Factorizations of invertible operators and 𝐾-theory of 𝐶*-algebras

Zhang, Shuang

doi:10.1090/S0273-0979-1993-00334-3

Factorizations of invertible operators and -theory of -algebras

Author: Shuang Zhang
Journal: Bull. Amer. Math. Soc. 28 (1993), 75-83
MSC: Primary 46L80; Secondary 19K33, 46L05, 46M20
DOI: https://doi.org/10.1090/S0273-0979-1993-00334-3
MathSciNet review: 1164064
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal{A}$ be a unital ${\text{C}}^{\ast}$ -algebra. We describe K-skeleton factorizations of all invertible operators on a Hilbert ${\text{C}}^{\ast}$ -module $\mathcal{H}_\mathcal{A}$ , in particular on $\mathcal{H}={l^2}$ , with the Fredholm index as an invariant. We then outline the isomorphisms ${K_0}(\mathcal{A}) \cong {\pi _{2k}}({[p]_0}) \cong {\pi _{2k}}({GL}_r^p(\mathcal{A}))$ and ${{K}_{1}}(\mathcal{A})\cong {{\pi }_{2k+1}}({[p]_0})\cong {{\pi }_{2k+1}}({GL}_r^p(\mathcal{A}))$ for $k \geq 0$ , where ${[p]_0}$ denotes the class of all compact perturbations of a projection p in the infinite Grassmann space ${Gr}^{\infty}(\mathcal{A})$ and ${GL}_r^p(\mathcal{A})$ stands for the group of all those invertible operators on ${\mathcal{H}_\mathcal{A}}$ essentially commuting with p.

[APT] C. A. Akemann, G. K. Pedersen, and J. Tomiyama, Multipliers of $C^{\ast}$ -algebras, J. Funct. Anal. 13 (1973), 277-301. MR 0470685 (57:10431)
[At] M. F. Atiyah, K-theory, Benjamin, New York, 1967. MR 0224083 (36:7130)
[Ar] W. Arveson, Notes on extensions of $C^{\ast}$ -algebras, Duke Math. J. 44 (1977), 329-355. MR 0438137 (55:11056)
[B1] B. Blackadar, K-theory for operator algebras, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, and Tokyo, 1987. MR 859867 (88g:46082)
[Br1] L. G. Brown, Stable isomorphism of hereditary subalgebras of $C^{\ast}$ -algebras, Pacific J. Math. 71 (1977), 335-348. MR 0454645 (56:12894)
[Br2] L. G. Brown, Semicontinuity and multipliers of $C^{\ast}$ -algebras, Canad. J. Math. 40 (1989), 769-887. MR 969204 (90a:46148)
[BDF1] L. G. Brown, R. G. Douglas, and P. A. Fillmore,, Unitary equivalence modulo the compact operators and extensions of $C^{\ast}$ -algebras, Proc. Conf. on Operator Theory, Lecture Notes in Math., vol. 345, Springer-Verlag, Heidelberg, 1977. MR 0380478 (52:1378)
[BDF2] -, Extensions of $C^{\ast}$ -algebras and K-homology, Ann. of Math. (2) 105 (1977), 265-324. MR 0458196 (56:16399)
[Co] A. Connes, Non-commutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1986), 257-360.
[Cu1] J. Cuntz, A class of $C^{\ast}$ -algebras and topological Markov chains II: Reducible chains and the $\operatorname{Ext}$ -functor for $C^{\ast}$ -algebras, Invent. Math. 63 (1981), 25-40. MR 608527 (82f:46073b)
[Cu2] -, K-theory for certain $C^{\ast}$ -algebras, Ann. of Math. (2) 131 (1981), 181-197.
[EK] E. G. Effros and J. Kaminker, Some homotopy and shape calculations for $C^{\ast}$ -algebras, Group Representations, Ergodic Theory, Operator Algebras, And Mathematical Physics, MSRI Publication No. 6, Springer-Verlag, New York, 1987. MR 880373 (88j:46064)
[E1] G. A. Elliott, Derivations of matroid $C^{\ast}$ -algebras. II, Ann. of Math. (2) 100 (1974), 407-422. MR 0352999 (50:5485)
[Ho] P. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, NJ, 1967. MR 0208368 (34:8178)
[Ka] M. Karoubi, K-theory: an introduction, Springer-Verlag, Berlin, Heidelberg, and New York, 1978. MR 0488029 (58:7605)
[Kas] G. G. Kasparov, Hilbert $C^{\ast}$ -modules: theorems of Stinespring and Voiculescu, J. Operator Theory 3 (1980), 133-150. MR 587371 (82b:46074)
[L] H. Lin, Simple $C^{\ast}$ -algebras with continuous scales and simple corona algebras, Proc. Amer. Math. Soc. 112 (1991), 871-880. MR 1079711 (92e:46118)
[MF] A. Miscenko and A. Fomenko, The index of elliptic operators over $C^{\ast}$ -algebras, Math. USSR Izv. 15 (1980), 87-112.
[Mi] J. A. Mingo, K-theory and multipliers of stable $C^{\ast}$ -algebras, Trans. Amer. Math. Soc. 299 (1987), 255-260. MR 869419 (88f:46136)
[Pe1] G. K. Pedersen, $SAW^{\ast}$ -algebras and corona $C^{\ast}$ -algebras, contributions to non-commutative topology, J. Operator Theory 15 (1986), 15-32. MR 816232 (87a:46095)
[Pe2] -, $C^{\ast}$ -algebras and their automorphism groups, Academic Press, London, New York, and San Francisco, 1979. MR 548006 (81e:46037)
[Ph] N. C. Phillips, Classifying algebras for the K-theory of $\sigma - C^{\ast}$ -algebras, Canad. J. Math. 41 (1989), 1021-1089. MR 1018451 (91h:46119)
[PS] A. Pressley and G. Segal, Loop groups, Oxford Science Publications, Clarendon Press, Oxford, 1986. MR 900587 (88i:22049)
[PPV] M. Pimsner, S. Popa, and D. Voiculescu, Homogeneous $C^{\ast}$ -extensions of $C(X) \otimes K(H)$ , J. Operator Theory 1 (1979), 55-108. MR 526291 (82e:46093a)
[OP] C. L. Olsen and G. K. Pedersen, Corona $C^{\ast}$ -algebras and their applications to lifting problems, Math. Scand. (to appear). MR 1036429 (91g:46064)
[SSU] A. Sheu, N. Salinas, and H. Upmerier, Toeplitz operators on pseudoconvex domains and foliation $C^{\ast}$ -algebras, Ann. of Math. (2) 130 (1989), 531-565. MR 1025166 (91e:47026)
[Ta] M. Takesaki, Theory of operator algebras. I, Springer-Verlag, Berlin, Heidelberg, and New York, 1979. MR 548728 (81e:46038)
[Zh1] S. Zhang, Certain $C^{\ast}$ -algebras with real rank zero and their corona and multiplier algebras, Part II, K-theory (to appear). MR 1186771 (94i:46094)
[Zh2] -, On the homotopy type of the unitary group and the Grassmann space of purely infinite simple $C^{\ast}$ -algebras, K-Theory (to appear). MR 1876798 (2002m:46088)
[Zh3] -, Exponential rank and exponential length of operators on Hilbert $C^{\ast}$ -module, Ann. of Math. (2) (to appear).
[Zh4] -, K-theory, K-skeleton factorizations and bi-variable index Index, Part I, Part II, Part III, preprints.
[Zh5] -, K-theory and bi-variable index Index(x, ${[p]_e}$ ): properties, invariants and applications, Part I, Part II, Part III, preprints.
[Zh6] -, K-theory and homotopy of certain groups and infinite Grassmann spaces associated with $C^{\ast}$ -algebra, preprint.
[Zh7] -, Torsion of K-theory, bi-variable index and certain invariants of the essential commutant of $M_{n}(C)$ . I, II, preprints.