From Wikipedia, the free encyclopedia
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,
the sequence
where ⊗min denotes the minimum
tensor product, is also exact.
Properties
- Every sub-C*-algebra and every quotient of an exact C*-algebra is exact. An extension of exact C*-algebras is not exact in general.
Characterizations
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 C*-algebra is exact if and only if every separable sub-C*-algebra is exact.
- A separable C*-algebra A is exact if and only if it is isomorphic to a subalgebra of the
Cuntz algebra .
References
- Brown, Nathanial P.; Ozawa, Narutaka (2008). C*-algebras and Finite-Dimensional Approximations. Providence: AMS.
ISBN
978-0-8218-4381-9.