}_{\Sigma}^T}\setminus\operatorname{int}\widehat{\mathfrak{E}_{\Sigma}^T}$ of all points in the clusters is finite (so, almost all points of the spectrum are interior), the term coordinates seems motivated.

If $\widehat\pi\in S_l\subset\mathscr{S}_l$ (see (5.3)), then $\mathbf U_*\widehat\pi$ is an interior point of the spectrum $\widehat{\mathbf U_*\mathfrak{E}_{\Sigma}^T}$ and, by the first representation in (5.1), it corresponds to a certain value of the parameter $r\in (0,\varepsilon_l)$. Then we denote the point $\widehat\pi$ by $\widehat\pi_r$. We shall now find its $\gamma$-coordinates.

Suppose that $\varphi\in C(\sigma_{\mathrm{ac}}( E^T_\gamma))$. Among the equivalent representations constituting $\widehat\pi_r$, we choose a representation $\pi_r$ such that

When $\varphi(t)=t$, the relation (5.6) takes the form

We see from (5.7) that when the point $\widehat\pi$ varies in $S_l$, its coordinates $t_{\gamma l}^k(\widehat\pi)$ sweep out the time cells $\psi_{\gamma l}^k\subset\sigma_{\mathrm{ac}}( E^T_\gamma)$ and we have

When $k$ is fixed, the correspondence $\operatorname{int}\psi^k_{\gamma l}\ni t^k_{\gamma l}(\widehat\pi)\leftarrow\widehat\pi\in S_l$ is bijective and determines a natural parameterization^9[x]9Or rather, one of the two possible parameterizations. of the interval $\mathscr{S}_l$. For its interior points, we put

Having chosen the parameterization in the way described above, we obviously parameterize the remaining cells $\psi^{k'}_{\gamma l}$, $k'\ne k$, and define functions $\tau^k_{\gamma}(r)$, $r\in(0,\varepsilon_l)$, $k=1,\dots,n_{\gamma l}$, according to (5.7). They are also invariants of the eikonal algebra.

5.3. The transformation

The canonical form (5.1) was obtained by “reformatting” the parametric representation (4.8). We now show how to arrive at it by starting from the original algebra $\mathfrak{E}^T_\Sigma$ and using its invariants.

So we have the eikonals $E^T_\gamma$, $\gamma\in\Sigma$, and the algebra $\mathfrak{E}^T_\Sigma$ generated by them.

Step 1. Find the spectrum $\widehat{ \mathfrak{E}^T_\Sigma}$, equip it with the (Jacobson) topology and distinguish its closed intervals and their components according to (5.3).

Step 2. Find the spectra $\sigma_{\mathrm{ac}}( E^T_\gamma)$ and introduce the $\gamma$-coordinates on $\operatorname{int}\widehat{\mathfrak{E}^T_\Sigma}$. Determine the cells $\psi^k_{\gamma l}$ by (5.9). Parameterize the closed intervals by (5.10).

Step 3. For each $l$, choose representations $\pi_r\colon \widehat{\mathfrak{E}^T_\Sigma}\to\mathbb M^{\varkappa_l}$, $\pi_r\,{\in}\,\widehat\pi_r\,{\in}\, S_l$, $r\in(0,\varepsilon_l)$, such that the $\mathbb M^{\varkappa_l}$-valued functions $\pi_r(e)$ are continuous with respect to $r\in(0,\varepsilon_l)$ for all $e\in\mathfrak{E}^T_\Sigma$ and extend them by continuity^10[x]10This choice is possible already because a canonical representation exists. to $[0,\varepsilon_l]$.

Step 4. Determine the eigenvalues $\tau^k_{\gamma l}(r)$ and eigenprojectors $P^k_{\gamma l}$ of the matrices $\pi_r(E^T_\gamma)$ (see (5.8)). Using them, define a map

The map $\mathbf U$ transforms $\mathfrak{E}^T_\Sigma$ to a canonical form (or rather one of its versions: see the comment after Theorem 2).

All our results concern the shifted eikonals $\dot E^T_\gamma$ (see Convention 2). Their reformulation for the original eikonals $E^T_\gamma=\dot E^T_\gamma-P^T_\gamma$ is obvious. It suffices to replace the functions $\tau^k_{\gamma l}$ in (5.1) by $\tau^k_{\gamma l}-1$.

Comments

The map ${\mathbf V}\colon \mathfrak{E}^T_\Sigma\to\bigoplus_{l=1}^\mathcal L \dot C(\mathscr{S}_l;\mathbb M^{\varkappa_l})$, which is defined on the generators by the formula

The following observation may be useful in the inverse problem. Define a relation between points of the spectrum $\widehat{\mathfrak{E}^T_\Sigma}$ by declaring that $\widehat\pi\stackrel{\gamma}{\sim_0}\widehat\pi'$ if some $\gamma$-coordinates of these points coincide. i.e. $\tau^k_{\gamma l}(\widehat\pi)=\tau^{k'}_{\gamma l'}(\widehat\pi')$. Then define an equivalence by writing $\widehat\pi\sim\widehat\pi'$ if there are points $\widehat\pi_1,\dots,\widehat\pi_p\in\widehat{\mathfrak{E}^T_\Sigma}$ and vertices $\gamma_1,\dots, \gamma_{p+1}\in\Sigma$ such that $\widehat\pi\stackrel{\gamma_1}\sim_0\widehat\pi_1\stackrel{\gamma_2}\sim_0\cdots \stackrel{\gamma_p}\sim_0\widehat\pi_p\stackrel{\gamma_{p+1}}{\sim_0}\widehat\pi'$. We can show that taking the quotient of the spectrum with respect to the relation $\widehat\pi\sim\widehat\pi'$ identifies only the points lying in the clusters. Moreover, this quotient of $\widehat{\mathfrak{E}^T_\Sigma}$ is homeomorphic to a graph. Examples show that the resulting graph is homeomorphic to the quotient of the wave-filled region $\Omega^T_\Sigma$ by a certain relation which has a simple geometric meaning.

In known examples [6], [8], the clusters arise only when an interior vertex of $\Omega$ is overlapped by waves coming from at least two boundary vertices. We conjecture that this is a general fact. It is also interesting whether the presence of cycles in $\Omega^T_\Sigma$ can be characterized in terms of the algebra $\mathfrak{E}^T_\Sigma$ (see [6]). This question is open.

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Citation in format AMSBIB

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\by M.~I.~Belishev, A.~V.~Kaplun
\paper Canonical form of the $C^*$-algebra of eikonals related to a~metric graph
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