Organization
Graduate School of Mathematics Division of Mathematics Computational Mathematics Professor
Graduate School
Graduate School of Mathematics
Undergraduate School
School of Science Department of Mathematics
Title
Professor
UEDA Yoshimichi
Degree 1
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博士(数理学) （ 1999.9 九州大学 ）
Research Interests 2
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Theory of operator algebras
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Free probability theory
Research Areas 1
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Natural Science / Mathematical analysis / Operator algebras
Current Research Project and SDGs 1
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Noncommutative Analysis
Research History 4
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Nagoya University Professor
2017.10
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Faculty of Mathematics, Kyushu University Associate professor
2002.4 - 2017.9
Country：Japan
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Graduate School of Science, Hiroshima University Assistant
1999.4 - 2002.3
Country：Japan
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Hiroshima University Graduate School of Science Assistant
1999.4 - 2002.3
Education 1
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Kyushu University Graduate School of Maehtmatics
1996.4 - 1999.3
Country： Japan
Papers 43
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Spherical representations of C*-flows, I Reviewed
Yoshimichi Ueda
Münster Journal of Mathematics Vol. 16 ( 1 ) page： 201 - 263 2023.3
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Pusz–Woronowicz Functional Calculus and Extended Operator Convex Perspectives Reviewed
Fumio Hiai, Yoshimichi Ueda, Shuhei Wada
Integral Equations and Operator Theory Vol. 94 ( 1 ) 2022.3
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Spherical Representations of C*-Flows II: Representation System and Quantum Group Setup Reviewed
Yoshimichi Ueda
Symmetry, Integrability and Geometry: Methods and Applications Vol. 18 ( 050 ) 2022
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Matrix Liberation Process II: Relation to Orbital Free Entropy Reviewed
Yoshimichi Ueda
Canadian Journal of Mathematics Vol. 73 ( 2 ) page： 493 - 541 2021.4
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Pusz--Woronowicz's functional calculus revisited Reviewed
Kanae Hatano and Yoshimichi Ueda
Acta Scientiarum Mathematicarum Vol. 87 ( 3-4 ) page： 485 - 503 2021
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Matrix Liberation Process I: Large Deviation Upper Bound and Almost Sure Convergence Reviewed
Ueda Yoshimichi
JOURNAL OF THEORETICAL PROBABILITY Vol. 32 ( 2 ) page： 806 - 847 2019.6
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A free product pair rigidity result in von Neumann algebras Reviewed
Ueda Yoshimichi
JOURNAL OF NONCOMMUTATIVE GEOMETRY Vol. 13 ( 2 ) page： 587 - 607 2019
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FREE PRODUCTS IN AQFT Reviewed
Longo Roberto, Tanimoto Yoh, Ueda Yoshimichi
ANNALES DE L INSTITUT FOURIER Vol. 69 ( 3 ) page： 1229 - 1258 2019
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On Arveson's boundary theorem Reviewed
Kei Hasegawa and Yoshimichi Ueda
Mathematical Proceedings of the Royal Irish Academy Vol. 119A ( 1 ) page： 1 - 1 2019
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A remark on orbital free entropy Reviewed
Yoshimichi Ueda
ARCHIV DER MATHEMATIK Vol. 108 ( 6 ) page： 629 - 638 2017.6
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Rigidity of free product von Neumann algebras Reviewed
Cyril Houdayer, Yoshimichi Ueda
COMPOSITIO MATHEMATICA Vol. 152 ( 12 ) page： 2461 - 2492 2016.12
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Asymptotic structure of free product von Neumann algebras Reviewed
Cyril Houdayer, Yoshimichi Ueda
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY Vol. 161 ( 3 ) page： 489 - 516 2016.11
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A characterization of the fullness of continuous cores of type III1 free product factors Reviewed
Reiji Tomatsu, Yoshimichi Ueda
KYOTO JOURNAL OF MATHEMATICS Vol. 56 ( 3 ) page： 599 - 610 2016.9
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DISCRETE CORES OF TYPE III FREE PRODUCT FACTORS Reviewed
Yoshimichi Ueda
AMERICAN JOURNAL OF MATHEMATICS Vol. 138 ( 2 ) page： 367 - 394 2016.4
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REMARKS ON FREE MUTUAL INFORMATION AND ORBITAL FREE ENTROPY Reviewed
Masaki Izumi, Yoshimichi Ueda
NAGOYA MATHEMATICAL JOURNAL Vol. 220 page： 45 - 66 2015.12
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Orbital Free Pressure and Its Legendre Transform Reviewed
Fumio Hiai, Yoshimichi Ueda
COMMUNICATIONS IN MATHEMATICAL PHYSICS Vol. 334 ( 1 ) page： 275 - 300 2015.2
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Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras Reviewed
Yoshimichi Ueda
PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES Vol. 90 ( 10 ) page： 151 - 155 2014.12
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On the geometry of von Neumann algebra preduals Reviewed
Miguel Martin, Yoshimichi Ueda
POSITIVITY Vol. 18 ( 3 ) page： 519 - 530 2014.9
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Orbital Free Entropy, Revisited Reviewed
Yoshimichi Ueda
INDIANA UNIVERSITY MATHEMATICS JOURNAL Vol. 63 ( 2 ) page： 551 - 577 2014
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Some analysis of amalgamated free products of von Neumann algebras in the non-tracial setting Reviewed
Yoshimichi Ueda
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES Vol. 88 page： 25 - 48 2013.8
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Factoriality, type classification and fullness for free product von Neumann algebras Reviewed
Yoshimichi Ueda
ADVANCES IN MATHEMATICS Vol. 228 ( 5 ) page： 2647 - 2671 2011.12
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On the predual of non-commutative H-infinity Reviewed
Yoshimichi Ueda
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY Vol. 43 page： 886 - 896 2011.10
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On type III1 factors arising as free products Reviewed
Yoshimichi Ueda
Mathematical Research Letters Vol. 18 ( 5 ) page： 909 - 920 2011
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A log-Sobolev type inequality for free entropy of two projections Reviewed
Fumio Hiai, Yoshimichi Ueda
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES Vol. 45 ( 1 ) page： 239 - 249 2009.2
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ORBITAL APPROACH TO MICROSTATE FREE ENTROPY Reviewed
Fumio Hiai, Takuho Miyamoto, Yoshimichi Ueda
INTERNATIONAL JOURNAL OF MATHEMATICS Vol. 20 ( 2 ) page： 227 - 273 2009.2
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On peak phenomena for non-commutative H infinity Reviewed
Yoshimichi Ueda
MATHEMATISCHE ANNALEN Vol. 343 ( 2 ) page： 421 - 429 2009.2
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Remarks on HNN extensions in operator algebras Reviewed
Ueda Yoshimichi
Illinois Journal of Mathematics Vol. 52 ( 3 ) page： 705 - 725 2008.12
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Notes on microstate free entropy of projections Reviewed
Fumio Hiai, Yoshimichi Ueda
Publications of the Research Institute for Mathematical Sciences Vol. 44 ( 1 ) page： 49 - 89 2008
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A free logarithmic Sobolev inequality on the circle Reviewed
Fumio Hiai, Denes Petz, Yoshimichi Ueda
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES Vol. 49 ( 3 ) page： 389 - 406 2006.9
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Free transportation cost inequalities for noncommutative multi-variables Reviewed
Fumio Hiai, Yoshimichi Ueda
INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS Vol. 9 ( 3 ) page： 391 - 412 2006.9
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Notes on treeability and costs for discrete groupoids in operator algebra framework Reviewed
Yoshimichi Ueda
Operator Algebras Vol. 1 page： 259 - 279 2006
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HNN extensions of von Neumann algebras Reviewed
Y Ueda
JOURNAL OF FUNCTIONAL ANALYSIS Vol. 225 ( 2 ) page： 383 - 426 2005.8
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Free transportation cost inequalities via random matrix approximation Reviewed
F Hiai, D Petz, Y Ueda
PROBABILITY THEORY AND RELATED FIELDS Vol. 130 ( 2 ) page： 199 - 221 2004.10
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Amalgamated free product over Cartan subalgebra, II Supplementary Results & Examples Invited Reviewed
Yoshimichi Ueda
Operator Algebras and Applications 2004
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Automorphisms of free product-type and their crossed-products Reviewed
Fumio Hiai, Yoshimichi Ueda, Yoshimichi Ueda
Journal of Operator Theory Vol. 50 page： 119 - 130 2003.6
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FREE PRODUCT ACTIONS AND THEIR APPLICATIONS Invited Reviewed
YOSHIMICHI UEDA
Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads 2003.1
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Y Ueda
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 355 ( 1 ) page： 349 - 371 2003
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Irreducible subfactors of L(double-struck F sign<inf>∞</inf>) of index λ > 4 Reviewed
Dimitri Shlyakhtenko, Yoshimichi Ueda
Journal fur die Reine und Angewandte Mathematik page： 149 - 166 2002.12
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Remarks on free products with respect to non-tracial states Reviewed
Y Ueda
MATHEMATICA SCANDINAVICA Vol. 88 ( 1 ) page： 111 - 125 2001
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A relation between certain interpolated Cuntz algebras and interpolated free group factors Reviewed
Y Ueda, Y Watatani
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 128 ( 5 ) page： 1397 - 1404 2000
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On the fixed-point algebra under a minimal free product-type action of the quantum group SUq (2) Reviewed
Y Ueda
INTERNATIONAL MATHEMATICS RESEARCH NOTICES Vol. 2000 ( 1 ) page： 35 - 56 2000
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A minimal action of the compact quantum group SUq(n) on a full factor Reviewed
Yoshimichi Ueda
Journal of the Mathematical Society of Japan Vol. 51 ( 2 ) page： 449 - 461 1999
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Amalgamated free product over Cartan subalgebra Reviewed
Yoshimichi Ueda
Pacific Journal of Mathematics Vol. 191 ( 2 ) page： 359 - 392 1999
MISC 12
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行列解析：函数解析・作用素環論の立場から Invited
植田 好道
数理科学 ( 60 ) 2022.11
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植田好道
数理科学 Vol. 55 ( 10 ) page： 22‐28 2017.10
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書評 Alexandru Nica and Roland Speicher : Lectures on the Combinatorics of Free Probability
植田 好道
数学 Vol. 69 ( 3 ) page： 329 - 333 2017
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測度とは何か : それは数学の世界の精巧な測定器 (特集 測度とは何か)
植田 好道
数学セミナー Vol. 55 ( 10 ) page： 8 - 13 2016.10
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ルベーグ積分は難しいか? (特集 初学者を悩ます数理の概念--理解のためのヒントを探る)
植田 好道
数理科学 Vol. 49 ( 5 ) page： 54 - 59 2011.5
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QUICK REVIEW ON PROPERTY (X) (Recent results of Banach and Function spaces and its applications)
UEDA YOSHIMICHI
RIMS Kokyuroku Vol. 1615 page： 28 - 33 2008.10
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ON ORBITAL FREE ENTROPY DIMENSION (New development of Operator Algebras)
UEDA YOSHIMICHI
RIMS Kokyuroku Vol. 1587 page： 116 - 124 2008.4
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フォンノイマン環の最近の発展についての紹介 (作用素環論における最近の分類問題について)
植田 好道
数理解析研究所講究録 Vol. 1435 page： 31 - 38 2005.5
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Hiai Fumio, Ueda Yoshimichi
RIMS Kokyuroku Vol. 1354 page： 10 - 19 2004.1
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Ueda Yoshimichi
RIMS Kokyuroku Vol. 1177 page： 40 - 43 2000.11
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TYPE $III$ FACTORS ARISING AS AMALGAMATED FREE PRODUCTS (Recent Topics in Operator Algebras)
Ueda Yoshimichi
RIMS Kokyuroku Vol. 1077 page： 79 - 87 1999.2
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Q$_\beta$の端数$\beta$とL(F$_r$)の端数r(作用素環における双加群と量子群の研究)
綿谷 安男, 植田 好道
数理解析研究所講究録 Vol. 1003 page： 66-80 1997.6
Presentations 4
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Yoshimichi Ueda
Interactions between Noncommutative Analysis and Quantum Information Theory 2019.6.1
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Matrix liberation process Invited
Yoshimichi Ueda
Workshop 1: Free probability - the theory, its extensions 2019.3.8 Centre de Recherches Mathematiques
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Matrix liberation process and a free probability question Invited
Yoshimichi Ueda
Workshop “Random Matrices and their applications” 2018.5.22
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Matrix liberation process Invited International conference
Yoshimichi Ueda
Workshop "Probabilistic techniques and Quantum Information Theory” 2017.10.24 Henri Poincare Institute
KAKENHI (Grants-in-Aid for Scientific Research) 19
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Research on comlex dynamical systems and C*-algebras
Grant number：25287019 2013.4 - 2018.3
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Watatani Yasuo, Matsui Taku, Ueda Yoshimichi, Masuda Toshihiko, Kajiwara Tsuyoshi, Matsumoto Kengo
Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid)
In this research, we associate C*-algebras with complex dynamical systems of iterations of rational functions R using Hilbert bimodule, and we have found interesting relations between operator algebras and complex dynamical systems. We can recover some properties of complex dynamical systems of R from the associated C*-algebras with the gauge action. We explain the structure of the singularity of branched points in terms of K-theory and KMS states for gauge actions. We determine the extreme points of the traces on the cores of the C*-algebras associated with self-similar maps. We find their ideal structure. We showed that the dimension group of the tent map is the free abelian infinite group and the shift on it.
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A new development of operator algebras and mathematical physics
Grant number：23244016 2011.4 - 2016.3
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
Kawahigashi Yasuyuki, IZUMI Masaki, OZAWA Narutaka, MATUI Hiroki, UEDA Yoshimichi
Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid)
Vertex operator algebras and local conformal nets (of operator algebras) are two mathematical frameworks to study chiral conformal field theory. Many analogies between the two have been known, but no directi relations were known so far. With Carpi, Longo and Weiner, we have proved that we can construct a local conformal net from a strongly local vertex operator algebra and we can also recover the original vertex operator algebra from this resulting local conformal net. We have further given a simple sufficient condition for strong locality. This gives a solution to an important problem studied over more than ten years. We have also shown that the automorpshim group of a vertex operator algebra and that of the corresponding local conformal net coincide. This result applies to many examples including the Moonshine vertex operator algebra. (The case of the Moonshine vertex operator algebra was known before.)
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A csomprehensive study of symmetries of operator algebras
Grant number：22340032 2010.4 - 2015.3
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
IZUMI Masaki, KAWAHIGASHI Yasuyuki, UEDA Yoshimichi, MATUI Hiroki, OZAWA Narutaka, OKAYASU Rui, TOMATSU Reiji, KIDA Yoshikata, YAMAGAMI Shigeru
Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid)
I studied the structure of symmetries of operator algebras. With Hiroki Matui, we studied discrete group actions on C*-algebras, and partially obtained a classification invariant by using topological properties of discrete groups and the automorphism groups of C*-algebras.
With Vaughan Jones, Scott Morrison, David Penneys, Emily Peters, Noah Snyder, we completely classified subfactors of index less than or equal to 5. With Pinhas Grossman and Noah Snyder, we gave detailed description of the structure of the Morita equivalence class of the fusion categories arising from the Asaeda-Haagerup subfactors. -
Study of inequalities and variational principle in free probability theory
Grant number：21540208 2009 - 2012
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
HIAI Fumio, UEDA Yoshimichi
Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid)
In the course of this research project we obtained various results in the three subjects of free probability, quantum probability and operator/matrix analysis. We studied free probability theory related to free entropy and free Fisher information. In quantum probability and quantum information, our study was directed to quantum relative entropy, quantum hypothesis testing and monotone Riemannian metrics. Several results in matrix analysis were also obtained concerning operator monotone and convex functions, matrix trace and norm functions, operator means and symmetric (anti) norms, etc.
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Synthetic Studies on Operator Algebras and Mathematical Physics
Grant number：19204015 2007 - 2010
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
KAWAHIGASHI Yasuyuki, OZAWA Narutaka, IZUMI Masaki, KISHIMOTO Akitaka, HIAI Fumio, KOSAKI Hideki, MATSUI Taku, UEDA Yoshimichi, MASUDA Toshihiko, MATUI Hiroki
Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid)
It has been known that when we pursue the fundamental problems in physics such as the origin of the universe, spacetime and materials, then we need the highest level of modern mathematics. These mathematical problems are also highly interesting from a purely mathematical viewpoint. I study conformal field theory among such problems using operator algebras. I have obtained results on classification problems, construction of examples and relations to other mathematical fields.
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Study on Complex dynamical systems and operator algebras
Grant number：19340040 2007 - 2010
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
WATATANI Yasuo, KOSAKI Hideki, MATSUI Taku, KAJIWARA Tsuyoshi, UEDA Yoshimichi, MASUDA Toshihiko
Authorship：Coinvestigator(s)
In the research we constructed C*-algebras of operators on Hilbert spaces for complex dynamical systems on a complex plane by the iteration of meromorphic functions like rational functions and transcendental entire functions. We found unexpected relations between complex dynamical systems and operator algebras. Self-similarity of Julia sets of complex dynamical systems corresponds to purely infinite simplicity of the associated C*-algebras. The branch structure of complex dynamical systems is related with the structure of KMS states (equilibrium states) for the gauge actions on the associated C*-algebras. We showed that we can recover the degree of the rational function, the number of branched points and the number of exceptional points.
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Study on operator theory and operator means
Grant number：19340035 2007 - 2010
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
KOSAKI Hideki, WATATANI Yasuo
Authorship：Coinvestigator(s)
Operator means can be constructed from certain scalar means in a natural fashion, and comparison of unitarily invariant norms of such operator means requires positive definiteness of ratios of corresponding scalar means. Positive definiteness for many such ratios was determined, and consequently many very precise norm inequalities (such as an abundance of generalizations of the Heinz inequality and monotonicity in a parameter of norms of operator binomial means) were obtained. Moreover, many such positive definite functions arising in this way were shown to be infinitely divisible. Also, Jensen-type majorization results and the equality condition for the trace Jensen inequality were proved in the setting of von Neumann algebras.
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Study of free probability and operator algebras based on random matrices
Grant number：17340043 2005 - 2008
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
HIAI Fumio, OBATA Nobuaki, URAKAWA Hajime, OZAWA Masanao, KOSAKI Hideki, YAMAGAMI Shigeru, HORA Akihito, UEDA Yoshimichi, NAKAMURA Makoto
Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid)
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Functional Analysis of Quantum Systems with Infinite Degree ofFir Freedom
Grant number：16340050 2004 - 2007
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
MATSUI Taku, TASAKI Shuichi, HIROKAWA Masako, NAKAYASHIKI Atsushi, UEDA Yoshimichi, HIROSHIMA Fumio
Authorship：Coinvestigator(s)
(1) We proved the Haag duality for half-infinite regions in pure states of 1-dim quantum spin chain.
As a corollary, we see that translationally invariant pure ground states of 1-dim finite range Hamiltonians have the split property.
(2)A translationally invariant pure state of a 1-dim quantum spin chain contains one copy of infinite entanglement if and only if the state does not have the split property provided that Alice and Bob systems are half-infinite regions of the 1-dim lattice.
We also considered the case when the state is not pure but factor. In such cases, the state contains one copy of infinite entanglement if and only if the Bell's constant attains its maximal value for any normal states relative to the given one. In particular, the Gibbs states for short range interactions cannot contain one copy of infinite entanglement because the Bell's constant takes its minimum value 1. New examples of states which are not pure can be provided by non-equilibrium steady states(NESS).
(3)We consider an analogue-of NESS for Bose systems with condensation. We found that Josephson current is non-vanishing for coupled Bosons with the same temperature.
(4)We studied fluctuation property of entropy production of NESS in the sense of D.Ruelle and obtained the law of large number under the assumption that the time evolution of the system has certain mixing property(Asymptotic Abelian).
We also investigated an analogue of the central limit theorem for entropy production. For some cases, the limit is not Gaussian. -
Study of noncommutative analysis and free probability theory on operator algebras
Grant number：14540198 2002 - 2003
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
HIAI Fumio, YAMAGAMI Shigeru, OBATA Nobuaki, URAKAWA Hajime, UEDA Yoshimnichi
Authorship：Coinvestigator(s)
We studied operator theory and operator algebra theory on Hubert spaces with applications to noncommutative probability theory. In recent years, free probability theory initiated by D.Voiculescu has been extensively developed as a branch of noncommutative probability theory. We investigated large deviation theory for random matrices and free relative entropy related to free probability theory. We investigated automorphsims on (interpolated) free group factors by the method of free products. Moreover, we studied norm inequalities for means of operators and obtained a number of inequalities extending the arithmetic-geometric norm inequalities for operators. More concretely,
(1)We obtained a new norm inequality for matrices involving operator monotone functions and unitarily invariant norms.
(2)We investigated free relative entropy extending free entropy into two variables We gave its two definitions by double logarithmic integral and by matrix approximation, and proved their coincidence by making use of large deviation for the empirical eigenvalue distribution of relevant random matrices.
(3)We showed that there are continuously many different aperiodic automorphisms on each (interpolated) free group factor by the method of free products and clarified the structure of their crossed-products.
(4)Based on theory of double integral transformation and the method of Schur multipliers, we obtained norm inequalities for unitarily invariant nroms of operators, which strengthen the arithmetic-geometric mean inequality.
(5)We gave an inequality for positive semidefinite matrices, comparing submultiplicativity with subadditivity for unitarily invariant norms.
(6)We re-proved the free probabilistic analog of Talagrand transportation cost inequality for measures due to Biane and Voiculescu by means of random matrix approximation. -
Deformation of 2-dimensional diffusion processes which preserves recurrence
Grant number：12640127 2000 - 2002
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
IWATA Koichiro, TAKEDA Masayoshi, KUBO Izumi
Authorship：Coinvestigator(s)
The uniformization theorem foe Riemann surfaces claims that complex structures on non-compact simply connected surfaces are equivalent to either the standard complex structure on the complex plain or the one on the unit disk. In terms of Brownian motions this classification is completely described by the recurrence of the process on the domain. On the complex plain the L^∞-norm of Beltrami-coefficients must be 1 as for those complex structures not equivalent to the standard one, in other words, if the corresponding Brownian motion is transient then the L^∞-norm of Beltrami-coefficient is 1. However this does not mean that every complex structure on the complex plain whose Beltrami-coefficient has unit L^∞-norm corresponds to transient Brownian motions. The aim of the present research project is to study such complex structures. One can construct a one-parameter family of complex structures with critical point and at the critical point the recurrence and the transience switch to the other. The model of the complex structures discussed in the present research is parameterized by the closed unit disk and the modulus of the parameter describes the L^∞-norm of the corresponding Beltrami-coefficient. When the parameter lies in the open unit disk the prescribed complex structure is equivalent to the standard one. On the boundary of the unit disk except for one point the complex structure is equivalent to the standard one on the unit disk and thus the Brownian motion is transient. At the critical point the complex structure is recurrent but the freedom of deformation that preserves recurrence is relatively low. A natural direction for further direction is as follows : Study the action of the modular group on the space of Beltrami-coefficients and the action of the modular group on the boundary.
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Systematic study of quantum groups from the viewpoint of operator algebras
Grant number：12640199 2000 - 2002
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
YAMANOUCHI Takehiko, UEDA Yoshimichi, ARAI Asao, KISHIMOTO Akitaka
Authorship：Coinvestigator(s)
Yamanouchi has made an intensive study of actions of locally compact quantum groups on von Neumann algebras. He introduced a notion of a Galois group for such an action. He has succeeded in topologically identifying the Galois group of an intergrable, minimal action of a general locally compact quantum group as the intrinsic group of the dual quantum group.
Kishimoto has made a very unique research on automorphisms or one-parameter automorphism groups on nontype I separable C^*-algebras. He showed that, for any irreducible representation of such a C^*-algebra without nonzero intersection with compact operators, there exists an approximately inner one-parameter automorphism group so that the representation is covariant with respect to this one-parameter automorphism group.
Arai has made a progress in his research in mathematical physics. He considered a model of quantum particles coupled to a massless quantum scalar field, and showed that the model has a ground state for all values of the coupling constant.
Ueda has made an anlysis on amalgamated free products of von Neumann algebras over Cartan subalgebras. By utilizing an irreducible inclusion of type III factors coming from his free-product type action of the quantum SU_q(2), he has been ble to solve the long-standing problem that the free group factor of Radulescu possesses irreducible subfactors of arbitrary index greater than 4. -
Noncommutative analysis based on operator algebras
Grant number：18H01122 2018.4 - 2023.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B) Grant-in-Aid for Scientific Research (B)
Authorship：Principal investigator
Grant amount：\10140000 （ Direct Cost: \7800000 、 Indirect Cost：\2340000 ）
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An approach to free independence based on mutual information
Grant number：16K13762 2016.4 - 2019.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research Grant-in-Aid for Challenging Exploratory Research
UEDA Yoshimichi
Authorship：Principal investigator
Grant amount：\2990000 （ Direct Cost: \2300000 、 Indirect Cost：\690000 ）
This research aimed at developing a theory of mutual information in free probability theory to understand the so-called free independence, the central concept of free probability theory. Two kinds of free probabilistic mutual information have been known so far; one is due to Voiculescu in 1999 and the other is due to Hiai, Miyamoto and myself in 2009. However, only a little about them, including their relationship, is known. In this research program, I first wrote a short note indicating an issue that I think is of importance, and then studied the question of large deviations for the matrix liberation process that I introduced as a key to develop the theory of free probabilistic mutual information.
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Study on von Neumann algebras, free probability and non-commutative function spaces
Grant number：24540214 2012.4 - 2016.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
UEDA Yoshimichi
Authorship：Principal investigator
Grant amount：\5200000 （ Direct Cost: \4000000 、 Indirect Cost：\1200000 ）
I clarified the structure of free product von Neumann algebras from various viewpoints. Especially, I identified the resulting free product von Neumann algebras with a certain explicit one under some conditions. Also, we computed several invariants for free product von Neumann algebras in the general setup, and obtained a result on the positional rigidity for subalgebras as well as a uniqueness result for the free product decomposition. I studied the orbital free entropy. In particular, we made some nontrivial computation of it. We also introduced the concept of pressure related to the orbital free entropy motivated by statistical physics and investigated its relation to random matrices. Moreover, I generalized the previous definition of orbital free entropy and discussed its general properties. Finally, we studied von Neumann algebra preduals and investigated a strong property, which is rather opposite to uniform convexity, for them.
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Study on free probability and operator algebras
Grant number：20540213 2008 - 2011
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
UEDA Yoshimichi
Authorship：Principal investigator
Grant amount：\4290000 （ Direct Cost: \3300000 、 Indirect Cost：\990000 ）
I studied operator algebras and related topics in relation to free probability theory. More precisely, I obtained the following results :(1) I gave a necessary and sufficient condition for an arbitrary free product von Neumann algebra to be a factor. More strongly, I described its central decomposition explicitly, gave an explicit description of the type I part, proved that the non-type I part always appears and becomes a type II1 or III factor, and gave an explicit algorithm for determining the type of that factor. Moreover I proved that that factor is always full.(2) I computed the Sd-andτ-invariants of any type III1 factor arising as the non-type I part of a free product von Neumann algebra.(3) I proved a very special property for weakly compact subsets in the preduals of non-commutative bounded Hardy spaces.(4) I studied the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras, and gave several partial answers to those questions.
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自由確率論と非従順フォン・ノイマン環の研究
Grant number：17740096 2005 - 2007
日本学術振興会 科学研究費助成事業 若手研究(B) 若手研究(B)
植田 好道
Authorship：Principal investigator
Grant amount：\3700000 （ Direct Cost: \3700000 ）
昨年度プレプリントアーカイブに公表した自由確率論における相互情報量の研究でやり残した問題を解決すべく努力した.具体的には相互情報量の自由確率論版に基づいて定義したエントロピー次元と元々の自由エントロピーに基づいて定義されたエントロピー次元の間に期待される恒等式を証明することに力を注いだ.紆余曲折はしたが最終的に期待通りの恒等式を証明することに成功し,紆余曲折した過程で得た発見も合わせて自由エントロピー次元に対する下半連続性を弱い形ながら証明することに成功した.この下半連続性の問題は有名な自由群因子環の同型問題に密接に関わっており多くの専門家に注目されていた問題ではあるが,以前に知られた肯定的な結果は1変数(=可換の場合)の場合だけであり,我々の得た結果は弱い形でかつ上述の同型問題を解くには不十分なものではあるが本質的に非可換な初めての肯定的結果である.これらの成果を既にプレプリントアーカイブにおいて発表済みでかつ投稿中であった論文に新たに加え大幅に書き直すことができた.書き直して再投稿の後,すぐに論文はアクセプトされた.
これまでとは本質的に違う方向の研究として非可換ハーディー空間の研究を行った.ハーディー空間に対する興味は元々本研究課題の柱である自由確率論の研究から来ているが,問題としては完全に独立である.具体的な研究成果としては円盤上で有界正則関数のなすハナッハ環が一意的な前共役をもつという安藤の定理の非可換版を証明した.これは特別な場合として安藤の定理はもちろん,安藤の定理の一般化として知られるいくつかの抽象ハーディー空間に対する結果をすべて含む.また,バナッハ空間論の興味からは関数空間に対するグロタンディエクの結果,フォンノイマン環に対するディクシミェ,境の結果,そして安藤の結果に続く新たな一意的な前共役をもつクラスの提出という観点から興味が持たれることと思う.この結果は極最近プレプリントアーカイブに発表することで公表済みではあるがまだ投稿はしていない. -
フォン・ノイマン環及びエルゴード変換群に関する非アメナブル現象の研究
Grant number：14740118 2002 - 2004
日本学術振興会 科学研究費助成事業 若手研究(B) 若手研究(B)
植田 好道
Authorship：Principal investigator
Grant amount：\4000000 （ Direct Cost: \4000000 ）
本研究課題の最終年度である今年度はDamien Gaboriau氏のエルゴード変換群に対する新しい不変量「コスト」に対する最近の仕事の作用素環論での枠組みでの理解、及びその手法に基づく可測離散擬群までの拡張を与えることに少々の時間を費やした。当初の見込み通りこの試みは概ね成功し、秋に東京大学大学院数理科学研究科にて行った集中講義でその概略を解説した。また、この試みを発表することはGaboriau氏の仕事に関係するいくつかの話題に新しい視点を提示するものとして重要であると考えられることから論文を現在準備中です。
一方、昨年論文にまとめた作用素環に対するHNN構成法の研究を進めた。今年度問題にしたのは自己準同型から生じるHNN構成で、これはもとの群論ではascending HNN構成と呼ばれるものです。具体的には、ある種の場合、一般的なHNN構成とは大きく異なって従順性を持ったものが生じ易いと想像出来るのですが、一般的な条件の下でascending HNN構成から生じるフォン・ノイマン環が適当な部分環に対する相対従順性をもつことを示すことにより、このことを明らかにしました。また、1、2年程前にMonod-Popa及びPestovにより独立に、群の相対従順性についてのある問題に対してやや異なる反例が与えられていたのですが、今回の結論と少しの計算によりこれらの反例の間の具体的な関係を明らかにすることが出来、実際には本質的に同じ例を与えていることが説明出来ます。この成果は、9月にオスロで行われたアーベルシンポジウムでの講演で発表しました。この内容はまだ更に研究が続いているのもあって論文の形では発表はしていません。
また、自由確率論関係では自由エントロピー理論の研究を前年に引き続き行いました。多くの部分は東北大学の日合文雄氏との共同研究の中で行いました。成果の一部は日合氏との共著論文の形で発表しました。 -
自由確率論の視点からのフォンノイマン環の研究
Grant number：12740108 2000 - 2001
日本学術振興会 科学研究費助成事業 奨励研究(A) 奨励研究(A)
植田 好道
Authorship：Principal investigator
Grant amount：\2200000 （ Direct Cost: \2200000 ）
1.ここ数年来行ってきた量子群作用の仕事で導入した自由積型自己同型のアイディアを発展させることを考えた。具体的には、自由群因子環上の自己同型で種々の特殊な性質をもつものの構成に応用した。結論の一つとして、接合積をとると区別できなくなるような外部同値ではない外的自己同型を非可算無限個構成した。これは分類可能な因子環で起こる現象と全く異なる今回初めて示されたものである。この結果は日合文雄氏との共著論文の一部として発表した。
2.自由確率論の応用として、以前に行われた様々な散発的結果の再考察を行った。具体的には,Haagerupにより導入された近似性質をある種の融合積に対して調べた。少し弱めた相対版と言うべき性質はいつでも成り立つが、そのものは成り立ったり成り立たなかったりすることを種々の具体例を構成することにより示した。一方、ある種のエルゴード理論に関連した融合積の自己同型群を調べた。完全に一般の自己同型を調べることは望むべくもないが、ある種の性質を満たす自己同型のなす部分群ならエルゴード理論との関連で詳しく調べることができることを確認した。
3.真に非可換なエルゴード理論を探究した。エルゴード理論の本質的なアイディアの一つとして写像の局所的な切り張りによる議論があるが、非可換測度空間である作用素環で同じことを実行しようとすると本質的な困難がある。しかし、ごく最近発表された仕事の中にかなり限定された状況下ではあるが、非可換な場合での切り張り議論に相当すると考えられなくもないものを見い出した。これは,研究目的に本質的に関わるものであり,今後もこの方向の研究を継続したい。