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.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On Transitive Systems of Subspaces in a Hilbert Space

Methods of *-representations in Hilbert space are applied to study of systems of $n$ subspaces in a linear space. It is proved that the problem of description of $n$-transitive subspaces in a finite-dimensional linear space is *-wild for $n \geq 5$.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On C*-algebras generated by pairs of q-commuting isometries

We consider the C*-algebras O_2^q and A_2^q generated, respectively, by isometries s_1, s_2 satisfying the relation s_1^* s_2 = q s_2 s_1^* with |q| < 1 (the deformed Cuntz relation), and by isometries s_1, s_2 satisfying the relation s_2 s_1 = q s_1 s_2 with |q| = 1. We show that O_2^q is isomorphic to the Cuntz-Toeplitz C*-algebra O_2^0 for any |q| < 1. We further prove that A_2^{q_1} is isomorphic to A_2^{q_2} if and only if either q_1 = q_2 or q_1 = complex conjugate of q_2. In the second part of our paper, we discuss the complexity of the representation theory of A_2^q. We show that A_2^q is *-wild for any q in the circle |q| = 1, and hence that A_2^q is not nuclear for any q in the circle.Comment: 18 pages, LaTeX2e "article" document class; submitted. V2 clarifies the relationships between the various deformation systems treate

On Transitive Systems of Subspaces in a Hilbert Space

Methods of *-representations in Hilbert space are applied to study of systems of n subspaces in a linear space. It is proved that the problem of description of n-transitive subspaces in a finite-dimensional linear space is *-wild for n ≥ 5