Let’s talk about idempotent completeness!
This blog will deal with idempotent completeness, as it relates to the UCT class in KK-theory. For an outside observer (i.e. a non category theorist) like myself working mostly with operator algebras it may seem abelian categories as quite elusive. The usual categories of Banach/C*-algebras is not even additive even if we use completely positive maps as morphisms; the category of vector bundles, though additive, does not admit kernels; even kasparovs KK-category whose objects are separable C*-algebras, and whose morphisms 
are the KK-groups, may lack kernels and cokernels for an arbitrary morphism.
. The main reference for these notes will be the book of Sadun [1]. In particular pages 27-29 and Theorem 2.1. We adopt the following conventions: The Fibonacci substitution rule with alphabet ![\[\{a, b\}\]](https://www.mathblog.realhjelp.no/wp-content/ql-cache/quicklatex.com-1b08c47efc23aad37432ce088473ae91_l3.png "Rendered by QuickLaTeX.com")
towards irregular points on the geodesic boundary 
of a non-compact symmetric space