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Masaru Nagisa

Publications and source records attributed to Masaru Nagisa.

14 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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arXiv2023

The operator $(p, q)$-norm of some matrices

We compute the operator $(p,q)$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators from the $n$ dimensional Banach space $\ell^p(n)$ to $\ell^q(n)$. We have shown that a special matrix $A=\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ which corresponds to a magic square has $\|A\|_{p,p} = \max \{\|Aξ\|_p : ξ\in\ell^p(n), \|ξ\|_p=1\} =15$ for any $p\in [1,\infty]$. In this paper, we extend this result and we compute $\|A\|_{p,q}$ for $1\le q \le p \le \infty$.

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arXiv2022

The p-norm of some matrices

We compute the operator $p$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators on the $n$ dimensional Banach space $\ell^p(n)$. The notion of logarithmic affine matrices is defined, and for such a matrix its $p$-norm is computed exactly. In particular, a matrix $A=\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ which corresponds to a magic square belongs to the class of logarithmic affine matrices, and its $p$-norm is equal to $15$ for any $p\in [1,\infty]$.

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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

Stable rank for inclusions of Banach algebras

We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.

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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

The order-$n$ minors of certain $(n+k) \times n$ matrices

We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an application we construct subspaces of $\C^m \otimes \C^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $k$ and which has a basis of Schmidt rank $k$ for $k=2,3,4$.

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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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arXiv2016

Some families of operator norm inequalities

It is known that the function $f(e^x)/g(e^x)$ is positive definite for some functions $f,g$ implies the operator norm inequality related to $f,g$. We treat functions which have the following form: $f(t) = t^{(1-\sum_{i=1}^n (a_i-b_i))/2}\prod_{i=1}^n \frac{b_i(t^{a_i}-1)}{a_i(t^{b_i}-1)}$.

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arXiv2012

Transforms on operator monotone functions

For an operator monotone function $f(t)$ on the positive real line, we show the operator monotonicity of the type of the functions $(t-a)(t-b)/(f(t)-f(a))(f^\sharp(t)-f^\sharp(b))$.

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arXiv2004

The numerical radius Haagerup norm and Hilbert space square factorizations

We study a factorization of bounded linear maps from an operator space $A$ to its dual space $A^*$. It is shown that $T : A \longrightarrow A^*$ factors through a pair of a column Hilbert spaces $\mathcal{H}_c$ and its dual space if and only if $T$ is a bounded linear form on $A \otimes A$ by the canonical identification equipped with a numerical radius type Haagerup norm. As a consequence, we characterize a bounded linear map from a Banach space to its dual space, which factors through a pair of Hilbert spaces.

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arXiv2004

Numerical Radius Norms on Operator Spaces

We introduce a numerical radius operator space $(X, \mathcal{W}_n)$. The conditions to be a numerical radius operator space are weaker than the Ruan's axiom for an operator space $(X, \mathcal{O}_n)$. Let $w(\cdot)$ be the numerical radius norm on $\mathbb{B}(\mathcal{H})$. It is shown that if $X$ admits a norm $\mathcal{W}_n(\cdot)$ on the matrix space $\mathbb{M}_n(X)$ which satisfies the conditions, then there is a complete isometry, in the sense of the norms $\mathcal{W}_n(\cdot)$ and $w_n(\cdot)$, from $(X, \mathcal{W}_n)$ into $(\mathbb{B}(\mathcal{H}), w_n)$. We study the relationship between the operator space $(X, \mathcal{O}_n)$ and the numerical radius operator space $(X, \mathcal{W}_n)$. The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.

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