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SPHERICAL REPRESENTATIONS FOR $\boldsymbol C^{\boldsymbol {*}}$-FLOWS III: WEIGHT-EXTENDED BRANCHING GRAPHS

Published online by Cambridge University Press:  15 April 2024

YOSHIMICHI UEDA*
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan

Abstract

We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Robert Yuncken

This work was supported by Grant-in-Aid for Scientific Research (B) JP18H01122.

References

Borodin, A. and Olshanski, G., Representations of the Infinite Symmetric Group, Cambridge Studies in Advanced Mathematics, 160 (Cambridge University Press, Cambridge, 2017).Google Scholar
Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics. 1, 2nd edn, Texts and Monographs in Physics (Springer-Verlag, New York, 1987).CrossRefGoogle Scholar
Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics. 2, 2nd edn, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1997).CrossRefGoogle Scholar
Connes, A., ‘Almost periodic states and factors of type III ${}_1$ ’, J. Funct. Anal. 16 (1974), 415445.CrossRefGoogle Scholar
De Commer, K., ‘Algebraic quantum hypergroups imbedded in algebraic quantum groups’, unfinished manuscript. http://kdecommer.u-cergy.fr/Pub.html.Google Scholar
Effros, E. G., Dimensions and ${C}^{\ast }$ -Algebras, CBMS Regional Conference Series in Mathematics, 46 (Conference Board of the Mathematical Sciences, Washington, DC, 1981).Google Scholar
Gorin, V., ‘The $q$ -Gelfand–Tsetlin graph, Gibbs measures and $q$ -Toeplitz matrices’, Adv. Math. 229 (2012), 201266.CrossRefGoogle Scholar
Haagerup, U., ‘The standard form of von Neumann algebras’, Math. Scand. 37 (1975), 271283.CrossRefGoogle Scholar
Kerov, S. V., Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, Translations of Mathematical Monographs, 219 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
Neshveyev, S. and Tuset, L., Compact Quantum Groups and Their Representation Categories, Cours Spécialisés, 20 (Société Mathétique de France, Paris, 2013).Google Scholar
Reich, H., ‘On the $K$ - and $L$ -theory of the algebra of operators affiliated to a finite von Neumann algebra’, K-Theory 24 (2001), 303326.CrossRefGoogle Scholar
Sato, R., ‘Quantized Vershik–Kerov theory and quantized central measures on branching graphs’, J. Funct. Anal. 277 (2019), 25222557.CrossRefGoogle Scholar
Takesaki, M., ‘Duality for crossed products and the structure of von Neumann algebras of type III’, Acta Math. 131 (1973), 249310.CrossRefGoogle Scholar
Takesaki, M., Theory of Operator Algebras I, Encyclopedia of Mathematical Sciences, 124, Operator Algebras and Non-commutative Geometry, 5 (Springer, Berlin, 2002).Google Scholar
Takesaki, M., Theory of Operator Algebras II, Encyclopedia of Mathematical Sciences, 125, Operator Algebras and Non-commutative Geometry, 6 (Springer, Berlin, 2003).CrossRefGoogle Scholar
Tomiyama, J., ‘On the tensor products of von Neumann algebras’, Pacific J. Math. 30 (1969), 263270.CrossRefGoogle Scholar
Ueda, Y., ‘Discrete cores of type III free product factors’, Amer. J. Math. 138 (2016), 367394.CrossRefGoogle Scholar
Ueda, Y., ‘Spherical representations of ${C}^{\ast }$ -flows II: Representation systems and quantum group setup’, SIGMA Symmetry Integrability Geom. Methods Appl. 18 (2022), 050, 43 pages.Google Scholar
Ueda, Y., ‘Spherical representations of ${C}^{\ast }$ -flows I’, Münster J. Math. 16 (2023), 201263.Google Scholar