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C*-algebra の例

C * -algebra としてまず なのは , なもの , つまり (locally) compact Hausdorff C * -algebra である

なものは , および から される まず (locally compact) groupoid などから られるものがある

  • group C * -algebra
  • reduced group C * -algebra
  • groupoid C * -algebra
  • reduced groupoid C * -algebra

Groupoid C * -algebra については , Renault [ Ren80 ] Paterson [ Pat99 ] がある Debord Lescure [ DL10 ] にまとま ている

Mesland [ Mes11 ] によると groupoid C * -algebra , group C * -algebra 以下 C * -algebra

Paterson には , inverse semigroup C * -algebra についても しく いてある semigroup についても Li [ Li12 ] している

  • semigroup C * -algebra

Cuntz Echterhoff Li [ CEL15 ] , その K -theory している Li [ Li14 Li16 ] , semigroup C * -algebra K -theory しようとして いる

Cuntz Li , から C * -algebra ることも えている [ Cun08 CL10 Li10 CL11 ] などである

  • ring C * -algebra

また , Cuntz [ CDL13 ] , から C * -algebra ることも えて いる

K 0 ring of algebraic integers になるような C * -algebra もある Nikolaev [ Nikb ] では , Effros [ Eff81 ] されている また , Nikolaev [ Nika ] K 0 cluster algebra になるような AF-algebra えて , cluster C * -algebra んで いる

  • cluster C * -algebra

, CW したものとして , Villadsen algebra というもの がある Villadsen [ Vil98 Vil99 ] により 導入 された

  • Villadsen algebra of the first type
  • Villadsen algebra of the second type

Villadsen algebra of the first type , seed space ばれる compact Hausdorff いて されるが , seed space CW であるときの について Toms Winter [ TW09 ] 調 べている

References

[CDL13]     Joachim Cuntz, Christopher Deninger, and Marcelo Laca. C * -algebras of Toeplitz type associated with algebraic number fields. Math. Ann. , 355(4):1383–1423, 2013, arXiv:1105.5352 .

[CEL15]     Joachim Cuntz, Siegfried Echterhoff, and Xin Li. On the K-theory of the C * -algebra generated by the left regular representation of an Ore semigroup. J. Eur. Math. Soc. (JEMS) , 17(3):645–687, 2015, arXiv:1201.4680 .

[CL10]     Joachim Cuntz and Xin Li. The regular C * -algebra of an integral domain. In Quanta of maths , volume 11 of Clay Math. Proc. , pages 149–170. Amer. Math. Soc., Providence, RI, 2010, arXiv:0807.1407 .

[CL11]     Joachim Cuntz and Xin Li. C * -algebras associated with integral domains and crossed products by actions on adele spaces. J. Noncommut. Geom. , 5(1):1–37, 2011, arXiv:0906.4903 .

[Cun08]     Joachim Cuntz. C * -algebras associated with the ax + b -semigroup over . In K -theory and noncommutative geometry , EMS Ser. Congr. Rep., pages 201–215. Eur. Math. Soc., Zürich, 2008, arXiv:math/0611541 .

[DL10]     Claire Debord and Jean-Marie Lescure. Index theory and groupoids. In Geometric and topological methods for quantum field theory , pages 86–158. Cambridge Univ. Press, Cambridge, 2010, arXiv:0801.3617 .

[Eff81]     Edward G. Effros. Dimensions and C * -algebras , volume 46 of CBMS Regional Conference Series in Mathematics . Conference Board of the Mathematical Sciences, Washington, D.C., 1981.

[Li10]     Xin Li. Ring C * -algebras. Math. Ann. , 348(4):859–898, 2010, arXiv:0905.4861 .

[Li12]     Xin Li. Semigroup C * -algebras and amenability of semigroups. J. Funct. Anal. , 262(10):4302–4340, 2012, arXiv:1105.5539 .

[Li14]     Xin Li. On K-theoretic invariants of semigroup C * -algebras attached to number fields. Adv. Math. , 264:371–395, 2014, arXiv:1212.3199 .

[Li16]     Xin Li. On K-theoretic invariants of semigroup C * -algebras attached to number fields, Part II. Adv. Math. , 291:1–11, 2016, arXiv:1503.01708 .

[Mes11]     Bram Mesland. Groupoid cocycles and K-theory. M ünster J. Math. , 4:227–249, 2011, arXiv:1005.3677 .

[Nika]     Igor Nikolaev. On cluster C * -algebras, arXiv:1508.00591 .

[Nikb]     Igor Nikolaev. Remark on arithmetic topology, arXiv:1706.06398 .

[Pat99]     Alan L. T. Paterson. Groupoids, inverse semigroups, and their operator algebras , volume 170 of Progress in Mathematics . Birkhäuser Boston, Inc., Boston, MA, 1999, http://dx.doi.org/10.1007/978-1-4612-1774-9 .

[Ren80]     Jean Renault. A groupoid approach to C * -algebras , volume 793 of Lecture Notes in Mathematics . Springer, Berlin, 1980.

[TW09]     Andrew S. Toms and Wilhelm Winter. The Elliott conjecture for Villadsen algebras of the first type. J. Funct. Anal. , 256(5):1311–1340, 2009, arXiv:math/0611059 .

[Vil98]     Jesper Villadsen. Simple C * -algebras with perforation. J. Funct. Anal. , 154(1):110–116, 1998, http://dx.doi.org/10.1006/jfan.1997.3168 .

[Vil99]     Jesper Villadsen. On the stable rank of simple C * -algebras. J. Amer. Math. Soc. , 12(4):1091–1102, 1999, http://dx.doi.org/10.1090/S0894-0347-99-00314-8 .