On the Existence of Configurations of Subspaces in a Hilbert Space with Fixed Angles

For a class of $*$-algebras, where $*$-algebra $A_{\Gamma,\tau}$ is generated by projections associated with vertices of graph $\Gamma$ and depends on a parameter $\tau$ $(0<\tau \leq 1)$, we study the sets $\Sigma_\Gamma$ of values of $\tau$ such that the algebras $A_{\Gamma,\tau}$ have nontrivial $*$-representations, by using the theory of spectra of graphs. In other words, we study such values of $\tau$ that the corresponding configurations of subspaces in a Hilbert space exist.


Introduction
A number of papers is devoted to the study of n-tuples of subspaces in a Hilbert space. The interest to this problem arose in many respects due to its applications to problems of mathematical physics (see, e.g., [2] and the bibliography therein).
In the present paper we study the configurations of subspaces H i (i = 1, . . . , n) associated with the vertices of graph Γ, where an angle between any two of subspaces is fixed (see Section 3). It is convenient to consider such configurations of subspaces as * -representations of algebras generated by projections with relations of Temperley-Lieb type (see [3,6,9]). For tree Γ the set Σ Γ (of those values of an "angle" τ where the corresponding configurations exist) is described in Section 4. This result is obtained by using the theory of graph spectra (needed notions and results are given in Section 2). For graphs containing cycles the situation is more complicated (see Remark 2).

Necessary facts from theory of spectra of graphs
We give some facts necessary for the exposition below, which can be found in [1]. Let Γ be a finite undirected graph without multiple edges and loops. The adjacency matrix of a graph Γ, with vertex set {1, . . . , n}, is n × n matrix A Γ = a i,j n i,j=1 with a i,j = 1 if there is an edge between i and j, and a i,j = 0; otherwise a i,i = 0 ∀ i. The eigenvalues of A Γ and the spectrum of A Γ are also called the eigenvalues and the spectrum of a graph Γ, respectively. The eigenvalues of Γ are denoted by λ 1 , . . . , λ n ; they are real because A Γ is symmetric. We assume that r = λ 1 ≥ λ 2 ≥ · · · ≥ λ n = q. The largest eigenvalue r = λ 1 is called the index of a graph Γ. 2. If Γ is a connected graph then 2 cos π n+1 ≤ r.
Remark 1. If Γ is a tree then r = −q, as the tree is a bipartite graph.
We also need the following statement. Corollary 1 (For trees). Let Γ be a tree with index r. Then 1. r < 2 if and only if Γ is one of the following graphs: A n , D n , E 6 , E 7 , E 8 (see Fig. 1).
2. r = 2 if and only if Γ is one of the following graphs: D n , E 6 , E 7 , E 8 (see Fig. 2).

Conf igurations of subspaces in a Hilbert space with f ixed angles between them
Let H be a complex Hilbert space and let H i , H j ⊂ H be its closed subspaces. We say that an angle between H i and H j is fixed and equals to ϕ i,j ∈ [0; π 2 ] if for the orthogonal projections P H i , P H j on these subspaces we have Having a finite undirected graph Γ without multiple edges and loops with the numbers on its edges, we define the conditions on the configuration of subspaces in a Hilbert space as follows.
The subspaces correspond to the vertices of a graph and an angle between any two of them is given by the number τ i,j standing on the respective edge. If vertices are not adjacent we assume that the corresponding subspaces are orthogonal. We consider the following questions: 1. For which values of the parameters τ i,j the configuration associated with a graph Γ exists.
2. Give the description of all irreducible configurations (associated with a fixed graph Γ and an arrangement of numbers on its edges) up to a unitary transformation.
It should be noted that the subspaces corresponding to vertices from different components of Γ are orthogonal, so we will consider only connected graphs.
These problems can be reformulated in terms of finding * -representations of * -algebras associated with Γ with the numbers τ i,j on the edges. Let Γ be a finite, undirected, connected graph without multiple edges and loops, with Γ 0 (|Γ 0 | = n) and Γ 1 the sets of the vertices and the edges respectively. Let τ : Γ 1 → (0, 1) be the arrangement of numbers on its edges. We enumerate the vertices of Γ by numbers 1, . . . , n in any way and denote τ (i, j) =: τ i,j = τ j,i . Definition 1. A Γ,τ is an * -algebra with 1 over C generated by projections p 1 , . . . , p n (p 2 Results on dimension of the algebra A Γ,τ (in dependence on a graph Γ) can be found in [7,8]. Below we suppose that Γ is a tree. Then the * -algebra A Γ,τ is finite dimensional and, therefore, does not have infinite dimensional irreducible * -representations.  A(Γ, τ )).