1,562 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

### Spectral Properties of Schr\"odinger Operators on Perturbed Lattices

We study the spectral properties of Schr\"{o}dinger operators on perturbed
lattices. We shall prove the non-existence or the discreteness of embedded
eigenvalues, the limiting absorption principle for the resolvent, construct a
spectral representation, and define the S-matrix. Our theory covers the square,
triangular, diamond, Kagome lattices, as well as the ladder, the graphite and
the subdivision of square lattice

### Notes on the Krupa-Zawisza Ultrapower of Self-Adjoint Operators

It is known that there is a difficulty in constructing the ultrapower of
unbounded operators. Krupa and Zawisza gave a rigorous definition of the
ultrapower A^{omega} of a selfadjoint operator A. In this note, we give
alternative description of A^{omega} and the Hilbert space H(A) on which
A^{omega} is densely defined, which provides a criterion to determine to which
representing sequence (\xi_n)n of a given vector \xi in dom(A^{omega}) has the
property that A^{omega}\xi = (A\xi_n)_{omega} holds.Comment: 13page

### Structure of bicentralizer algebras and inclusions of type III factors (Mathematical aspects of quantum fields and related topics)

We investigate the structure of the relative bicentralizer algebra B(N ⊂ M, φ) for inclusions of von Neumann algebras with normal expectation where N is a type Ill₁ subfactor and φ ∈ N* is a faithful state. We first construct a canonical flow β[φ] : R*₊ ↷ B(N C M, ip) on the relative bicentralizer algebra and we show that the W*-dynamical system (B(N ⊂ M, φ), β[φ]) is independent of the choice of ip up to a canonical isomorphism. In the case when N = M, we deduce new results on the structure of the automorphism group of B(M, φ) and we relate the period of the flow β[φ] to the tensorial absorption of Powers factors. For general irreducible inclusions N ⊂ M, we relate the ergodicity of the flow β[φ] to the existence of irreducible hyperfinite subfactors in M that sit with normal expectation in N. When the inclusion N ⊂ M is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when N is amenable

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