Английская Википедия: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>0 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C \;\xrightarrow{}\; 0</math>

the sequence

<math>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

• Every nuclear C*-algebra is exact.

• 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.

• It follows that every sub-C*-algebra of a nuclear C*-algebra is exact.

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 <math>\mathcal{O}_2</math>.

References

• Шаблон:Cite book