Suvrajit Bhattacharjee
Groups, $C^{*}$-algebras and $K$-theory
Consider the first object in the title above, i.e., a group $G$, and suppose that there is a torsion element in $G$, i.e., an element $g$ of finite order, say, $n > 1$. For an $n$-th root of unity $\omega$, i.e., $\omega^{n}=1 \in \mathbb{C}$, the element \[ p=\frac{1}{n}\sum_{i=0}^{n-1}\omega^{i}g^{i} \] is an idempotent in the complex group algebra $\mathbb{C}G$, i.e., $p^{2}=p$. One then has the following conjecture which dates back at least to the first half of the last century: if $G$ is torsion free, i.e., if $G$ has no torsion element other than $e$, then $\mathbb{C}G$ has no idempotent other than $0$ and $1$. This seemingly algebraic statement is in fact of a deep geometric and analytic nature and providing the conjecture its natural home – which is the goal of this lecture series – will lead us to the rest of the objects in the title above. We will therefore spend the first one-third of the lectures in reviewing and collecting necessary information from the theory of $C^{*}$-algebras. We will equally spend another one-third of the lectures in doing the same for $K$-theory and $K$-homology. Having gathered the required background, the rest of the lectures will be devoted to the conjecture itself and in conveying the geometry hiding underneath.
We do not know whom the algebraic statement should be attributed to but its $C^{*}$-analogue is usually called the Kadison–Kaplansky conjecture. The geometric form – which is our aim for these lectures – is due, among many others, to Baum, Connes, Kasparov and Mishchenko, and, most importantly, Novikov.
Last but not the least, we will assume nothing more than rudimentary functional analysis for these lectures.
