On representations of the algebras generated by a finite resolution of the identity and a collection of jointly orthogonal projections

1Ashurova, EN, 2Ostrovskyi, VL, 2Samoilenko, Yu.S
1Chiltern Clinical Research in Ukraine, Kyiv
2Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2017, 10:3-9
https://doi.org/10.15407/dopovidi2017.10.003
Section: Mathematics
Language: Ukrainian
Abstract: 

We study properties of representations of the involutive algebra generated by self-adjoint idempotents, q1, . . ., qn and p1, . . ., pm, which satisfy the conditions q1 + . . . + qn = e, pjpk = 0, j ≠ k. The corresponding collections of projections in a Hilbert space arise in the study of the Fredholm properties of Toeplitz operators. In particular, for generic irredu cible representations with dim Pj = 1, j = 1 . . . , m, we have constructed a commuting family of normal operators, whose joint spectrum determines the representation up to unitary equivalence.

Keywords: families of orthoprojections, Toeplitz operators
References: 
  1. Vasilevski, N. L. (1998). C*-algebras generated by orthogonal projections and their applications. Integr. Equ. Oper. Theory, 31, pp. 113-132. https://doi.org/10.1007/BF01203459
  2. Karlovich, Yu. I. & Pessoa, L. V. (2007). C*-algebras of Bergmann type operators with piecewise continuous coefficients. Integr. Equ. Oper. Theory, 57, pp. 521-565. https://doi.org/10.1007/s00020-006-1473-x
  3. Strelets, A. V. & Feshchenko, I. S. (2012). On systems of subspaces of a Hilbert space that satisfy conditions on the angles between every pair of subspaces. St. Petersburg Math. J., 24, No. 5, pp. 823–846. https://doi.org/10.1090/S1061-0022-2013-01264-7
  4. Kruglyak, S. A. & Samoĭlenko, Ju. S. (1980). Unitary equivalence of sets of self-adjoint operators. Funct. Anal. Appl., 14, No. 1, pp. 48-50. https://doi.org/10.1007/BF01078420
  5. Ashurova, E.N. & Ostrovskyi, V.L. (2015). On representations of “all but two” algebras. Zbirnyk Prats Instytutu Matematyky NAN Ukrainy, 12, No. 1. pp. 8-21 (in Ukrainian).
  6. Samoilenko, Yu. S. & Strelets, A. V. (2009). On simple n-tuples of subspaces of a Hilbert space. Ukr. Math. J., 61, No. 12, pp. 1956-1994. https://doi.org/10.1007/s11253-010-0325-7
  7. Vasilevski, N.L. (1999). On the structure of Bergmann and poly-Bergmann spaces. Integr. Equ. Oper. Theory, 33, pp. 471-488. https://doi.org/10.1007/BF01291838