On Borel equivalence relations related to self-adjoint operators

In a recent work, the authors studied various Borel equivalence relations defined on the Polish space ${\rm{SA}}(H)$ of all (not necessarily bounded) self-adjoint operators on a separable infinite-dimensional Hilbert space $H$. In this paper we study the domain equivalence relation $E_{\rm{dom}}^{{\rm{SA}}(H)}$ given by $AE_{\rm{dom}}^{{\rm{SA}}(H)}B\Leftrightarrow {\rm{dom}}{A}={\rm{dom}}{B}$ and determine its exact Borel complexity: $E_{\rm{dom}}^{{\rm{SA}}(H)}$ is an $F_{\sigma}$ (but not $K_{\sigma}$) equivalence relation which is continuously bireducible with the orbit equivalence relation $E_{\ell^{\infty}}^{\mathbb{R}^{\mathbb{N}}}$ of the standard Borel group $\ell^{\infty}=\ell^{\infty}(\mathbb{N},\mathbb{R})$ on $\mathbb{R}^{\mathbb{N}}$. This, by Rosendal's Theorem, shows that $E_{\rm{dom}}^{{\rm{SA}}(H)}$ is universal for $K_{\sigma}$ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.Comment: 10 pages, added more detail of the proof of Proposition 3.8 after the referee's suggestio

On Polish Groups of Finite Type

Sorin Popa initiated the study of Polish groups which are embeddable into the unitary group of a separable finite von Neumann algebra. Such groups are called of finite type. We give necessary and sufficient conditions for Polish groups to be of finite type, and construct exmaples of such groups from semifinite von Neumann algebras. We also discuss permanence properties of finite type groups under various algebraic operations. Finally we close the paper with some questions concerning Polish groups of finite type.Comment: 20 page

Weyl-von Neumann Theorem and Borel Complexity of Unitary Equivalence Modulo Compacts of Self-Adjoint Operators

Weyl-von Neumann Theorem asserts that two bounded self-adjoint operators $A,B$ on a Hilbert space $H$ are unitarily equivalent modulo compacts, i.e., $uAu^*+K=B$ for some unitary $u\in \mathcal{U}(H)$ and compact self-adjoint operator $K$, if and only if $A$ and $B$ have the same essential spectra: $\sigma_{\rm{ess}}(A)=\sigma_{\rm{ess}}(B)$. In this paper we consider to what extent the above Weyl-von Neumann's result can(not) be extended to unbounded operators using descriptive set theory. We show that if $H$ is separable infinite-dimensional, this equivalence relation for bounded self-adjoin operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense $G_{\delta}$-orbit but does not admit classification by countable structures. On the other hand, apparently related equivalence relation $A\sim B\Leftrightarrow \exists u\in \mathcal{U}(H)\ [u(A-i)^{-1}u^*-(B-i)^{-1}$ is compact], is shown to be smooth. Various Borel or co-analytic equivalence relations related to self-adjoint operators are also presented.Comment: 36 page

Non-commutative hypergroup of order five

We prove that all hypergroups of order four are commutative and that there exists a non-comutative hypergroup of order five. These facts imply that the minimum order of non-commutative hypergroups is five even though the minimum order of non-commutative groups is six