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 be a unital
-algebra. We describe K-skeleton factorizations of all invertible operators on a Hilbert
-module
, in particular on
, with the Fredholm index as an invariant. We then outline the isomorphisms
and
for
, where
denotes the class of all compact perturbations of a projection p in the infinite Grassmann space
and
stands for the group of all those invertible operators on
essentially commuting with p.
- [APT]
C. A. Akemann, G. K. Pedersen, and J. Tomiyama, Multipliers of
-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
-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
-algebras, Pacific J. Math. 71 (1977), 335-348. MR 0454645 (56:12894)
- [Br2]
L. G. Brown, Semicontinuity and multipliers of
-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
-algebras, Proc. Conf. on Operator Theory, Lecture Notes in Math., vol. 345, Springer-Verlag, Heidelberg, 1977. MR 0380478 (52:1378)
- [BDF2]
-, Extensions of
-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
-algebras and topological Markov chains II: Reducible chains and the
-functor for
-algebras, Invent. Math. 63 (1981), 25-40. MR 608527 (82f:46073b)
- [Cu2]
-, K-theory for certain
-algebras, Ann. of Math. (2) 131 (1981), 181-197.
- [EK]
E. G. Effros and J. Kaminker, Some homotopy and shape calculations for
-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
-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
-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 3 (1980), 133-150. MR 587371 (82b:46074)
- [L]
H. Lin, Simple
-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
-algebras, Math. USSR Izv. 15 (1980), 87-112.
- [Mi]
J. A. Mingo, K-theory and multipliers of stable
-algebras, Trans. Amer. Math. Soc. 299 (1987), 255-260. MR 869419 (88f:46136)
- [Pe1]
G. K. Pedersen,
-algebras and corona
-algebras, contributions to non-commutative topology, J. Operator Theory 15 (1986), 15-32. MR 816232 (87a:46095)
- [Pe2]
-,
-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
-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
-extensions of
, J. Operator Theory 1 (1979), 55-108. MR 526291 (82e:46093a)
- [OP]
C. L. Olsen and G. K. Pedersen, Corona
-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
-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
-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
-algebras, K-Theory (to appear). MR 1876798 (2002m:46088)
- [Zh3]
-, Exponential rank and exponential length of operators on Hilbert
-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,
): 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
-algebra, preprint.
- [Zh7]
-, Torsion of K-theory, bi-variable index and certain invariants of the essential commutant of
. I, II, preprints.
Retrieve articles in Bulletin of the American Mathematical Society with MSC: 46L80, 19K33, 46L05, 46M20
Retrieve articles in all journals with MSC: 46L80, 19K33, 46L05, 46M20
Additional Information
DOI:
https://doi.org/10.1090/S0273-0979-1993-00334-3
Article copyright:
© Copyright 1993
American Mathematical Society