Noncommutative semialgebraic sets and associated lifting problems

Loring, Terry; Shulman, Tatiana

doi:10.1090/S0002-9947-2011-05313-4

Noncommutative semialgebraic sets and associated lifting problems
HTML articles powered by AMS MathViewer

by Terry A. Loring and Tatiana Shulman PDF

Trans. Amer. Math. Soc. 364 (2012), 721-744 Request permission

Abstract:

We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated $C^{*}$-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and Pedersen’s discovery of the norm adjusting power of quasi-central approximate units.

A projective $C^{*}$-algebra is the analog of an absolute retract. Thus we can say that various noncommutative semialgebraic sets turn out to be absolute retracts. In particular we show that a noncommutative absolute retract results from the intersection of the approximate locus of a noncommutative homogeneous polynomial with the noncommutative unit ball. By unit ball we are referring to the $C^{*}$-algebra of the universal row contraction. We show that various alternative noncommutative unit balls are also projective.

Sufficiently many $C^{*}$-algebras are now known to be projective so that we are able to show that the cone over any separable $C^{*}$-algebra is the inductive limit of $C^{*}$-algebras that are projective.

References

Charles A. Akemann, Left ideal structure of $C^*$-algebras, J. Functional Analysis 6 (1970), 305–317. MR 0275177, DOI 10.1016/0022-1236(70)90063-7
Charles A. Akemann and Gert K. Pedersen, Ideal perturbations of elements in $C^*$-algebras, Math. Scand. 41 (1977), no. 1, 117–139. MR 473848, DOI 10.7146/math.scand.a-11707
William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
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
B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR 2188261, DOI 10.1007/3-540-28517-2
Karol Borsuk, Theory of shape, Monografie Matematyczne, Tom 59, PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0418088
A. Chigogidze and AN Dranishnikov, Which compacta are noncommutative ARs?, Topology and its Applications 157 (2010), no. 4, 774–778.
Kenneth R. Davidson, Lifting positive elements in $C^*$-algebras, Integral Equations Operator Theory 14 (1991), no. 2, 183–191. MR 1090700, DOI 10.1007/BF01199904
Kenneth R. Davidson and David R. Pitts, Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), no. 3, 321–337. MR 1627901, DOI 10.1007/BF01195123
E. G. Effros and J. Kaminker, Homotopy continuity and shape theory for $C^\ast$-algebras, Geometric methods in operator algebras (Kyoto, 1983) Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 152–180. MR 866493
Søren Eilers and Ruy Exel, Finite-dimensional representations of the soft torus, Proc. Amer. Math. Soc. 130 (2002), no. 3, 727–731. MR 1866027, DOI 10.1090/S0002-9939-01-06150-0
Søren Eilers and Terry A. Loring, Computing contingencies for stable relations, Internat. J. Math. 10 (1999), no. 3, 301–326. MR 1688149, DOI 10.1142/S0129167X99000112
Søren Eilers, Terry A. Loring, and Gert K. Pedersen, Stability of anticommutation relations: an application of noncommutative CW complexes, J. Reine Angew. Math. 499 (1998), 101–143. MR 1631120
Ruy Exel, The soft torus and applications to almost commuting matrices, Pacific J. Math. 160 (1993), no. 2, 207–217. MR 1233352
Ruy Exel and Terry A. Loring, Finite-dimensional representations of free product $C^*$-algebras, Internat. J. Math. 3 (1992), no. 4, 469–476. MR 1168356, DOI 10.1142/S0129167X92000217
S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 449–474. MR 173265
Terry A. Loring, Projective $C^\ast$-algebras, Math. Scand. 73 (1993), no. 2, 274–280. MR 1269264, DOI 10.7146/math.scand.a-12471
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
Terry A. Loring, A projective $C^*$-algebra related to $K$-theory, J. Funct. Anal. 254 (2008), no. 12, 3079–3092. MR 2418619, DOI 10.1016/j.jfa.2008.03.004
Terry A. Loring, $C^*$-algebra relations, Math. Scand. 107 (2010), no. 1, 43–72. MR 2679392, DOI 10.7146/math.scand.a-15142
—, Weakly projective $C^ *$-algebras, Rocky Mountain J. Math. (to appear), http://arxiv.org/abs/0905.1520.
Terry A. Loring and Gert K. Pedersen, Projectivity, transitivity and AF-telescopes, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4313–4339. MR 1616003, DOI 10.1090/S0002-9947-98-02353-8
Catherine L. Olsen and Gert K. Pedersen, Corona $C^*$-algebras and their applications to lifting problems, Math. Scand. 64 (1989), no. 1, 63–86. MR 1036429, DOI 10.7146/math.scand.a-12248
Tatiana Shulman, Lifting of nilpotent contractions, Bull. Lond. Math. Soc. 40 (2008), no. 6, 1002–1006. MR 2471949, DOI 10.1112/blms/bdn084
V. A. Vassiliev, Applied Picard-Lefschetz theory, Mathematical Surveys and Monographs, vol. 97, American Mathematical Society, Providence, RI, 2002. MR 1930577, DOI 10.1090/surv/097