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Projectivity, transitivity and AF-telescopes


Authors: Terry A. Loring and Gert K. Pedersen
Journal: Trans. Amer. Math. Soc. 350 (1998), 4313-4339
MSC (1991): Primary 46L05
DOI: https://doi.org/10.1090/S0002-9947-98-02353-8
MathSciNet review: 1616003
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Abstract: Continuing our study of projective $C^{*}$-algebras, we establish a projective transitivity theorem generalizing the classical Glimm-Kadison result. This leads to a short proof of Glimm's theorem that every $C^{*}$-algebra not of type I contains a $C^{*}$-subalgebra which has the Fermion algebra as a quotient. Moreover, we are able to identify this subalgebra as a generalized mapping telescope over the Fermion algebra. We next prove what we call the multiplier realization theorem. This is a technical result, relating projective subalgebras of a multiplier algebra $M(A)$ to subalgebras of $M(E)$, whenever $A$ is a $C^{*}$-subalgebra of the corona algebra $C(E)=M(E)/E$. We developed this to obtain a closure theorem for projective $C^{*}$-algebras, but it has other consequences, one of which is that if $A$ is an extension of an MF (matricial field) algebra (in the sense of Blackadar and Kirchberg) by a projective $C^{*}$-algebra, then $A$ is MF. The last part of the paper contains a proof of the projectivity of the mapping telescope over any AF (inductive limit of finite-dimensional) $C^{*}$-algebra. Translated to generators, this says that in some cases it is possible to lift an infinite sequence of elements, satisfying infinitely many relations, from a quotient of any $C^{*}$-algebra.


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Additional Information

Terry A. Loring
Affiliation: Department of Mathematics, University of New Mexico, Albuquerque, New Mexico 87131
Email: loring@math.unm.edu

Gert K. Pedersen
Affiliation: Mathematics Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
Email: gkped@math.ku.dk

DOI: https://doi.org/10.1090/S0002-9947-98-02353-8
Keywords: Projectivity, transitivity, multipliers, telescopes, Bratteli diagram, Glimm's theorem, MF algebra
Received by editor(s): November 7, 1994
Additional Notes: This research was made possible through a NATO Collaboration Grant (# 920177). Both authors also acknowledge the support of their respective science foundations: NFS (# DMS–9215024) and SNF; and the second author recalls with gratitude the hospitality offered (twice!) by the Department of Mathematics at the University of New Mexico.
Article copyright: © Copyright 1998 American Mathematical Society

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