2018年度年会(於:東京大学)

SUMMARY: Quantum (non-commutative) probability, tracing back to von Neumann who originally aimed at statistical study of quantum mechanics, has been penetrated into other branches of mathematics. In this lecture we focus on the interaction with spectral analysis of (growing) graphs developed during the last decade and mention some current topics.

In general, quantum probability is discussed in terms of an algebraic probability space \((\mathcal {A},\varphi )\), where \(\mathcal {A}\) is a unital \(*\)-algebra and \(\varphi :\mathcal {A}\rightarrow \mathbb {C}\) is a state. Given a graph we consider the adjacency matrix (or other matrices such as Laplacian matrix, distance matrix, etc.) as a real random variable of an algebraic probability space, typically the adjacency algebra equipped with a state.

The first method is quantum decomposition. If the adjacency matrix \(A\) is decomposed into a sum of three operators \(A=A^++A^-+A^\circ \), where \(A^+,A^-,A^\circ \) are the creation, annihilation and preservation operators in an interacting Fock space, then the spectral distribution of \(A\) is obtained as the vacuum distribution of the canonical field operator of the interacting Fock space, where the theory of orthogonal polynomials is applied. This method is useful also for asymptotic spectral distributions of growing graphs.

The second method is based on various concepts of independence arising from non-commutative nature. If the adjacency matrix is expressible as a sum of independent random variables, typically when a graph admits a product structure, the spectral distribution is given by convolution of probability distributions. Hence for a growing graph, the asymptotic spectral distribution is obtained from the associated central limit theorem. The Cartesian, comb, star, free, Kronecker, lexicographic products of graphs are discussed along with quantum probability.

SUMMARY: The Laplace transform of a power function on the positive half line is equal to a power function again up to a constant multiple, where the Gamma function appears as the coefficient. Replacing the half line by a homogeneous convex cone, Gindikin showed that the Laplace transform of a relatively invariant function over the homogeneous cone equals a relatively invariant function on the dual cone with an explicit Gamma factor. On the other hand, it is known in mathematical statistics that the Laplace transform of a power of the determinant function over the cone of positive definite real symmetric matrices with prescribed zero components equals a product of powers of minors with an explicit Gamma factor provided that the zero pattern is associated to a chordal graph. Although such a cone is not necessarily homogeneous, the integral formula is quite similar to Gindikin’s formula eventually.

In this talk, we introduce a new cone consisting of positive definite real symmetric matrices with a specific block decompositions satisfying certain axioms. The class of this new cone contains all the homogeneous cones and the cones associated to chordal graphs. We establish Gamma-type integral formulas concerning the Laplace transform of a product of powers of minors over the new cones, so that the formulas mentioned above are obtained as special cases. Furthermore, considering the analytic continuation of the Gamma integral formula, we define the Riesz distribution whose Laplace transform equals a product of powers of minors of symmetric matrices, which gives a fundamental solution of a certain differential operator in a special case.

SUMMARY: The notion of (two-variable) operator means was introduced in an axiomatic way by Kubo and Ando (1980). A long-standing problem since then had been to generalize it to more than two variables. A breakthrough happened when the definitions of multivariate geometric means of positive definite matrices were found by the iteration method of Ando, Li and Mathias (2004) and in the Riemannian geometry approach by Moakher (2005) and by Bhatia and Holbrook (2006). In this talk I survey recent developments on multivariate means of positive definite matrices/operators in the Riemannian geometry approach, mainly on the multivariate version of the weighted geometric and power means. A significant feature of the multivariate geometric mean is that it is understood as the Cartan barycenter in the Riemannian trace metric, so it is also characterized by the Karcher equation (the gradient zero equation). This approach using Karcher type equations can be extended to an even more general setting of probability measures on the positive definite matrices (also operators), where the Wasserstein distance plays a crucial role. I furthermore explain log-majorization for the geometric mean in the setting of probability measures and a recent result on deformation of multivariate operator means by a fixed point method.

SUMMARY: Here, we will state simply a general meaning for reproducing kernels. We would like to answer for the general and essential question that: what are reproducing kernels? By considering the basic problem, we were able to obtain a general concept of the generalized delta function as a generalized reproducing kernel and, as a general reproducing kernel Hilbert space, we can consider all separable Hilbert spaces comprising functions.

SUMMARY: The general integral transforms in the framework of Hilbert spaces were combined with the general theory of reproducing kernels and many applications were developped. The basic assumption here that the integral kernels belong to some Hilbert spaces. However, as a very typical integral transform, in the case of Fourier integral transform, the integral kernel does not belong to \(L_2({\bf R})\), however, we can establish the isometric identity and inversion formula.

On the above situations, we will develop some general integral transform theory containing the Fourier integral transform case that the integral kernel does not belong to any Hilbert space, based on the general concept of generalized reproducing kernels.

SUMMARY: We consider the relationship between linear continuous operators acting on the space of entire functions of one variable of a given order and linear differential operators of infinite order satisfying certain growth conditions for the coefficients. We found that these two classes of operators are equivalent. Our results can be extended to the case of several variables.

SUMMARY: The operator algebra is introduced based on the framework of logarithmic representation of infinitesimal generators. In conclusion a set of generally-unbounded infinitesimal generators is characterized as a module over the Banach algebra.

SUMMARY: The asymptotic behavior of the effective mass \(m_{\rm eff}(\Lambda )\) of the so-called Nelson model in quantum field theory is considered, where \(\Lambda \) is an ultraviolet cutoff parameter of the model. Let \(m\) be the bare mass of the model. It is shown that for sufficiently small coupling constant \(|\alpha |\) of the model, \(m_{{\rm eff}}(\Lambda )/m\) can be expanded as \(m_{{\rm eff}}(\Lambda )/m= 1+\sum _{n=1}^\infty a_n(\Lambda ) \alpha ^{2n}\). A physical folklore is that \(a_n(\Lambda )=O( [\log \Lambda ]^{(n-1)})(\Lambda \to \infty )\). It is rigorously shown that \[0<\lim _{\Lambda \to \infty }a_1(\Lambda )<C,\quad C_1\leq \lim _{\Lambda \to \infty }a_2(\Lambda )/\log \Lambda \leq C_2\] with some constants \(C\), \(C_1\) and \(C_2\).

SUMMARY: We can construct the Gibbs measure associated with the renormalized Nelson model in scalar quantum field theory. By using this Gibbs measure we investigate properties of the ground state of the Nelson Hamiltonian.

SUMMARY: Kuramoto’s famous conjecture is that there is a positive constant \(K_c\) such that the system of oscillators becomes synchronized only when the coupling constant \(K>K_c\). H. Chiba proved this conjecture for the continuous version of Kuramoto model in 2015. His proof covers the cases that the initial distributions \(g(\omega )\) of frequencies of oscillators are Gaussian or Cauchy. His key tool is the precise analysis of the generalized eigenvalues of some unbounded linear operator \(T\) related to \(g(\omega )\) on some Hilbert space. We extended his method to the analytic distributions of form \(g=e^{-P(\omega )}/M\), where \(P(\omega )\) is an even polynomial which increases in \(\omega >0\), and M is a positive constant.

SUMMARY: Statistical analysis of styles of translated versions of literary works is more difficult than that of the originals, because most of the variables used in the latter case indicate only characteristics of the authors, not of the translators. We replace probability measures related with the text of a literary work by conditional expectations related with the text of the original and translated versions of the work, and compare the numerical values of these condional expectations. Results obtained by applying our framework to Mancu books on natural sciences, Neo-Confucianism or Catholicism published in the 17th and the 18th centuries are given.

SUMMARY: Consider the following property \((P)\) for a bounded closed convex set \(C\) in a Banach space \(X\): \((P)\) For every \(x\in X\), a positive-scalar multiple of \(x\) gives a nearest point in \(C\) to \(x\). Then it is clear that a closed ball with its center at the origin has this property. The converse to this assertion is the subject of this talk, and it is proved that a bounded closed convex set \(C\subset X\) with \(0\in \mathrm {Int}\,C\) possessing property \((P)\) is a closed ball with center \(0\), provided \(X\) is smooth and \(\dim \,X>1\). It is also proved that if a closed convex set \(C\subset X\) with \(0\in \mathrm {Int}\,C\) satisfies \((P)\), then \(X\) is smooth, provided the boundary of \(C\) is smooth in a weak sense.

SUMMARY: Our motivation of this research is based on a certain refinement of an inequality that is induced by an abstract Jensen’s inequality and on a characterization of a semigroup operations that is distributed by a vector addition in a plane. We determine cancellative, continuous operations on \(\mathbb R\) with the usual topology that is distributed by the usual multiplication and addition.

SUMMARY: We show that a surjective isometry with respect to the sum norm between the Banach algebras of Lipschitz maps with the values in unital commutative \(C^*\)-algebras is canonical. When the unital commutative \(C^*\)-algebra is the complex plane, the result confirms the statement of Example 8 of the paper by Jarosz and Pathak.

SUMMARY: Heckman–Opdam hypergeometric function for the root system of type \(A_{n-1}\) with a certain degenerate parameter can be expressed by the Lauricella hypergeometric function \(F_D\).

SUMMARY: For a real split Lie group \(G=KAN\), Vogan introduced the notion of fine \(K\)-types. We define a similar class of \(K\)-types for general connected real semisimple Lie groups. For a \(K\)-type in this class, we study matrix-valued elementary spherical functions. If there exists an invariant differential operator of the first order acting on the spherical functions, we can explicitly write the elementary spherical functions using Opdam’s non-symmetric hypergeometric functions.

SUMMARY: In this talk, we announce the result on a generalization of Cartan decomposition for symmetric spaces to spherical homogeneous spaces of reductive type. Moreover, we deal with some examples concerning to our result and explain how to find abelian subgroups.

SUMMARY: Let \(X\) be a homogeneous space of a real reductive Lie group \(G\). Then it is proved by T. Kobayashi and T. Oshima that the regular representation \(C^{\infty }(X)\) contains each irreducible representation of \(G\) at most finitely many times if a minimal parabolic subgroup \(P\) of \(G\) has an open orbit on \(X\), or equivalently, if the number of \(P\)-orbits on \(X\) is finite. Moreover, Kobayashi proved that for a general parabolic subgroup \(Q\) of \(G\), there is a degenerate principal series representation induced from \(Q\) contained in \(C^{\infty }(X)\) with infinite multiplicity if \(Q\) has no open orbit on \(X\). In this article, we prove that there is a degenerate principal series representation induced from \(Q\) contained in \(C^{\infty }(X)\) with infinite multiplicity if the number of orientable (or transverse orientable) \(Q\)-orbits on \(X\) is infinite even when there exists an open \(Q\)-orbit on \(X\).

SUMMARY: We introduce the notion of wreath algebra with trace. This can be regarded as an abstraction of the wreath product of an algebra with trace and the infinite symmetric group. Using this notion, we describe an invariant theory simply.

SUMMARY: We introduce a dimension group for a self-similar map as the \(K_0\)-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 the countably generated free abelian group together with the unilatral shift.

SUMMARY: We will characterize topologically conjugate two-sided topological Markov shifts \((\bar {X}_A,\bar {\sigma }_A)\) in terms of the associated asymptotic Ruelle \(C^*\)-algebras \({\mathcal {R}}_A\) with its commutative \(C^*\)-subalgebras \(C(\bar {X}_A)\) and the canonical circle actions. We will also show that extended Ruelle algebras \({\tilde {\mathcal {R}}}_A\), which are purely infinite version of the asymptotic Ruelle algebras, with its \(C^*\)-subalgebras \(C(\bar {X}_A)\) and the canonical torus actions \(\gamma ^A\) are complete invariants for topological conjugacy of two-sided topological Markov shifts.

SUMMARY: Let \(G\) be a finite group, \(A\) a unital separable finite simple nuclear C*-algebra, and \(\alpha \) an action of \(G\) on \(A\). Assume that \(A\) absorbs the Jiang–Su algebra \(\mathcal {Z}\), the extremal boundary of the trace space of \(A\) is compact and finite dimensional and that \(\alpha \) fixes any tracial state of \(A\). Then \(\mathrm {tsr}(A \rtimes _\alpha G) = 1\). In particular, when \(A\) has a unique tracial state, we conclude \(\mathrm {tsr}(A \rtimes _\alpha G) = 1\) without above conditions on a tracial state space of \(A\).

SUMMARY: We prove a boundary rigidity result for the embedding of a reduced free product C\(^*\)-algebra into its associated “crossed product” C\(^*\)-algebra. This provides new examples of rigid embeddings of exact C\(^*\)-algebras into purely infinite simple nuclear C\(^*\)-algebras.

SUMMARY: There is a one-to-one correspondence between quasi-free states on a self-dual CAR algebra and covariance operators. The problem of when the correspondence preserve convex combinations is solved in the case when the Hilbert space which we treat has a finite dimansion and covariance operators commutes.

SUMMARY: We discuss on the classification of Bost–Connes systems. We present that two Bost–Connes C*-algebras for number fields are isomorphic if and only if the original semigroups actions are conjugate. Together with recent reconstruction results in number theory by Cornelissen–de Smit–Li–Marcolli–Smit, we conclude that two Bost–Connes C*-algebras are isomorphic if and only if the original number fields are isomorphic. This is a joint work with Y. Kubota.

SUMMARY: We discuss the Tannaka–Kreĭn duality theorem of Woronowicz from the viewpoint of Q-systems in the sense of Fidaleo–Isola.

SUMMARY: In this talk, we study a unique prime factorization property for tensor product factors with infinitely many tensor components. We provide several examples of type II and III factors which satisfy this property, including all free product factors with diffuse free product components. Our proof is based on Popa’s intertwining techniques and the study of relative amenability on the continuous cores.

SUMMARY: Let a finitely generated amenable group \(G\) and a probability measure \(\mu \) on it (that is finitely-supported, symmetric, and non-degenerate) be given. I will present a construction, via the \(\mu \)-random walk on \(G\), of a harmonic cocycle and the associated orthogonal representation of \(G\). Then I describe when the constructed orthogonal representation contains a non-trivial finite-dimensional subrepresentation (and hence an infinite virtually abelian quotient), and some sufficient conditions for \(G\) to satisfy Shalom’s property \(H_{\mathrm {FD}}\). (joint work with A. Erschler, arXiv:1609.08585)

SUMMARY: We prove some formulas for operator norm inequalities related to operator means. As an example, we can get the following inequality (proved by H. Kosaki in 2014): For any positive integer \(n\geq m\), it holds \[ \frac {1}{2^n} ||| \sum _{i=0}^n {}_{n}C_{i} H^{i/n}XK^{(n-i)/n} ||| \le \frac {1}{2^m} ||| \sum _{j=0}^m {}_{m}C_{j} H^{j/m}XK^{(m-j)/m} |||, \] where \(||| \cdot |||\) is an arbitrary unitarily invariant norm on \(\mathbb {M}_N(\mathbb {C})\), \(H,K,X \in \mathbb {M}_N(\mathbb {C})\) and \(H,K\ge 0\).

SUMMARY: Hopf algebra is one of a important structure to consider objects in the quantum information theory. But the Sweedler’s convention, which is a basic tool of the calculation, is a little complicated for beginners. So, as another tool, we observe the graphical calculus of Hopf algebras. Though it is often used partially, we extend it for instance to the quantum double or universal R-matrix.

SUMMARY: Furuta showed a weighted version of a mixed Schwarz inequality for any square matrices. In this talk, we show the following matrix version based on the Cauchy–Schwarz inequality for matrices: Let \(A\), \(X\) and \(Y\) be matrices in \(\mathbb {M}_{n}\) and \(U\in \mathbb {M}_n\) a unitary matrix in a polar decomposition of \(Y^*AX=U|Y^*AX|\). Then \[ |Y^*AX|\le X^*|A|^{2\alpha }X\# U^*Y^*|A^*|^{2\beta }YU \] holds for all \(\alpha ,\beta \in [0,1]\) with \(\alpha +\beta =1\).

SUMMARY: In this talk, by virtue of the Cauchy–Schwarz operator inequality due to J. I. Fujii, we show a weighted mixed Schwarz operator inequality in terms of the geometric operator mean. As applications, we show Wielandt type operator inequalities via the geometric operator mean.

SUMMARY: Pálfia has been obtained that the Generalized Karcher Equation (GKE) has a unique positive solution, and he obtained a lot of nice properties of it as an operator mean. In this talk, we shall introduce relations among a solution of the Generalized Karcher Equation (GKE), representing function of an operator mean and relative operator entropy, firstly. Next, we shall introduce further extensions of the Ando–Hiai inequality.

SUMMARY: Let \(\alpha \) be in \((0,1)\) and \(r>0\) and \(\#_\alpha \) stand for the weighted operator geometric mean. We consider the following statement: \begin{equation*} A,B>0, \quad A\#_\alpha B\ge I \Rightarrow A^r\#_\alpha B^r\ge I. \end{equation*} Ando and Hiai show that if \(r\ge 1\), then this holds. In the present paper, we prove that the above statement holds only if \(r\ge 1\). We try to find a characterization of the indices \(p,q\in (0,1)\) and \(\mu ,\lambda >0\) satisfying \begin{equation*} A,B>0,\quad A\#_p B \ge I \Rightarrow A^\mu \#_q B^\lambda \ge I . \end{equation*}

SUMMARY: Let \(A\) and \(B\) be bounded positive invertible operators on a Hilbert space \(\mathcal {H}\). For each \(n\in \mathbb {N}\), let \(\Psi ^{[1]}_{A,B}(x,y)\equiv \frac {\Psi _{A,B}(x) - \Psi _{A,B}(y)}{x-y}\) and \(\Psi ^{[n]}_{A,B}(x,y)\equiv \frac {\Psi ^{[n-1]}_{A,B}(x,y) - \Psi ^{[n-1]}_{A,B}(y,y)}{x-y}\) (\(n\geq 2\)), where \(\Psi _{A,B}(t)\equiv A^{\frac {1}{2}} ( A^{-\frac {1}{2}}BA^{-\frac {1}{2}})^t A^{\frac {1}{2}}\). So far, we have introduced the notions of the \(n\)-th operator divergences \(D_{FK}^{[n]}(A|B)\) and \(\Delta _1^{[n]}(x)\) which are generalizations of the Petz–Bregman divergence \(D_{FK}(A|B)\equiv B-A-S(A|B)\) and \(\Delta _1\equiv T_{x}(A|B) - S(A|B)\), respectively. In this talk, we introduce the notions of the \(n\)-th relative operator entropies \(T_{x}^{[n]}(A|B) \equiv \Psi _{A,B}^{[n]}(x,0)\) and \(S^{[n]}(A|B) \equiv \Psi _{A,B}^{[n]}(0,0)\), and show some relations among the \(n\)-th relative operator entropies and the \(n\)-th operator divergences.