nLab KK-bootstrap category





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Special notions


Extra structure



Operator algebra

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The thick subcategory

KK bootKK KK_{boot} \hookrightarrow KK

of the triangulated category KKKK 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 KUModKU Mod

KK bootHo(KUMod). 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.


See (Blackadar, def. 22.3.4).

Examples of objects in the bootstrap category


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).


Künneth theorem

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

Embedding into KUKU-module spectra


On the bootstrap category the operator K-theory spectrum functor KKHo(KUMod)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.


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

The corresponding localizing subcategories are discussed in

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

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)

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

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

Last revised on August 15, 2013 at 06:17:17. See the history of this page for a list of all contributions to it.