This is an announcement for the paper "Borel equivalence relations in the space of bounded operators" by Iian B. Smythe.

Abstract: We consider various notions of equivalence in the space of bounded operators on a Hilbert space, including modulo finite rank operators, modulo Schatten $p$-classes, and modulo compact operators. Using Hjorth's theory of turbulence, the latter two are shown to be not classifiable by countable structures, while the first cannot be reduced to the orbit equivalence relation of any Polish group action. The results for modulo finite rank and modulo compact operators are also shown for the restrictions of these equivalence relations to the space of projection operators. Families of non-classifiable equivalence relations on sequence spaces are described and utilized in these results.

Archive classification: math.LO math.OA

Mathematics Subject Classification: Primary 03E15, 47B10, Secondary 47C15, 46A45

Remarks: 36 pages

Submitted from: ibs24@cornell.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1407.5325

or

http://arXiv.org/abs/1407.5325