nLab KK-bootstrap category

Idea

The thick subcategory

$$KK_{boot} \hookrightarrow KK$$

of the triangulated category $KK$ of KK-theory generated by the tensor unit is called the bootstrap category (Rosenberg-Schochet 87), because on this subcategory KK-theory has a good axiomatic description (see Blackadar 22.3).

The canonical functor from KK to (the homotopy category of) KU-module spectra restricts to a full and faithful functor on the bootstrap category (prop. below), and so the KK-bootstrap category may also be thought of as a full subcategory of $KU Mod$

$$KK_{boot} \hookrightarrow Ho(KU Mod) \,.$$

Many C*-algebras that “appear in practice” are in the bootstrap category, notably the groupoid convolution algebras of amenable Lie groupoids (prop. below). Generally, a C*-algebra is in the bootstrap category precisely if it is KK-equivalent to a commutative C*-algebra.

Definition

See (Blackadar, def. 22.3.4).

Examples of objects in the bootstrap category

Proposition

The groupoid convolution algebra of an amenable topological groupoid is in the bootstrap category.

This is due to (Tu 99, prop. 10.7), recalled for instance as (Uuye 11, example 3.6).

Properties

Künneth theorem

In the bootstrap category a Künneth theorem for operator K-theory is true. (Rosenberg-Schochet 87).

Embedding into $KU$-module spectra

Proposition

On the bootstrap category the operator K-theory spectrum functor $KK \to Ho(KU Mod)$ is

1. a fully faithful functor,

2. a strong monoidal functor

The first statement is (DEKM 11, section 3), the second is (DEKM 11, prop. 4.2).

See at KK-theory the section Triangulated and spectrum-enriched structure for more details.

References

The notion originates in

• Jonathan Rosenberg, Claude Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55 (1987), no. 2, 431–474.

A texbook account is around def. 22.3.4 of

• Bruce Blackadar, K-Theory for Operator Algebras

The corresponding localizing subcategories are discussed in

• Ivo Dell'Ambrogio, Localizing subcategories in the Bootstrap category of separable C-algebras_, Journal of K-Theory 8 (2011), 493-50 (arXiv:1003.0183)

The full embedding into the homotopy category of KU-modules is discussed in section 3 of

• Ivo Dell'Ambrogio, Heath Emerson, Tamaz Kandelaki, Ralf Meyer, A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula (arXiv:1104.3441)

The inclusion of groupoid convolution algebras of amenable topological groupoids into the bootstrap category is discussed in

• Jean-Louis Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, K-Theory 17 (1999), no. 3, 215–264. MR 1703305 (2000g:19004)

• Otgonbayar Uuye, A note on the Künneth theorem for nonnuclear $C^\ast$-algebras (arXiv:1111.7228)

Discussion of the bootstrap category in equivariant K-theory is in

• Heath Emerson, Localization techniques in circle-equivariant KK-theory (arXiv:1004.2970)

This show that there are algebras in the plain bootstrap category with circle group-equivariant structure which are not $U(1)$-equivariant KK-equivalent to a commutative $U(1)$-$C^\ast$-alegbra.