Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T22:38:03.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential length and traces

Published online by Cambridge University Press:  14 November 2011

N.Christopher Phillips
N.Christopher Phillips
Affiliation:
Department of Mathematics, University of Oregon, Eugene OR 97403-1222, U.S.A.(current address); and Department of Mathematics, University of Georgia, Athens GA 30602, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a quantity called the reduced C* exponential rank rcel (A) of a C*-algebra A, which satisfies rcel (A) ≦ cel (A). We show that rcel (A) = ∞ whenever A has two distinct normalised traces which agree on K0(A), and we prove a partial converse. This gives some understanding of why cel (A) = π cer (A) for some C*-algebras A but not for others. We also characterise rcel (A) as the supremum of the rectifiable distances from unitaries in the identity component of the unitary group to the commutator subgroup of this component.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 13 - 29
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Blackadar, B.. A simple unital projectionless C *-algebra. J. Operator Theory 5 (1981), 63–71.Google Scholar
2Blackadar, B.. K-Theory for Operator Algebras, MSRI publications no. 5 (New York: Springer, 1986).CrossRefGoogle Scholar
3Blackadar, B., Dădărlat, M. and Rørdam, M.. The real rank of inductive limit C *-algebras, Math. Scand. 69 (1991), 211–216.CrossRefGoogle Scholar
4Blackadar, B. and Kumjian, A.. Skew products of relations and the structure of simple C *-algebras. Math. Z. 189(1985), 55–63.CrossRefGoogle Scholar
5Blackadar, B., Kumjian, A. and Rørdam, M.. Approximately central matrix units and the structure of noncommutative tori, K-Theory 6 (1992) 267–284.CrossRefGoogle Scholar
6Cuntz, J. and Pedersen, G. K.. Equivalence and traces on C*-algebras. J. Funct. Anal. 33 (1979), 135–164.CrossRefGoogle Scholar
7Harpe, P. de la and Skandalis, G.. Déterminant associé à une trace sur une algèbre de Banach. Ann. Inst. Fourier (Grenoble) 34 no. 1 (1984), 241–260.CrossRefGoogle Scholar
8Goodearl, K.. Riesz decomposition in inductive limit C*-algebras, Rocky Mountain J. Math. (to appear).Google Scholar
9Kuiper, N. H.. The homotopy type of the unitary group of Hilbert space. Topology 3 (1965), 19–30.CrossRefGoogle Scholar
10Lin, H.. Exponential rank of C *-algebras with real rank 0 and Brown-Pedersen's conjectures. J. Funct. Anal. 114 (1993), 1–11.CrossRefGoogle Scholar
11Phillips, N. C.. Simple C *-algebras with the property weak (FU). Math. Scand. 69 (1991), 127–151.CrossRefGoogle Scholar
12Phillips, N. C.. Approximation by unitaries with finite spectrum in purely infinite C *-algebras J. Funct. Anal. 120 (1994), 98–106.CrossRefGoogle Scholar
13Phillips, N. C.. How many exponentials? Amer. J. Math, (to appear).Google Scholar
14Phillips, N. C.. The rectifiable metric on the space of projections in a C *-algebra. Internat. J. Math 3 (1992), 679–698.Google Scholar
15Phillips, N. C.. Reduction of exponential rank in direct limits of C *-algebras. Canadian J. Math. (to appear).Google Scholar
16Phillips, N. C. and Ringrose, J. R.. Exponential rank in operator algebras. In Current Topics in Operator Algebras, eds. Araki, H. et al. , pp. 395–413 (Singapore: World Scientific, 1991).Google Scholar
17Pimsner, M. and Voiculescu, D.. Embedding the irrational rotation algebra into an AF algebra. J. Operator Theory 4 (1980), 201–210.Google Scholar
18Ringrose, J. R.. Exponential length and exponential rank in operator algebras. Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), 55–71.CrossRefGoogle Scholar
19Zhang, S.. On the exponential rank and exponential length of C *-algebras. J. Operator Theory (to appear).Google Scholar
20Zhang, S.. Exponential rank and exponential length of operators on Hilbert C *-modules. Ann. of Math 137 (1993), 129–144.CrossRefGoogle Scholar