Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras

Motke Porat, Victor Vinnikov

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we generalize classical results regarding minimal realizations of non-commutative (nc) rational functions using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. We prove the existence and uniqueness of a minimal realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity. Moreover, we show that using this realization we can evaluate the function on all of its domain (of matrices of all sizes) and also with respect to any stably finite algebra. As a corollary we obtain a new proof of the theorem by Cohn and Amitsur, that equivalence of two rational expressions over matrices implies that the expressions are equivalent over all stably finite algebras. Applications to the matrix valued and the symmetric cases are presented as well.

Original languageEnglish
Pages (from-to)1250-1299
Number of pages50
JournalJournal of the London Mathematical Society
Volume104
Issue number3
DOIs
StatePublished - 1 Oct 2021

Keywords

  • 16K40
  • 16R50
  • 46L54
  • 47A56
  • 90C22 (secondary)
  • 93C35 (primary)

ASJC Scopus subject areas

  • Mathematics (all)

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