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Positive combinations and sums of projections in purely infinite simple $ C^*$-algebras and their multiplier algebras


Authors: Victor Kaftal, P. W. Ng and Shuang Zhang
Journal: Proc. Amer. Math. Soc. 139 (2011), 2735-2746
MSC (2010): Primary 46L05; Secondary 47C15
DOI: https://doi.org/10.1090/S0002-9939-2011-10995-X
Published electronically: March 24, 2011
MathSciNet review: 2801613
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Abstract: Every positive element in a purely infinite simple $ \sigma$-unital $ C^*$-algebra $ \mathscr{A}$ is a finite linear combination of projections with positive coefficients. Also, every positive $ a$ in the multiplier algebra $ \mathscr M(\mathscr{A})$ of a purely infinite simple $ \sigma$-unital $ C^*$-algebra $ \mathscr{A}$ is a finite linear combination of projections with positive coefficients. Furthermore, if the essential norm $ \Vert a\Vert _{ess} > 1$, then $ a$ is a finite sum of projections in $ \mathscr M(\mathscr{A})$. As a consequence, any positive element in the generalized Calkin Algebra $ \mathscr M(\mathscr{A})/\mathscr{A}$ or in $ \mathscr M(\mathscr{A})$ but not in $ \mathscr{A}$ is a positive scalar multiple of a finite sum of projections.


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  • 1. B. Blackadar, Notes on the structure of projections in simple algebras, Semesterbericht Funktionalanalysis, Tubingen, Wintersemester (1982/83).
  • 2. L.G. Brown, Stable isomorphism of hereditary subalgebras of C*-algebras, Pacific J. Math., 71 (1977), 335-348. MR 0454645 (56:12894)
  • 3. L. Brown and G. Pedersen, $ C^*$-algebras of real rank zero J. Funct. Anal., 99 (1991), 131-149. MR 1120918 (92m:46086)
  • 4. M. D. Choi and P. Y. Wu, Sums of orthogonal projections, preprint.
  • 5. K. Dykema, D. Freeman, K. Kornelson, D. Larson, M. Ordower and E. Weber, Ellipsoidal tight frames and projection decompositions of operators, Illinois J. Math., 48 (2004), no. 2, 477-489. MR 2085421 (2005e:42092)
  • 6. T. Fack, Finite sums of commutators in $ C^*$-algebras, Ann. Inst. Fourier (Grenoble), 32 (1982), no. 1, 129-137. MR 658946 (83g:46051)
  • 7. P. Fillmore, On sums of projections, J. Funct. Anal., 4 (1969), 146-152. MR 0246150 (39:7455)
  • 8. P. Fillmore, Sums of operators with square zero, Acta Sci. Math. (Szeged), 28 (1967), 285-288. MR 0221301 (36:4353)
  • 9. C. K. Fong and G. J. Murphy, Averages of projections, J. Operator Theory, 13 (1985), no. 2, 219-225. MR 775994 (86h:47071)
  • 10. V. Kaftal, P. W. Ng and S. Zhang, Strong sums of projections in von Neumann factors, J. of Func. Anal., 257 (2009), no. 8, 2497-2529. MR 2555011 (2011b:46096)
  • 11. V. Kaftal, P. W. Ng and S. Zhang, Projection decomposition in multiplier algebras, Mathematische Annalen (2009), to appear.
  • 12. V. Kaftal, P. W. Ng and S. Zhang, Finite sums of projections in von Neumann algebras, (2010), preprint.
  • 13. K. Kornelson and D. Larson, Rank-one decomposition of operators and construction of frames, Contemp. Math., 345, Amer. Math. Soc., 2004, 203-214. MR 2066830 (2005e:42096)
  • 14. S. Kruglyak, V. Rabanovich and Y. Samoĭlenko, On sums of projections, Funct. Anal. and Appl., 36 (2002), 182-195. MR 1935900 (2004e:47021)
  • 15. H. Lin and S. Zhang, On infinite simple $ C^*$-algebras, J. Funct. Anal., 100 (1991), no. 1, 221-231. MR 1124300 (92m:46088)
  • 16. L. W. Marcoux, On the linear span of projections in certain simple $ C^*$-algebras, Indiana Univ. Math. J., 51 (2002), no. 3, 753-771. MR 1911053 (2003i:46057)
  • 17. L. W. Marcoux, Sums of small number of commutators, J. Operator Theory, 56 (2006), no. 1, 111-142. MR 2261614 (2008m:46107)
  • 18. L. W. Marcoux and G. J. Murphy, Unitarily-invariant linear subspaces in $ C^*$-algebras, Proc. Amer. Math. Soc., 126 (1998), 3597-3605. MR 1610753 (99b:46085)
  • 19. K. Matsumoto, Selfadjoint operators as a real span of $ 5$ projections, Math. Japon., 29 (1984), 291-294. MR 747934 (85g:47031)
  • 20. C. Pearcy and D. Topping, Sums of small numbers of idempotents, Michigan Math. J., 14 (1967), 453-465. MR 0218922 (36:2006)
  • 21. G. K. Pedersen, The linear span of projections in simple $ C^*$-alebras, J. Operator Theory, 4 (1980), no. 2, 289-296. MR 595417 (82b:46078)
  • 22. P. Y. Wu, Additive combinations of special operators, Funct. Anal. Oper. Theory. Banach Ctr. Publ. Inst. Math. Polish Acad. Sci., 30 (1994), 337-361. MR 1285620 (95d:47018)
  • 23. S. Zhang, On the structure of projections and ideals of corona algebras, Canad. J. Math., 41 (1989), no. 4, 721-742. MR 1012625 (90h:46094)
  • 24. S. Zhang, A property of purely infinite simple $ C^*$-algebras, Proc. Amer. Math. Soc., 109 (1990), no. 3, 717-720. MR 1010004 (90k:46134)
  • 25. S. Zhang, A Riesz decomposition property and ideal structure of multiplier algebras, J. Operator Theory, 24 (1990), no. 2, 209-225. MR 1150618 (93b:46116)
  • 26. S. Zhang, Matricial structure and homotopy type of simple C*-algebras with real rank zero, J. Operator Theory, 26 (1991), 283-312. MR 1225518 (94f:46075)
  • 27. S. Zhang, Certain $ C^*$-algebras with real rank zero and their corona and multiplier algebras. I, Pacific J. Math., 155 (1992), no. 1, 169-197. MR 1174483 (94i:46093)

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Additional Information

Victor Kaftal
Affiliation: Department of Mathematics, University of Cincinnati, P. O. Box 210025, Cincinnati, Ohio 45221-0025
Email: victor.kaftal@math.uc.edu

P. W. Ng
Affiliation: Department of Mathematics, University of Louisiana, 217 Maxim D. Doucet Hall, P.O. Box 41010, Lafayette, Louisiana 70504-1010
Email: png@louisiana.edu

Shuang Zhang
Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025
Email: zhangs@math.uc.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10995-X
Received by editor(s): July 1, 2010
Published electronically: March 24, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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