Introduction

Abstract

In this work we present some techniques and results which lead to new invariants of C*-algebras. The fundamental organizational principle of C*-homotopy theory infers there exists a homotopy theory of C*-algebras determined by short exact sequences, matrix invariance and by complex-valued functions on the topological unit interval. We shall make this precise by constructing model structures on certain spaces which are built up of C*-algebras in much the same way as every natural number acquires a prime factorization. Our approach combines a new take on C*-algebras dictated by category theory and the recently perfected homotopy theory of cubical sets. The idea of combining C*-algebras and cubical sets into a category of “cubical C*-spaces” may perhaps be perceived as quite abstract on a first encounter. However, these spaces arise naturally from a homotopy theoretic viewpoint. We observe next the failure of a more straightforward topological approach.