On universal 𝐶*-algebras generated by 𝑛 projections with scalar sum

Shulman, Tatiana

doi:10.1090/S0002-9939-08-09654-8

On universal $C^*$-algebras generated by $n$ projections with scalar sum
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by Tatiana Shulman

Proc. Amer. Math. Soc. 137 (2009), 115-122

DOI: https://doi.org/10.1090/S0002-9939-08-09654-8

Published electronically: August 14, 2008

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Abstract:

We study the universal $C^*$-algebras generated by $n$ projections $p_1,\dotsc ,p_n$ subject to the relation $p_1+\cdots +p_n=\lambda 1$, $\lambda \in \mathbb {R}$. The questions of when these $C^*$-algebras are type I, nuclear or exact are considered. It is proved also that among these $C^*$-algebras there is a continuum of mutually nonisomorphic ones.

References

Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
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
Eberhard Kirchberg, On restricted perturbations in inverse images and a description of normalizer algebras in $C^*$-algebras, J. Funct. Anal. 129 (1995), no. 1, 1–34. MR 1322640, DOI 10.1006/jfan.1995.1040
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
S. A. Kruglyak, Coxeter functors for a class of $\ast$-quivers, Ukraïn. Mat. Zh. 54 (2002), no. 6, 789–797 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 6, 967–978. MR 1956637, DOI 10.1023/A:1021764304170
S. A. Kruglyak, V. I. Rabanovich, and Yu. S. Samoĭlenko, On sums of projections, Funktsional. Anal. i Prilozhen. 36 (2002), no. 3, 20–35, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 36 (2002), no. 3, 182–195. MR 1935900, DOI 10.1023/A:1020193804109
Stanislav Kruglyak, Vyacheslav Rabanovich, and Yuriĭ Samoĭlenko, Decomposition of a scalar matrix into a sum of orthogonal projections, Linear Algebra Appl. 370 (2003), 217–225. MR 1994329, DOI 10.1016/S0024-3795(03)00390-2
S. A. Kruglyak and A. V. Roĭter, Locally scalar representations of graphs in the category of Hilbert spaces, Funktsional. Anal. i Prilozhen. 39 (2005), no. 2, 13–30, 94 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 39 (2005), no. 2, 91–105. MR 2161513, DOI 10.1007/s10688-005-0022-8
S. A. Krugljak and Ju. S. Samoĭlenko, Unitary equivalence of sets of selfadjoint operators, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 60–62 (Russian). MR 565103
Stanislav Krugliak and Yuriǐ Samoǐlenko, On complexity of description of representations of $\ast$-algebras generated by idempotents, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1655–1664. MR 1636978, DOI 10.1090/S0002-9939-00-05100-5
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
V. Ostrovskyi and Yu. Samoilenko, Introduction to the theory of representations of finitely presented *-algebras. I. Representations by bounded operators, Reviews in Mathematics and Mathematical Physics, vol. 11, Harwood Academic Publishers, Amsterdam, 1999. MR 1997101
V. I. Rabanovich and Yu. S. Samoĭlenko, When the sum of idempotents or projectors is a multiple of unity, Funktsional. Anal. i Prilozhen. 34 (2000), no. 4, 91–93 (Russian); English transl., Funct. Anal. Appl. 34 (2000), no. 4, 311–313. MR 1819651, DOI 10.1023/A:1004173827361
Simon Wassermann, Tensor products of free-group $C^*$-algebras, Bull. London Math. Soc. 22 (1990), no. 4, 375–380. MR 1058315, DOI 10.1112/blms/22.4.375