Appears in collection : 2025 - T1 - Representation theory and noncommutative geometry

In the early 1980’s, the question was raised of comparing the K-theory groups of the full and reduced $C^*$-algebras of a group $G$. J. Cuntz has described a condition ($K$-amenability) implying that they are isomorphic. Note that $K$-amenability is incompatible with Kazhdan’s property $T$, and is implied by the Haagerup property, a strong negation of property $T$. In this talk we shall explain that Cuntz’ condition relies on the construction of a $G$-Fredhom module. We shall give such a module in the example of $\mathrm{SL}_2$ on the fields of $p$-adics (Julg-Valette), of complex numbers (Kasparov) and of real numbers (Fox-Haskel and Julg-Kasparov).