Algebraic K-theory of stable operator algebras

Around 1978, it has been conjectured that algebraic and topological K-theories are isomorphic for stable complex $C^$*-algebras. This conjecture has been proved in 1990 by Suslin and Wodzicki, using the notion of H-unital ring due to Wodzicki, with other results due to Cuntz, Higson and Kasparov. In this lecture, we give a new proof of this theorem and extend it to real stable operator algebras. Besides the notion of H-unital ring, we use exact properties of suitable completed tensor product. We also use special Fourier series in order to define Bott elements in algebraic K-theory. This is joint work with Mariusz Wodzicki.