Borel equivalence relations in the space of bounded operators. (English) Zbl 1420.03124

Smythe, Iian B.

Fundam. Math. 237, No. 1, 31-45 (2017).

Summary: We consider various notions of equivalence in the space of bounded operators

•  Encyclopedia of Mathematics
•  nLab
•  Wikipedia
•  Wolfram MathWorld

on a Hilbert space, in particular modulo finite rank, modulo Schatten \(p\)-class, and modulo compact. Using Hjorth’s theory of turbulence, the latter two are shown to be not classifiable by countable structures, while the first is not reducible to the orbit equivalence relation

•  Encyclopedia of Mathematics
•  nLab
•  Wikipedia
•  Wolfram MathWorld

of any Polish group action

•  Encyclopedia of Mathematics
•  Wikipedia
•  Wolfram MathWorld

. The results for modulo finite rank and modulo compact operators

•  Encyclopedia of Mathematics
•  nLab
•  Wikipedia
•  Wolfram MathWorld

are also shown for the restrictions of these equivalence relations to the space of projection operators.

MSC:

03E15	Descriptive set theory
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)

Keywords:

Borel equivalence relations; turbulence; compact operators; Calkin algebra; Schatten \(p\)-class; projection operators