A Galois Correspondence for Reduced Crossed Products of Simple $\text{C}^{\ast }$-algebras by Discrete Groups

Cameron, Jan ; Smith, Roger R.

Canadian journal of mathematics, Tome 71 (2019) no. 5, pp. 1103-1125

Résumé

Let a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

Détail

DOI : 10.4153/CJM-2018-014-6

Mots-clés : C∗ -algebra, group, crossed product, bimodule, reduced, twisted

Bibliographie

[1] Bédos, E., Discrete groups and simple C∗-algebras . Math. Proc. Cam. Phil. Soc. 109(1991), 521–537. . Google Scholar | DOI

[2] Bonsall, F. F. and Duncan, J., Complete normed algebras . Ergebnisse der Mathematik und ihrer Grenzgebiete, 80, Springer-Verlag, New York-Heidelberg, 1973. Google Scholar

[3] Bryder, R. S. and Kennedy, M., Reduced twisted crossed products over C∗-simple groups . Int. Math. Res. Not. IMRN 2018 no. 6, 1638–1655. . Google Scholar | DOI

[4] Bures, D., Abelian subalgebras of von Neumann algebras . Memoirs of the American Mathematical Society, 110, American Mathematical Society, Providence, RI, 1971. Google Scholar

[5] Cameron, J. M., Hochschild cohomology of II factors with Cartan maximal abelian subalgebras . Proc. Edinb. Math. Soc. 52(2009), no. 2, 287–295. . Google Scholar | DOI

[6] Cameron, J., Pitts, D. R., and Zarikian, V., Bimodules over Cartan MASAs in von Neumann algebras, norming algebras, and Mercer’s theorem . New York J. Math. 19(2013), 455–486. Google Scholar

[7] Cameron, J. and Smith, R. R., Bimodules in crossed products of von Neumann algebras . Adv. Math. 274(2015), 539–561. . Google Scholar | DOI

[8] Cameron, J. and Smith, R. R., Intermediate subalgebras and bimodules for general crossed products of von Neumann algebras . Internat. J. Math. 27(2016), no. 11, 1650091. . Google Scholar | DOI

[9] Choda, H., A Galois correspondence in a von Neumann algebra . Tôhoku Math. J. 30(1978), 491–504. . Google Scholar | DOI

[10] Choda, H., A correspondence between subgroups and subalgebras in a discrete C∗-crossed product . Math. Japon. 24(1979/80), 225–229. Google Scholar

[11] Choi, M. D., Completely positive linear maps on complex matrices . Linear Algebra and Appl. 10(1975), 285–290. . Google Scholar | DOI

[12] Christensen, E. and Sinclair, A. M., Module mappings into von Neumann algebras and injectivity . Proc. London Math. Soc. 71(1995), 618–640. . Google Scholar | DOI

[13] Elliott, G. A., Some simple C∗-algebras constructed as crossed products with discrete outer automorphism groups . Publ. Res. Inst. Math. Sci. 16(1980), 299–311. . Google Scholar | DOI

[14] Glimm, J., A Stone-Weierstrass theorem for C∗-algebras . Ann. of Math. 72(1960), 216–244. . Google Scholar | DOI

[15] Haagerup, U. and Kraus, J., Approximation properties for group C∗-algebras and group von Neumann algebras . Trans. Amer. Math. Soc. 344(1994), 667–699. . Google Scholar | DOI

[16] Haagerup, U. and Zsidó, L., Sur la propriété de Dixmier pour les C∗-algèbres . C. R. Acad. Sci. Paris Sér. I Math. 298(1984), 173–176. Google Scholar

[17] Halpern, H., Essential central spectrum and range for elements of a von Neumann algebra . Pacific J. Math. 43(1972), 349–380. . Google Scholar | DOI

[18] Izumi, M., Inclusions of simple C∗-algebras . J. Reine Angew. Math. 547(2002), 97–138. . Google Scholar | DOI

[19] Izumi, M., Longo, R., and Popa, S., A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras . J. Funct. Anal. 155(1998), 25–63. . Google Scholar | DOI

[20] Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras. Vol. II. Advanced theory . Pure and Applied Mathematics, 100, Academic Press, Inc., Orlando, FL, 1986. . Google Scholar | DOI

[21] Kishimoto, A., Simple crossed products of C∗-algebras by locally compact abelian groups . Yokohama Math. J. 28(1980), 69–85. Google Scholar

[22] Kishimoto, A., Outer automorphisms and reduced crossed products of simple C∗-algebras . Comm. Math. Phys. 81(1981), 429–435. . Google Scholar | DOI

[23] Landstad, M. B., Olesen, D., and Pedersen, G. K., Towards a Galois theory for crossed products of C∗-algebras . Math. Scand. 43(1978), 311–321. . Google Scholar | DOI

[24] Mercer, R., Bimodules over Cartan subalgebras. Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987). Rocky Mountain J. Math. 20 (1990), 487–502. . Google Scholar | DOI

[25] Mercer, R., Isometric isomorphisms of Cartan bimodule algebras . J. Funct. Anal. 101(1991), 10–24. . Google Scholar | DOI

[26] Murray, F. J. and Von Neumann, J., On rings of operators . Ann. Math. 37(1936), 116–229. . Google Scholar | DOI

[27] Olesen, D., Inner ∗-automorphisms of simple C∗-algebras . Comm. Math. Phys. 44(1975), 175–190. . Google Scholar | DOI

[28] Packer, J. A. and Raeburn, I., Twisted crossed products of C∗-algebras . Math. Proc. Cam. Phil. Soc. 106(1989), 293–311. . Google Scholar | DOI

[29] Phillips, J., Outer automorphisms of separable C∗-algebras . J. Funct. Anal 70(1987), 111–116. . Google Scholar | DOI

[30] Pop, F., Sinclair, A. M., and Smith, R. R., Norming C∗-algebras by C∗-subalgebras . J. Funct. Anal. 175(2000), 168–196. . Google Scholar | DOI

[31] Quigg, J. C., Duality for reduced twisted crossed products of C∗-algebras . Indiana Univ. Math. J. 35(1986), 549–571. . Google Scholar | DOI

[32] Sakai, S., Derivations of simple C∗-algebras . J. Funct. Anal. 2(1968), 202–206. . Google Scholar | DOI

[33] Sinclair, A. M. and Smith, R. R., Finite von Neumann algebras and masas . London Math. Soc. Lecture Note Series, 351, Cambridge Univ. Press, Cambridge, 2008. . Google Scholar | DOI

[34] Strǎtilǎ, S., Central spectral theory in W∗-algebras, and applications . (Romanian) Stud. Cerc. Mat. 25(1973), 1167–1259. Google Scholar

[35] Strǎtilǎ, S. and Zsidó, L., An algebraic reduction theory for W∗-algebras. II . Rev. Roumaine Math. Pures Appl. 18(1973), 407–460. Google Scholar

[36] Sutherland, C. E., Cohomology and extensions of von Neumann algebras I . Publ. Res. Inst. Math. Sci. 16(1980), 105–133. . Google Scholar | DOI

[37] Takesaki, M., Theory of operator algebras. I . Springer-Verlag, New York-Heidelberg, 1979. Google Scholar

[38] Van Daele, A., Continuous crossed products and type III von Neumann algebras . London Math. Soc. Lecture Note Series, 31, Cambridge Univ. Press, Cambridge, 1978. Google Scholar

[39] Wright, F. B., A reduction for algebras of finite type . Ann. of Math. 60(1954), 560–570. . Google Scholar | DOI