arXiv subjects

Yasuo Watatani

Publications and source records attributed to Yasuo Watatani.

At least 19 recordsLinked to original sources

arXiv2025

Law of large numbers for non-linear traces of the Choquet type on finite factors

We introduced non-linear traces of the Choquet type and the Sugeno type on semi-finite factors M in [36] as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need a weighted dimension function on the projections of M, which is an analog of a monotone measure. In this paper, we study the law of large numbers for non-linear traces of the Choquet type on finite factors M. Since averages do not converge in general, we study the range of their accumulation points, that is, we estimate their limit supremum and limit infimum. We examine the trials of sequences consisting of self-adjoint operators, which appear in coin toss or Powers' binary shifts. We have also found some unexpected examples of Powers' binary shifts which satisfy what we call the uniform norm law of large numbers. This is an attempt at non-linear and non-commutative probability theory on matrix algebras and factors of type II_1.

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arXiv2024

Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers

We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor $\mathcal{M}$ as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need weighted dimension function $p \mapsto α(τ(p))$ for projections $p \in \mathcal{M}$, which is an analog of a monotone measure. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and Sugeno type respectively. Based on the notion of generalized eigenvalues and singular values, we show that non-linear traces of the Choquet type are closely related to the Lorentz function spaces and the Lorentz operator spaces if the weight functions $α$ are concave. For the algebras of compact operators and factors of type ${\rm II}$, we completely determine the condition that the associated weighted $L^p$-spaces for the non-linear traces become quasi-normed spaces in terms of the weight functions $α$ for any $0 < p < \infty$. We also show that any non-linear trace of the Sugeno type gives a certain metric on the factor. This is an attempt at non-linear and non-commutative integration theory on semifinite factors.

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arXiv2022

Non-linear traces on the algebra of compact operators and majorization

We study non-linear traces of Choquet type and Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both Choquet type and Sugeno type respectively. There exists a close relation between non-linear traces of Choquet type and majorization theory. We study trace class operators for non-linear traces of Choquet type. More generally we discuss Schatten-von Neumann $p$-class operators for non-linear traces of Choquet type. We determine when they form Banach spaces. This is an attempt of non-commutative integration theory for non-linear traces of Choquet type on the algebra of compact operators. We also consider the triangle inequality for non-linear traces of Sugeno type.

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arXiv2021

Non-linear traces on matrix algebras, majorization, unitary invariant norms and 2-positivity

We study non-linear traces of Choquet type and Sugeno type on matrix algebras. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both Choquet type and Sugeno type respectively. There exists a close relation among non-linear traces of Choquet type, majorization, unitary invariant norms and 2-positivity.

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arXiv2020

Non-linear monotone positive maps

We study several classes of general non-linear positive maps between C*-algebras, which are not necessary completely positive maps. We characterize the class of the compositions of *-multiplicative maps and positive linear mapsas the class of non-linear maps of boundedly positive type abstractly. We consider three classes of non-linear positive maps defined only on the positive cones, which are the classes of being monotone, supercongruent or concave. Any concave maps are monotone. The intersection of the monotone maps and the supercongruent maps characterizes the class of monotone Borel functional calculus. We give many examples of non-linear positive maps, which show that there exist no other relations among these three classes in general.

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arXiv2018

Systems of two subspases in a Hilbert space

We study two subspace systems in a separable infinite-dimensional Hilbert space up to (bounded) isomorphism. One of the main result of this paper is the following: Isomorphism classes of two subspace systems given by graphs of bounded operators are determined by unitarily equivalent classes of the operator ranges and the nullity of the original bounded operators giving graphs. We construct several non-isomorphic examples of two subspace systems in an infinite-dimensional Hilbert space.

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arXiv2018

Dimension groups for self-similar maps and matrix representations of the core of the associated C*-algebras

We introduce a dimension group for a self-similar map as the ${\rm K}_0$-group of the core of the $C^*$-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty}\cong {\mathbb Z}[t]$ together with the unilatral shift, i.e. the multiplication map by $t$ as an abstract group. Thus the canonical endomorphisms on the ${\rm K}_0$-groups are not automorphisms in geneal. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

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arXiv2016

Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces

We introduce unbounded strongly irreducible operators and transitive operators. These operators are related to a certain class of indecomposable Hilbert representations of quivers on infinite-dimensional Hilbert spaces. We regard the theory of Hilbert representations of quivers is a generalization of the theory of unbounded operators. A non-zero Hilbert representation of a quiver is said to be transitive if the endomorphism algebra is trivial. If a Hilbert representation of a quiver is transitive, then it is indecomposable. But the converse is not true. Let $Γ$ be a quiver whose underlying undirected graph is an extended Dynkin diagram. Then there exists an infinite-dimensional transitive Hilbert representation of $Γ$ if and only if $Γ$ is not an oriented cyclic quiver.

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arXiv2014

Relative position of three subspaces in a Hilbert space

We study the relative position of three subspaces in a separable infinite-dimensional Hilbert space. In the finite-dimensional case, Brenner described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space. We also give a partial result which gives a condition on a system to have a (dense) decomposition containing a pentagon.

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arXiv2013

Ideals of the core of C*-algebras associated with self-similar maps

We give a complete classification of the ideals of the core of the C*-algebras associated with self-similar maps under a certain condition. Any ideal is completely determined by the intersection with the coefficient algebra C(K) of the self-similar set K. The corresponding closed subset of K is described by the singularity structure of the self-similar map. In particular the core is simple if and only if the self-similar map has no branch point. A matrix representation of the core is essentially used to prove the classification.

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arXiv2013

Perturbations of intermediate C-subalgebras for simple C-algebras

We study uniform perturbations of intermediate C*-subalgebras of inclusions of simple C*-algebras. If a unital simple C*-algebra has a simple C*-subalgebra of finite index, then sufficiently close simple intermediate C*-subalgebras are unitarily equivalent. These C*-subalgebras need not to be nuclear. The unitary can be chosen in the relative commutant algebra. An imediate corollary is the following: If the relative commutant is trivial, then the set of intermediate C*-subagebras is a finite set.

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arXiv2013

Strongly irreducible operators and indecomposable representations of quivers on infinite-dimensional Hilbert spaces

We study several classes of indecomposable representations of quivers on infinite-dimensional Hilbert spaces and their relation. Many examples are constructed using strongly irreducible operators. Some problems in operator theory are rephrased in terms of representations of quivers. We shall show two kinds of constructions of quite non-trivial indecomposable Hilbert representations of the Kronecker quiver such that their endomorphism rings are trivial, which are called transitive. One is a perturbation of a weighted shift operator by a rank-one operator. The other one is a modification of an unbounded operator used by Harrison,Radjavi and Rosenthal to provide a transitive lattice.

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arXiv2012

C*-algebras associated with complex dynamical systems and backward orbit structure

Let $R$ be a rational function. The iterations $(R^n)_n$ of $R$ gives a complex dynamical system on the Riemann sphere. We associate a $C^*$-algebra and study a relation between the $C^*$-algebra and the original complex dynamical system. In this short note, we recover the number of $n$-th backward orbits counted without multiplicity starting at branched points in terms of associated $C^*$-algebras with gauge actions. In particular, we can partially imagine how a branched point is moved to another branched point under the iteration of $R$. We use KMS states and a Perron-Frobenius type operator on the space of traces to show it.

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arXiv2012

Traces on cores of C*-algebras associated with self-similar maps

We completely classify the extreme tracial states onthe cores of the C*-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.

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arXiv2010

KMS states on finite-graph C*-algebras

We study KMS states on finite-graph C*-algebras with sinks and sources. We compare finite-graph C*-algebras with C*-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature $β$ is large, then the set of extreme $β$-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational funcition from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshevyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

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arXiv2010

Fundamental group of simple $C^*$-algebras with unique trace II

We show that any countable subgroup of the multiplicative group $\mathbb{R}_+^{\times}$ of positive real numbers can be realized as the fundamental group $\mathcal{F}(A)$ of a separable simple unital $C^*$-algebra $A$ with unique trace. Furthermore for any fixed countable subgroup $G$ of $\mathbb{R}_+^{\times}$, there exist uncountably many mutually nonisomorphic such algebras $A$ with $G = \mathcal{F}(A)$.

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arXiv2009

Fundamental group of simple $C^*$-algebras with unique trace

We introduce the fundamental group ${\mathcal F}(A)$ of a unital simple $C^*$-algebra $A$ with a unique normalized trace. We compute fundamental groups ${\mathcal F}(A)$ of several nuclear or non-nuclear $C^*$-algebras $A$. K-theoretical obstruction enables us to compute the fundamental group easily. Our study is essentially based on the computation of Picard groups by Kodaka.

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arXiv2008

Toeplitz-composition C*-algebras for certain finite Blaschke products

Let R be a finite Blaschke product of degree at least two with R(0)=0. Then there exists a relation between the associated composition operator C_R on the Hardy space and the C*-algebra associated with the complex dynamical system on the Julia set of R. We study the C*-algebra generated by both the composition operator C_R and the Toeplitz operator T_z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C*-algebra associated with the complex dynamical system, which is simple and purely infinite.

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