# Lück’s Approximation Theorem

• Holger Kammeyer
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2247)

## Abstract

After motivating and presenting the statement of Lück’s approximation theorem, we prepare the proof with a detailed synopsis of spectral calculus for Banach-, C-, and von Neumann algebras. The proof is then presented in a manner that emphasizes the two main ingredients: weak convergence of spectral measures and the logarithmic bound on spectral distribution functions. We discuss in detail in how far some of the assumptions in the theorem can be relaxed. We explain the relation of the first 2-Betti number to rank gradient and cost and present a proof of the Abért–Nikolov theorem which builds a bridge between the two latter concepts. Next we discuss the approximation conjecture and show in what sense it is a consequence of the determinant conjecture. After proving the determinant conjecture for residually finite groups, we explain how the approximation conjecture gives further insight on the Atiyah conjecture and hence on the Kaplansky conjecture.

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