arXiv subjects

Koichi Shimada

Publications and source records attributed to Koichi Shimada.

7 recordsLinked to original sources

arXiv2016

Maximal amenable MASAs of the free group factor of two generators arising from the free products of hyperfinite factors

In this paper, we give examples of maximal amenable subalgebras of the free group factor of two generators. More precisely, we consider two copies of the hyperfinite factor $R_i$ of type $\mathrm{II}_1$. From each $R_i$, we take a Haar unitary $u_i$ which generates a Cartan subalgebra of it. We show that the von Neumann subalgebra generated by the self-adjoint operator $u_1+u_1^{-1}+u_2+u_2^{-1}$ is maximal amenable in the free product. This provides infinitely many non-unitary conjugate maximal amenable MASAs.

math.OA↗

arXiv2016

Maximal amenability of the generator subalgebra in $q$-Gaussian von Neumann algebras

In this article, we give explicit examples of maximal amenable subalgebras of the $q$-Gaussian algebras, namely, the generator subalgebra is maximal amenable inside the $q$-Gaussian algebras for real numbers $q$ with its absolute value sufficiently small. To achieve this, we construct a Riesz basis in the spirit of Rădulescu and develop a structural theorem for the $q$-Gaussian algebras.

math.OA↗

arXiv2015

Approximate Unitary Equivalence of Finite Index Endomorphisms of the AFD Factors

We characterize the condition for two finite index endomorphisms on an AFD factor to be approximately unitarily equivalent. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi--Sutherland--Takesaki and Masuda--Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.

math.OA↗

arXiv2014

Classification of actions of compact abelian groups on subfactors with index less than 4

We calssify actions of discrete abelian groups on some inclusions of von Neumann algebras, up to cocycle conjugacy. As an application, we classify actions of compact abelian groups on the inclusions of AFD factors of type II_1 with index less than 4, up to stable conjugacy.

math.OA↗

arXiv2014

A Classification of Flows on AFD Factors with Faithful Connes--Takesaki Modules

We classify flows on AFD factors with faithful Connes-Takesaki modules. This is a generalization of classification of trace-scaling flows on the AFD $\mathrm{II}_\infty$ factor, which is equivalent to the uniqueness of the AFD $\mathrm{III}_1$ factor. In order to do this, we show that a flow on an AFD factor with faithful Connes-Takesaki module has the Rohlin property, which gives a partial answer to a characterization problem of the Rohlin property posed by Masuda-Tomatsu. It is also possible to think of this result as an $\mathbf{R}$-version of Izumi's result about compact group actions on type III factors with faithful Connes-Takesaki modules.

math.OA↗

arXiv2014

Locally Compact Separable Abelian Group Actions on Factors with the Rohlin Property

We show a classification theorem for locally compact separable abelain group actions on factorsw with the Rohlin property. This is a generalization of Masuda and Tomatsu's work.

math.OA↗

arXiv2014

Rohlin Flows on Amalgamated Free Product Factors

We construct flows on an amalgamated free product factor and characterize the Rohlin property for them by freeness of $\mathbf{R}\times \mathbf{Z}$ actions on a measured space. As a corolally, it turns out that there exist Rohlin flows on an amalgamated free product factor. This shows that the classification theorem of Masuda and Tomatsu for the Rohlin flows is also applicable to flows on non-McDuff factors.

math.OA↗