nuclear C*-algebra

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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).

(…)

Every commutative C*-algebra is nuclear.

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

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.

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*

Last revised on September 26, 2016 at 18:48:11. See the history of this page for a list of all contributions to it.