28 research outputs found

### 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