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9. Généralisations de la propriété d'approximation de Haagerup pour les algèbres de von Neumann arbitraires

Author

Caspers, Martijn, Okayasu, Rui, Skalski, Adam, and Tomatsu, Reiji

Subjects
Mathematics - Functional Analysis, Mathematics::Operator Algebras, FOS: Mathematics, Mathematics - Operator Algebras, Operator Algebras (math.OA), Functional Analysis (math.FA)
Abstract

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced., La notion de propriété d'approximation de Haagerup, introduite à l'origine pour les algèbres de von Neumann ayant une trace finie, normale, et fidèle, est généralisée pour les algèbres de von Neumann arbitraires. Nous discutons deux caractérisations équivalentes : une du point de vue de la représentation standard et une autre du point de vue des applications linéaires approximantes liées à un poids fidèle, normal, semifini. Quelques propriétés de permanence, en particulier celles concernant les produits croisés, sont établies et certains exemples sont introduits.

Published
2014

12. Free group $C^*$-algebras associated with $\ell_p$

Author

Okayasu, Rui

Subjects
46L05, 22D25, FOS: Mathematics, Mathematics - Operator Algebras, Operator Algebras (math.OA)
Abstract

For every $p\geq 2$, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to the positive linear functionals on the free group $C^*$-algebra associated with the ideal $\ell_p$. This is a generalization of Haagerup's characterization for the case of the reduced free group $C^*$-algebra. As a consequence, the associated $C^*$-algebras are mutually non-isomorphic, and they have a unique tracial state.

Published
2012

14. Free group C*-algebras associated with ℓp.

Author

Okayasu, Rui

Subjects
*FREE groups, *FREE algebras, *GENERATORS of groups, *POSITIVE systems, *MATHEMATICAL mappings, *FINITE groups
Abstract

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state. [ABSTRACT FROM AUTHOR]

Published
2014
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15. Free group C*-algebras associated with ℓp.

Author

Okayasu, Rui

Subjects
*FREE groups, *FINITE fields, *MATHEMATICAL functions, *GENERALIZATION
Abstract

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state. [ABSTRACT FROM AUTHOR]

Published
2014
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16. Free group C*-algebras associated with ℓp.

Author

Okayasu, Rui

Subjects
FREE groups, FINITE fields, MATHEMATICAL functions, GENERALIZATION
Abstract

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state. [ABSTRACT FROM AUTHOR]

Published
2014
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17. RELATIVE ENTROPY FOR ABELIAN SUBALGEBRAS.

Author

OKAYASU, RUI

Subjects
*ALGEBRA, *ABELIAN semigroups, *ENTROPY, *MATHEMATICAL symmetry, *VON Neumann algebras, *HILBERT space, *MATHEMATICS
Abstract

For finite dimensional abelian subalgebras of a finite von Neumann algebra, we obtain the value of conditional relative entropy defined by Choda. We also consider the modified invariant defined by Pimsner and Popa. [ABSTRACT FROM AUTHOR]

Published
2010
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