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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 7th 2013
• (edited Aug 7th 2013)
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 14th 2013

I have expanded a little bit at bootstrap category and gave the entry this Idea-section:

The thick subcategory 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). Many if not most types of C*-algebras that “appear in practice” are actually in the bootstrap category, notably the groupoid convolution algebras of amenable Lie groupoids.

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. \ref{StrongMonoidalFullInclusionIntoKUMod} below), and so the KK-bootstrap category may also be thought of as a full subcategory of $KU Mod$.