Operator Theory – That Can't Be Right

January 30, 2026February 7, 2026

Idempotent completeness of KK (and why we care)

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 ${Hom (A, B) = KK(A, B)}$ are the KK-groups, may lack kernels and cokernels for an arbitrary morphism.

January 10, 2026January 29, 2026

K-theory of Fibonacci tilings of R

In the following post, we go through an example computation of the K-theory of the tiling space of the Fibonacci tiling of $\R$ . 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

$\{a, b\}$

is given by

September 25, 2024February 25, 2025

A note on torsion in K-theory

In this post we will look at the correspondence between torsion in cohomology of finite CW-complexes and torsion in their K-theory. The title could just as well have been “a use case for Atiyah-Hirzebruch spectral sequence” as this will be our primary tool.

March 22, 2022March 23, 2022

Theorem of the month – Bott Periodicity for operator K-theory

In this post we review the proof of the Bott periodicity theorem for C*-algebras, generalizing the original Bott periodicity theorem from homotopy theory (and topological K-theory).

March 2, 2021March 2, 2021

Exactness of the reduced crossed product functor

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July 24, 2018August 26, 2021

Hilbert C*-modules

An increasingly prominent tool in operator theory is the Hilbert C*-module, which are (loosely speaking) Hilbert spaces where the inner product takes values in a C*-algebra. The next level of generalization is that of Hilbert modules over locally C*-algebras (we briefly mentioned locally C*-algebras in this post), and much of the following theory extends to this setting as well.
Here I give the definition of a Hilbert C*-module and collect some of it’s properties, mostly as a reference for personal use. I will likely update this post with new material later on, hopefully without making it too bloated. The theory is now well developed in the literature so the proofs will kept to a bare minimum. For references I will mostly use [1] and [2].

Continue reading “Hilbert C*-modules”