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

Playful expressions of one-year-old chimpanzee infants in social and solitary play contexts

Knowledge of the context and development of playful expressions in chimpanzees is limited because research has tended to focus on social play, on older subjects, and on the communicative signaling function of expressions. Here we explore the rate of playful facial and body expressions in solitary and social play, changes from 12- to 15-months of age, and the extent to which social partners match expressions, which may illuminate a route through which context influences expression. Naturalistic observations of seven chimpanzee infants (Pan troglodytes) were conducted at Chester Zoo, UK (n = 4), and Primate Research Institute, Japan (n = 3), and at two ages, 12 months and 15 months. No group or age differences were found in the rate of infant playful expressions. However, modalities of playful expression varied with type of play: in social play, the rate of play faces was high, whereas in solitary play, the rate of body expressions was high. Among the most frequent types of play, mild contact social play had the highest rates of play faces and multi-modal expressions (often play faces with hitting). Social partners matched both infant play faces and infant body expressions, but play faces were matched at a significantly higher rate that increased with age. Matched expression rates were highest when playing with peers despite infant expressiveness being highest when playing with older chimpanzees. Given that playful expressions emerge early in life and continue to occur in solitary contexts through the second year of life, we suggest that the play face and certain body behaviors are emotional expressions of joy, and that such expressions develop additional social functions through interactions with peers and older social partners