Strongly self-absorbing property for inclusions of 𝐶*-algebras with a finite Watatani index

Osaka, Hiroyuki; Teruya, Tamotsu

doi:10.1090/S0002-9947-2013-05907-7

Strongly self-absorbing property for inclusions of $\mathrm {C}^*$-algebras with a finite Watatani index
HTML articles powered by AMS MathViewer

by Hiroyuki Osaka and Tamotsu Teruya PDF

Trans. Amer. Math. Soc. 366 (2014), 1685-1702 Request permission

Abstract:

Let $P \subset A$ be an inclusion of unital $\mathrm {C}^*$-algebras and $E\colon A \rightarrow P$ be a conditional expectation of index-finite type. We introduce a Rokhlin property for $E$ and discuss the $\mathcal {D}$-absorbing property, where $\mathcal {D}$ is a separable, unital, strongly self-absorbing $\mathrm {C}^*$-algebra, i.e., $\mathcal {D} \not = \mathbb {C}$, and there exists an isomorphism $\varphi \colon \mathcal {D} \rightarrow \mathcal {D} \otimes \mathcal {D}$ such that $\varphi$ is approximately unitarily equivalent to the embedding $d \mapsto d \otimes 1_D$. UHF algebras of infinite type, the Jiang–Su algebra $\mathcal {Z}$, and Cuntz algebras $O_2$ and $O_\infty$ are typical examples of strongly self-absorbing $\mathrm {C}^*$-algebras. In this paper, we consider permanence properties for a strongly self-absorbing property under inclusions of unital $\mathrm {C}^*$-algebras with a finite Watatani index. We show the following:

Let $P \subset A$ be an inclusion of unital $\mathrm {C}^*$-algebras, $E$ a conditional expectation from $A$ onto $P$ with a finite index, and $\mathcal {D}$ be a separable unital strongly self-absorbing $\mathrm {C}^*$-algebra.

If $A$ is separable and $\mathcal {D}$-absorbing (i.e., $A \otimes \mathcal {D} \cong A$), and $E$ has the Rokhlin property, then $P$ is $\mathcal {D}$-absorbing.
If $\mathcal {D}$ is the universal UHF algebra $\mathcal {U}_\infty$ and $\alpha \colon G \rightarrow \mathrm {Aut}(D)$ has the Rokhlin property, then $\mathcal {U}_\infty \rtimes _\alpha G \cong \mathcal {U}_\infty$.
If $A = \mathcal {D}$ is an inductive limit of weakly semiprojective $\mathrm {C}^*$-algebras and $E$ has the Rokhlin property, then $P \cong \mathcal {D}$. In particular, this is true when $\mathcal {D}$ is a $\mathrm {UHF}$ algebra of infinite type, $\mathcal {O}_2$, or $\mathcal {O}_\infty$.

As an application, if $A$ is a unital $\mathcal {D}$-absorbing $\mathrm {C}^*$-algebra and $\alpha$ is an action from a finite group $G$ on $A$ with the Rokhlin property, then we have

the crossed product algebra $A \rtimes _\alpha G$ is also $\mathcal {D}$-absorbing,
for any subgroup $H \subset G$, $A^H$ is $\mathcal {D}$-absorbing,
if $A$ is $\mathcal {O}_2$, then $A \rtimes _\alpha G \cong \mathcal {O}_2$.

References

Bruce Blackadar, Shape theory for $C^\ast$-algebras, Math. Scand. 56 (1985), no. 2, 249–275. MR 813640, DOI 10.7146/math.scand.a-12100
Bruce Blackadar, Symmetries of the CAR algebra, Ann. of Math. (2) 131 (1990), no. 3, 589–623. MR 1053492, DOI 10.2307/1971472
Bruce Blackadar, Semiprojectivity in simple $C^*$-algebras, Operator algebras and applications, Adv. Stud. Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 1–17. MR 2059799, DOI 10.2969/aspm/03810001
Bruce Blackadar, Alexander Kumjian, and Mikael Rørdam, Approximately central matrix units and the structure of noncommutative tori, $K$-Theory 6 (1992), no. 3, 267–284. MR 1189278, DOI 10.1007/BF00961466
Man Duen Choi and Edward G. Effros, The completely positive lifting problem for $C^*$-algebras, Ann. of Math. (2) 104 (1976), no. 3, 585–609. MR 417795, DOI 10.2307/1970968
Man Duen Choi and Edward G. Effros, Nuclear $C^*$-algebras and the approximation property, Amer. J. Math. 100 (1978), no. 1, 61–79. MR 482238, DOI 10.2307/2373876
Marius Dadarlat, N. Christopher Phillips, and Andrew S. Toms, A direct proof of $\scr Z$-stability for approximately homogeneous $C^*$-algebras of bounded topological dimension, Bull. Lond. Math. Soc. 41 (2009), no. 4, 639–653. MR 2521359, DOI 10.1112/blms/bdp037
Edward G. Effros and Jonathan Rosenberg, $C^{\ast }$-algebras with approximately inner flip, Pacific J. Math. 77 (1978), no. 2, 417–443. MR 510932
David E. Evans and Akitaka Kishimoto, Trace scaling automorphisms of certain stable AF algebras, Hokkaido Math. J. 26 (1997), no. 1, 211–224. MR 1432548, DOI 10.14492/hokmj/1351257815
James G. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. MR 112057, DOI 10.1090/S0002-9947-1960-0112057-5
Guihua Gong, Xinhui Jiang, and Hongbing Su, Obstructions to $\scr Z$-stability for unital simple $C^*$-algebras, Canad. Math. Bull. 43 (2000), no. 4, 418–426. MR 1793944, DOI 10.4153/CMB-2000-050-1
Richard H. Herman and Vaughan F. R. Jones, Period two automorphisms of $\textrm {UHF}$ $C^{\ast }$-algebras, J. Functional Analysis 45 (1982), no. 2, 169–176. MR 647069, DOI 10.1016/0022-1236(82)90016-7
Richard H. Herman and Vaughan F. R. Jones, Models of finite group actions, Math. Scand. 52 (1983), no. 2, 312–320. MR 702960, DOI 10.7146/math.scand.a-12008
Ilan Hirshberg, Mikael Rørdam, and Wilhelm Winter, $\scr C_0(X)$-algebras, stability and strongly self-absorbing $C^*$-algebras, Math. Ann. 339 (2007), no. 3, 695–732. MR 2336064, DOI 10.1007/s00208-007-0129-8
Ilan Hirshberg and Wilhelm Winter, Rokhlin actions and self-absorbing $C^*$-algebras, Pacific J. Math. 233 (2007), no. 1, 125–143. MR 2366371, DOI 10.2140/pjm.2007.233.125
Masaki Izumi, Inclusions of simple $C^\ast$-algebras, J. Reine Angew. Math. 547 (2002), 97–138. MR 1900138, DOI 10.1515/crll.2002.055
Masaki Izumi, Finite group actions on $C^*$-algebras with the Rohlin property. I, Duke Math. J. 122 (2004), no. 2, 233–280. MR 2053753, DOI 10.1215/S0012-7094-04-12221-3
Masaki Izumi, Finite group actions on $C^*$-algebras with the Rohlin property. II, Adv. Math. 184 (2004), no. 1, 119–160. MR 2047851, DOI 10.1016/S0001-8708(03)00140-3
Ja A. Jeong and Gi Hyun Park, Saturated actions by finite-dimensional Hopf $*$-algebras on $C^*$-algebras, Internat. J. Math. 19 (2008), no. 2, 125–144. MR 2384896, DOI 10.1142/S0129167X08004583
Xinhui Jiang and Hongbing Su, On a simple unital projectionless $C^*$-algebra, Amer. J. Math. 121 (1999), no. 2, 359–413. MR 1680321
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127
E. Kirchberg, The classification of purely infinite $C^*$-algebras using Kasparov’s theory, Preliminary preprint (3rd draft).
Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
Akitaka Kishimoto, On the fixed point algebra of a UHF algebra under a periodic automorphism of product type, Publ. Res. Inst. Math. Sci. 13 (1977/78), no. 3, 777–791. MR 0500177, DOI 10.2977/prims/1195189607
Terry A. Loring, Lifting solutions to perturbing problems in $C^*$-algebras, Fields Institute Monographs, vol. 8, American Mathematical Society, Providence, RI, 1997. MR 1420863, DOI 10.1090/fim/008
Hiroyuki Osaka, Kazunori Kodaka, and Tamotsu Teruya, The Rohlin property for inclusions of $C^*$-algebras with a finite Watatani index, Operator structures and dynamical systems, Contemp. Math., vol. 503, Amer. Math. Soc., Providence, RI, 2009, pp. 177–195. MR 2590623, DOI 10.1090/conm/503/09900
Hiroyuki Osaka and N. Christopher Phillips, Crossed products by finite group actions with the Rokhlin property, Math. Z. 270 (2012), no. 1-2, 19–42. MR 2875821, DOI 10.1007/s00209-010-0784-4
N. Christopher Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Doc. Math. 5 (2000), 49–114. MR 1745197
N. Christopher Phillips, Recursive subhomogeneous algebras, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4595–4623. MR 2320643, DOI 10.1090/S0002-9947-07-03850-0
N. Christopher Phillips, The tracial Rokhlin property for actions of finite groups on $C^\ast$-algebras, Amer. J. Math. 133 (2011), no. 3, 581–636. MR 2808327, DOI 10.1353/ajm.2011.0016
N. C. Phillips, Finite cyclic group actions with the tracial Rokhlin property, to appear in Trans. Amer. Math. Soc. (arXiv:mathOA/0609785).
M. Rørdam, Classification of nuclear, simple $C^*$-algebras, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR 1878882, DOI 10.1007/978-3-662-04825-2_{1}
Mikael Rørdam, A simple $C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109–142. MR 2020420, DOI 10.1007/BF02392697
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
Tamotsu Teruya, A characterization of normal extensions for subfactors, Proc. Amer. Math. Soc. 120 (1994), no. 3, 781–783. MR 1207542, DOI 10.1090/S0002-9939-1994-1207542-7
Tamotsu Teruya, Normal intermediate subfactors, J. Math. Soc. Japan 50 (1998), no. 2, 469–490. MR 1613172, DOI 10.2969/jmsj/05020469
Andrew Toms, On the independence of $K$-theory and stable rank for simple $C^*$-algebras, J. Reine Angew. Math. 578 (2005), 185–199. MR 2113894, DOI 10.1515/crll.2005.2005.578.185
Andrew S. Toms and Wilhelm Winter, Strongly self-absorbing $C^*$-algebras, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3999–4029. MR 2302521, DOI 10.1090/S0002-9947-07-04173-6
Andrew S. Toms and Wilhelm Winter, $\scr Z$-stable ASH algebras, Canad. J. Math. 60 (2008), no. 3, 703–720. MR 2414961, DOI 10.4153/CJM-2008-031-6
Yasuo Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117. MR 996807, DOI 10.1090/memo/0424
Wilhelm Winter, Decomposition rank and $\scr Z$-stability, Invent. Math. 179 (2010), no. 2, 229–301. MR 2570118, DOI 10.1007/s00222-009-0216-4
Wilhelm Winter, Strongly self-absorbing $C^*$-algebras are $\scr Z$-stable, J. Noncommut. Geom. 5 (2011), no. 2, 253–264. MR 2784504, DOI 10.4171/JNCG/74