Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-02-05T18:08:04.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Partial ∗-Automorphisms, Normalizers, and Submodules in Monotone Complete C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Masamichi Hamana
Masamichi Hamana*
Affiliation:
Department of Mathematics, Faculty of Science, University of Toyama, Toyama 930-8555, Japan e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For monotone complete ${{C}^{*}}$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed ${{C}^{*}}$-subalgebra, the relation $X=AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX+XA\subset X$ and $X{{X}^{*}}+{{X}^{*}}X\subset A$ and a set of certain partial isometries $s$ in the “normalizer” of $A$ in $B$, and similarly for the map $s\mapsto \text{Ad }s$ between the latter set and a set of certain “partial $*$-automorphisms” of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type $\text{I}{{\text{I}}_{1}}$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the ${{C}^{*}}$-algebra version of these results.

Keywords

46L0546L0846L4020M18
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 6 , 01 December 2006 , pp. 1144 - 1202
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Berberian, S. K., Baer *-Rings. Grundlehren der Mathematischen Wissenschaften 195, Springer-Verlag, New York, 1972.Google Scholar
[2] Blackadar, B., K-Theory for Operator Algebras. Mathematical Sciences Research Institute Publications 5, Springer-Verlag, New York, 1986.Google Scholar
[3] Brown, L. G., Green, P., and Rieffel, M. A., Stable isomorphism and strong Morita equivalence of C*-algebras. Pacific J. Math. 71(1977), no. 2, 349–363.Google Scholar
[4] Cuntz, J., Simple C*-algebras generated by isometries. Comm. Math. Phys. 57(1977), no. 2, 173–185.Google Scholar
[5] Dixmier, J., Sur certains espaces considérés par M. H. Stone. Summa Brasil. Math. 2(1951), 151–182.Google Scholar
[6] Dixmier, J., Sous-anneaux abélian maximaux dans les factuers de type fini. Ann. of Math. 59(1954), 279–286.Google Scholar
[7] Exel, R., Circle actions on C*-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence. J. Funct. Anal. 122(1994), no. 2, 361–401.Google Scholar
[8] Exel, R., Twisted partial actions: a classification of regular C*-algebra bundles. Proc. London Math. Soc.(3) 74(1997), no. 2, 417–443.Google Scholar
[9] Exel, R., Amenability for Fell bundles. J. Reine Angew. Math. 492(1997), 41–73.Google Scholar
[10] Hamana, M., Tensor products for monotone complete C*-algebras. I. Japan. J. Math. (N.S.) 8(1982), no. 2, 259–283.Google Scholar
[11] Hamana, M., Dynamical systems based on monotone complete C*-algebras. In: Current Topics in Operator Algebras. World Scientific. Publishing, River Edge, NJ, 1991, pp. 282–296.Google Scholar
[12] Hamana, M., Modules over monotone complete C*-algebras. Intern. J. Math. 3(1992), no. 2, 185–204.Google Scholar
[13] Hamana, M., Infinite. σ-finite, non-W*, AW*-factors. Internat. J. Math. 12(2001), no. 1, 81–95.Google Scholar
[14] Hamana, M., Coactions of discrete groups on monotone complete C*-algebras, in preparation.Google Scholar
[15] Hewitt, E. and Ross, K. A., Abstract harmonic analysis. II. Grundlehren der Mathematischen Wissenschaften 152, Springer-Verlag, New York, 1970.Google Scholar
[16] Johnson, B. E., AW*-algebras are QW*-algebras. Pacific J. Math. 23(1967), 97–99.Google Scholar
[17] Kadison, R. V., Operator algebras with a faithful weakly-closed representation. Ann. of Math. 64(1956), 175–181.Google Scholar
[18] Kadison, R V. and Pedersen, G. K., Equivalence in operator algebras. Math. Scand. 27(1970), 205–222.Google Scholar
[19] Kaplansky, I., Projections in Banach algebras. Ann. of Math. 53(1951), 235–249.Google Scholar
[20] Lawson, M. V., Inverse Semigroups. The Theory of Partial Symmetries. World Scientific Publishing, River Edge, NJ, 1998.Google Scholar
[21] Murray, F. J. and von Neumann, J., Rings of operators. IV. Ann. of Math. 44(1943), 716–808.Google Scholar
[22] Nakagami, Y. and Takesaki, M., Duality for crossed products of von Neumann algebras. Lecture Notes in Mathematics 731, Springer-Verlag, Berlin, 1979.Google Scholar
[23] Ozawa, M., Nonuniqueness of the cardinality attached to homogeneous AW*-algebras. Proc. Amer. Math. Soc. 93(1985), no. 4, 681–684.Google Scholar
[24] Paterson, A. L. T., Groupoids, Inverse Semigroups, and Their Operator Algebras. Progress in Math. 170, Birkhäuser Boston, Boston, MA, 1999.Google Scholar
[25] Pedersen, G. K., C*-Algebras and Their Automorphism Groups. London Mathematical Society Monographs 14, Academic Press, London, 1979.Google Scholar
[26] Power, S. C., Limit Algebras: An Introduction to Subalgebras of C*-Algebras. Pitman Research Notes in Mathematics 278, Longman Scientific and Technical, Harlow, 1992.Google Scholar
[27] Reid, G. A., A generalisation ofW*-algebras. Pacific J. Math. 15(1965), 1019–1026.Google Scholar
[28] Rieffel, M. A., Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner. In: Studies in Analysis, Adv. in Math. Suppl. Stud. 4, Academic Press, New York, 1979, pp. 43–82.Google Scholar
[29] Saitō, K. and Wright, J. D. M., All AW*-factors are normal. J. London Math. Soc.(2) 44(1991), no. 1, 143–154.Google Scholar
[30] Sakai, S., C*-algebras and W*-algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete 60, Springer-Verlag, New York, 1971.Google Scholar
[31] Takesaki, M., The structure of a von Neumann algebra with a homogeneous periodic state. Acta Math. 131(1973), 79–121.Google Scholar
[32] Takesaki, M., Theory of operator algebras. I. Springer-Verlag, New York, 1979.Google Scholar
[33] Tomiyama, J., Tensor products and projections of norm one in von Neumann algebras. Lecture Notes, University of Copenhagen, 1970.Google Scholar
[34] Wright, J. D. M., On some problems of Kaplansky in the theory of rings of operators. Math. Z. 172(1980), no. 2, 131–141.Google Scholar
[35] Youngson, M. A., Completely contractive projections on C*-algebras. Quart. J. Math. Oxford 34(1983), 507-511.Google Scholar
[36] Zettl, H. H., A characterization of ternary rings of operators. Adv. in Math. 48(1983), no. 2, 117–143.Google Scholar