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 | References | Similar Articles | Additional Information

Abstract: Continuing our study of projective -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 -algebra not of type I contains a -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 to subalgebras of , whenever is a -subalgebra of the corona algebra . We developed this to obtain a closure theorem for projective -algebras, but it has other consequences, one of which is that if is an extension of an MF (matricial field) algebra (in the sense of Blackadar and Kirchberg) by a projective -algebra, then 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) -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 -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