Exact C*-algebra
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In mathematics, an exact C*-algebra is a C*-algebra that preserves exact sequences under the minimum tensor product.
Definition
A C*-algebra E is exact if, for any short exact sequence,
- [math]\displaystyle{ 0 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C \;\xrightarrow{}\; 0 }[/math]
the sequence
- [math]\displaystyle{ 0\;\xrightarrow{}\; A \otimes_\min E\;\xrightarrow{f\otimes \operatorname{id}}\; B\otimes_\min E \;\xrightarrow{g\otimes \operatorname{id}}\; C\otimes_\min E \;\xrightarrow{}\; 0, }[/math]
where ⊗min denotes the minimum tensor product, is also exact.
Properties
Exact C*-algebras have the following equivalent characterizations:
- A C*-algebra A is exact if and only if A is nuclearly embeddable into B(H), the C*-algebra of all bounded operators on a Hilbert space H.
- A separable C*-algebra A is exact if and only if it is isomorphic to a subalgebra of the Cuntz algebra [math]\displaystyle{ \mathcal{O}_2 }[/math].
All nuclear C*-algebras and their C*-subalgebras are exact.
References
- Brown, Nathanial P.; Ozawa, Narutaka (2008). C*-algebras and Finite-Dimensional Approximations. Providence: AMS. ISBN 978-0-8218-4381-9.
Original source: https://en.wikipedia.org/wiki/Exact C*-algebra.
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