nLab nuclear C*-algebra

Idea

A type of C*-algebra.

Many equivalent definitions, not all of which were known to be equivalent when they were made. Usually credited to Takesaki (1964). An important characterization in terms of an appropriate “approximation property” was given by Lance (1972).

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Example

Every commutative C*-algebra is nuclear.

If $H$ is an infinite-dimensional Hilbert space then $B(H)$ is not nuclear.

Properties

Relation to KK-theory

On K-nuclear $C^\ast$-algebras $C$, the KK-theory functor $KK(C,-)$ preserves short exact sequences in the middle (satisfies excision). (Kasparov 80, Skandalis 88).

Hence restricted to nuclear $C*$-algebras the canonical functor $KK \to E$ from KK-theory to E-theory (see there) is a full and faithful functor.

References

The notion was studied as a condition for excision in KK-theory in section 7 of

• Gennady Kasparov, The operator $K$-functor and extensions of $C^{\ast}$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719, MR81m:58075, Zbl, abstract, english doi, free Russian original: pdf

The generalization to K-nuclearity was introduced and discussed in

• Georges Skandalis, Une notion de nuclearité en K-theorie, K-Theory 1 (1988) 549-574.

• Wikipedia, Nuclear C-star-algebra