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Factorization in the Invertible Group of a C*-Algebra

Published online by Cambridge University Press:  20 November 2018

Michael J. Leen
Michael J. Leen*
Affiliation:
Department of Mathematics Whittier College Whittier, CA USA
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Abstract

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In this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

Keywords

46L05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 6 , 01 December 1997 , pp. 1188 - 1205
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Blackadar, B., K-Theory for Operator Algebras. MSRI Publications No. 5, Springer-Verlag, New York, 1986.Google Scholar
2. Brown, L.G. and Pedersen, G.K., C*-algebras of real rank zero. J. Functional Analysis 99(1991), 131–149.Google Scholar
3. Cuntz, J., The structure of multiplication and addition in simple C*-algebras. Math. Scand. 40(1977). 215–233.Google Scholar
4. Cuntz, J., K-theory for certain C*-algebras. Ann. of Math. 113(1981), 181–197.Google Scholar
5. Fack, T., Finite sums of commutators in C*-algebras. Ann. Inst. Fourier 32-1(1984), 169–202.Google Scholar
6. de la Harpe, P. and Skandalis, G., Produits finis de commutateurs dans les C*-algebras. Ann. Inst. Fourier 34-4(1984), 169–202.Google Scholar
7. Pedersen, G.K., C*-Algebras and their Automorphism Groups. Academic Press, London, 1979.Google Scholar
8. Phillips, N.C., The rectifiable metric on the space of projections in C*-algebra. International J. Math. 3(1992), 679–698.Google Scholar
9. Phillips, N.C., How many exponentials?. Amer J. Math., to appear.Google Scholar
10. Phillips, N.C., Factorization problems in the invertible group of a homogeneous C*-algebra. Preprint.Google Scholar
11. Phillips, N.C., A survey of exponential rank. Preprint.Google Scholar
12. Wu, P.Y., The operator factorization problems. Linear Alg. Appl. 117(1989), 35–63.Google Scholar
13. Zhang, S., Certain C*-algebras with real rank zero and their corona and multiplier algebras, Part I. Pac. J. Math. 155(1992), 169–197.Google Scholar