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Yusuke Sawada

Publications and source records attributed to Yusuke Sawada.

8 recordsLinked to original sources
arXiv2020

The adjacency matrices and the transition matrices related to random walks on graphs

A pointed graph $(Γ,v_0)$ induces a family of transition matrices in Wildberger's construction of a hermitian hypergroup via a random walk on $Γ$ starting from $v_0$. We will give a necessary condition for producing a hermitian hypergroup as we assume a weaker condition than the distance-regularity for $(Γ,v_0)$. The condition obtained in this paper connects the transition matrices and the adjacency matrices associated with $Γ$.

math.CO↗
arXiv2020

Hypergroup structures of open quantum random walks on distance sets

Wildberger has introduced the method to construct a hermitian discrete hypergroup from a random walk on a graph. We will apply his method to an open quantum random walk (OQRW) on a distance set, and show that any discrete hypergroup which is not necessary hermitian is realized by an OQRW on a distance set. We will investigate distributions of OQRWs on distance sets in the view point of hypergroups.

math-ph↗
arXiv2020

Hypergroups and distance distributions of random walks on graphs

Wildberger's construction enables us to obtain a hypergroup from a special graph via random walks. We will give a probability theoretic interpretation to products on the hypergroup. The hypergroup can be identified with a commutative algebra whose basis is transition matrices. We will estimate the operator norm of such a transition matrix and clarify a relationship between their matrix products and random walks.

math.PR↗
arXiv2019

E$_0$-semigroups and product systems of W$^*$-bimodules

Product systems have been originally introduced to classify E$_0$-semigroups on type I factors by Arveson. We develop the classification theory of E$_0$-semigroups on a general von Neumann algebra and the dilation theory of CP$_0$-semigroups in terms of W$^*$-bimodules. For this, we provide a notion of product system of W$^*$-bimodules. This is a W$^*$-bimodule version of Arveson's and Bhat-Skeide's product systems. There exists a one-to-one correspondence between CP$_0$-semigroups and units of product systems of W$^*$-bimodules. The correspondence implies a construction of a dilation of a given CP$_0$-semigroup, a classification of E$_0$-semigroups on a von Neumann algebra up to cocycle equivalence and a relationship between Bhat-Skeide's and Muhly-Solel's constructions of minimal dilations of CP$_0$-semigroups.

math.OA↗
arXiv2018

A remark on the minimal dilation of the semigroup generated by a normal UCP-map

There are known three ways to construct the minimal dilation of the discrete semigroup generated by a normal unital completely positive map on a von Neumann algebra, which are given by Arveson, Bhat-Skeide and Muhly-Solel. In this paper, we clarify the relation of the constructions by Bhat-Skeide and Muhly-Solel.

math.OA↗
arXiv2017

Hypergroups derived from random walks on some infinite graphs

Wildberger gave a method to construct a finite hermitian discrete hypergroup from a random walk on a certain kind of graphs. In this article, we reveal that his method is applicable to a random walk on a certain kind of infinite graphs. Moreover, we make some observations of finite or infinite graphs on which a random walk produces a hermitian discrete hypergroup.

math.PR↗