## Abstract and Applied Analysis

### The Linear Span of Projections in AH Algebras and for Inclusions of ${C}^{\ast}$-Algebras

#### Abstract

In the first part of this paper, we show that an AH algebra $A=\underset{\to }{\mathrm{lim}}({A}_{i},{\varphi }_{i})$ has the LP property if and only if every element of the centre of ${A}_{i}$ belongs to the closure of the linear span of projections in $A$. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unital ${C}^{\ast}$-algebras $P\subset A$ with a finite Watatani index, if a faithful conditional expectation $E:A\to P$ has the Rokhlin property in the sense of Kodaka et al., then $P$ has the LP property under the condition that$A$ has the LP property. As an application, let $A$ be a simple unital ${C}^{\ast}$-algebra with the LP property, $\alpha$ an action of a finite group $G$ onto $\text{Aut}(A)$. If $\alpha$ has the Rokhlin property in the sense of Izumi, then the fixed point algebra ${A}^{G}$ and the crossed product algebra $\mathrm{A }{\rtimes}_{\alpha }\mathrm{ G}$ have the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2012), Article ID 204319, 12 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449939

Digital Object Identifier
doi:10.1155/2013/204319

Mathematical Reviews number (MathSciNet)
MR3039147

Zentralblatt MATH identifier
1282.46057

#### Citation

Hoa, Dinh Trung; Ho, Toan Minh; Osaka, Hiroyuki. The Linear Span of Projections in AH Algebras and for Inclusions of ${C}^{\ast}$ -Algebras. Abstr. Appl. Anal. 2013, Special Issue (2012), Article ID 204319, 12 pages. doi:10.1155/2013/204319. https://projecteuclid.org/euclid.aaa/1393449939

#### References

• G. K. Pedersen, “The linear span of projections in simple ${C}^{\ast\,\!}$-algebras,” Journal of Operator Theory, vol. 4, no. 2, pp. 289–296, 1980.
• B. Blackadar, O. Bratteli, G. A. Elliott, and A. Kumjian, “Reduction of real rank in inductive limits of ${C}^{\ast\,\!}$-algebras,” Mathematische Annalen, vol. 292, no. 1, pp. 111–126, 1992.
• O. Bratteli and G. A. Elliott, “Small eigenvalue variation and real rank zero,” Pacific Journal of Mathematics, vol. 175, no. 1, pp. 47–59, 1996.
• G. A. Elliott, T. M. Ho, and A. S. Toms, “A class of simple ${C}^{\ast\,\!}$-algebras with stable rank one,” Journal of Functional Analysis, vol. 256, no. 2, pp. 307–322, 2009.
• T. M. Ho, “On the property SP of certain AH algebras,” Comptes Rendus Mathématiques de l'Académie des Sciences. La Société Royale du Canada, vol. 29, no. 3, pp. 81–86, 2007.
• K. R. Goodearl, “Notes on a class of simple ${C}^{\ast\,\!}$-algebras with real rank zero,” Publicacions Matemàtiques, vol. 36, no. 2, pp. 637–654, 1992.
• A. S. Toms and W. Winter, “The Elliott conjecture for Villadsen algebras of the first type,” Journal of Functional Analysis, vol. 256, no. 5, pp. 1311–1340, 2009.
• A. S. Toms, “On the classification problem for nuclear ${C}^{\ast\,\!}$-algebras,” Annals of Mathematics. Second Series, vol. 167, no. 3, pp. 1029–1044, 2008.
• G. A. Elliott, “On the classification of inductive limits of sequences of semisimple finite-dimensional algebras,” Journal of Algebra, vol. 38, no. 1, pp. 29–44, 1976.
• G. A. Elliott and G. Gong, “On the classification of ${C}^{\ast\,\!}$-algebras of real rank zero. II,” Annals of Mathematics. Second Series, vol. 144, no. 3, pp. 497–610, 1996.
• G. A. Elliott, G. Gong, and L. Li, “On the classification of simple inductive limit ${C}^{\ast\,\!}$-algebras. II. The isomorphism theorem,” Inventiones Mathematicae, vol. 168, no. 2, pp. 249–320, 2007.
• B. Blackadar, M. Dădărlat, and M. Rørdam, “The real rank of inductive limit ${C}^{\ast\,\!}$-algebras,” Mathematica Scandinavica, vol. 69, no. 2, pp. 267–276, 1991.
• R. V. Kadison, “Diagonalizing matrices,” American Journal of Mathematics, vol. 106, no. 6, pp. 1451–1468, 1984.
• L. G. Brown and G. K. Pedersen, “${C}^{\ast\,\!}$-algebras of real rank zero,” Journal of Functional Analysis, vol. 99, no. 1, pp. 131–149, 1991.
• G. A. Elliott, “A classification of certain simple ${C}^{\ast\,\!}$-algebras,” in Quantum and Non-Commutative Analysis (Kyoto, 1992), vol. 16 of Mathematical Physics Studies, pp. 373–385, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1993.
• K. Thomsen, “Inductive limits of interval algebras: the tracial state space,” American Journal of Mathematics, vol. 116, no. 3, pp. 605–620, 1994.
• Y. Watatani, Index for C$^{\ast\,\!}$-Subalgebras, Memoirs of the American Mathematical Society, American Mathematical Society, Providence, RI, USA, 1990.
• M. Izumi, “Finite group actions on ${C}^{\ast\,\!}$-algebras with the Rohlin property. I,” Duke Mathematical Journal, vol. 122, no. 2, pp. 233–280, 2004.
• K. Kodaka, H. Osaka, and T. Teruya, “The Rohlin property for inclusions of C$^{\ast\,\!}$-algebras with a finite Watatani index,” Contemporary Mathematics, vol. 503, pp. 177–195, 2009.
• H. Osaka and T. Teruya, “Strongly self-absorbing property for inclusions of C$^{\ast\,\!}$-algebras with a finite Watatani index,” Transactions of the American Mathematical Society. In press, http://arxiv.org/abs/1002.4233.
• J. A. Jeong and G. H. Park, “Saturated actions by finite-dimensional Hopf$^{\ast\,\!}$ -algebras on ${C}^{\ast\,\!}$-algebras,” International Journal of Mathematics, vol. 19, no. 2, pp. 125–144, 2008.
• H. Osaka and N. C. Phillips, “Crossed products by finite group actions with the Rokhlin property,” Mathematische Zeitschrift, vol. 270, no. 1-2, pp. 19–42, 2012.
• N. C. Phillips, “The tracial Rokhlin property for actions of finite groups on ${C}^{\ast\,\!}$-algebras,” American Journal of Mathematics, vol. 133, no. 3, pp. 581–636, 2011.
• I. Hirshberg and W. Winter, “Rokhlin actions and self-absorbing ${C}^{\ast\,\!}$-algebras,” Pacific Journal of Mathematics, vol. 233, no. 1, pp. 125–143, 2007.
• C. Pasnicu and C. N. Phillips, “Permanence properties for crossed products and fixed point algebras of finite groups,” http://arxiv.org/abs/1208.3810.
• S. Echterhoff, W. Lück, N. C. Phillips, and S. Walters, “The structure of crossed products of irrational rotation algebras by finite subgroups of ${SL}_{2}(Z)$,” Journal für die Reine und Angewandte Mathematik, vol. 639, pp. 173–221, 2010.