arXiv subjects

Hiroshi Ando

Publications and source records attributed to Hiroshi Ando.

28 records · Page 2Linked to original sources

arXiv2017

When does the Weyl-von Neumann Theorem hold?

A famous theorem due to Weyl and von Neumann asserts that two bounded self-adjoint operators are unitarily equivalent modulo the compacts, if and only if their essential spectrum agree. The above theorem does not hold for unbounded operators. Nevertheless, there exist closed subsets $M$ of $\mathbb{R}$ on which the Weyl--von Neumann Theorem hold: all (not necessarily bounded) self-adjoint operators with essential spectrum $M$ are unitarily equivalent modulo the compacts. In this paper, we determine exactly which $M$ satisfies this property.

math.SP↗

arXiv2016

Non-commutativity of the central sequence algebra for separable non-type I C$^{\ast}$-algebras

We show that if $A$ is a separable, simple and non-type I C$^{\ast}$ algebra, then for every properly infinite hyperfinite von Neumann algebra $M$ with separable predual, its Ocneanu ultrapower $M'\cap M^ω$ arises as a sub-quotient of the central sequence algebra $F(A)$ defined by the second named author. In particular, this answers affirmatively the question of the second named author (Abel Symposium '04): the central sequence C$^{\ast}$-algebra of the reduced free group C$^{\ast}$-algebra $C_{\rm{red}}^*(\mathbb{F}_2)$ is non-commutative.

math.OA↗

arXiv2014

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 $F_σ$ (but not $K_σ$) 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 $K_σ$ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.

math.LO↗

arXiv2014

Ultraproducts of von Neumann algebras

We study several notions of ultraproducts of von Neumann algebras from a unifying viewpoint. In particular, we show that for a sigma-finite von Neumann algebra $M$, the ultraproduct $M^ω$ introduced by Ocneanu is a corner of the ultraproduct $\prod^ωM$ introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state $φ$ of $M$ on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ($σ_t^{φ^ω}=(σ_t^φ)^ω$). Applying these results, we obtain several phenomena of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower $M^ω$ of a Type III$_0$ factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of $M$ in $M^ω$ and Connes' asymptotic centralizer algebra $M_ω$.

math.OA↗

arXiv2014

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: $σ_{\rm{ess}}(A)=σ_{\rm{ess}}(B)$. 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 $G_δ$-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.

math.FA↗

arXiv2013

Ultraproducts, QWEP von Neumann Algebras, and the Effros-Maréchal Topology

Based on the analysis on the Ocneanu/Groh-Raynaud ultraproducts and the Effros-Maréchal topology on the space vN(H) of von Neumann algebras acting on a separable Hilbert space H, we show that for a von Neumann algebra M in vN(H), the following conditions are equivalent: (1) M has the Kirhcberg's quotient weak expectation property (QWEP). (2) M is in the closure of the set F_{inj of injective factors on H with respect to the Effros-Maréchal topology. (3) M admits an embedding i into the Ocneanu ultrapower R_{infty}^{omega} of the injective III_1 factor R_{\infty} with a normal faithful conditional expectation epsilon: R_{infty}^{omega} to i(M). (4) For every epsilon>0, natural number n, and xi_1,...,xi_n in P_M^{natural}, there is a natural number k and a_1,...,a_nin M_k(C)_+, such that | -tr_k(a_ia_j)|<epsilon (1<=i,j<=n) holds, where tr_k is the tracial state on M_k(C), and P_M^{natural} is the natural cone in the standard form of M.

math.OA↗

arXiv2013

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 (ξ_n)n of a given vector ξin dom(A^{omega}) has the property that A^{omega}ξ= (Aξ_n)_{omega} holds.

math.FA↗

arXiv2011

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.

math.OA↗

arXiv2010

Lie Group-Lie Algebra Correspondences of Unitary Groups in Finite von Neumann Algebras

We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group $U(\mathcal{H})$ in a Hilbert space $\mathcal{H}$ with $U(\mathcal{H})$ equipped with the strong operator topology. More precisely, for any strongly closed subgroup $G$ of the unitary group $U(\mathfrak{M})$ in a finite von Neumann algebra $\mathfrak{M}$, we show that the set of all generators of strongly continuous one-parameter subgroups of $G$ forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra $\mathfrak{M}$ of all densely defined closed operators affiliated with $\mathfrak{M}$ from the viewpoint of a tensor category.

math.OA↗

arXiv2009

On the local structure of the representation of a local gauge group

This paper clarifies the local structure of the energy representation of a local gauge group. The group to be considered is a smooth map from a manifold into a compact Lie group. It acts on a Boson Fock spaces generated by connection 1-forms on the manifold. The irreduciblity depends on the dimensionality of a manifold. However, we showed that even if the representation is reducible(for example, dim=1 case) the commutant is very small. The discussion is based on the support property of the function, which is analogous to the algebraic Quantum Field Theory.

math-ph↗