Noncommutative simplicial complexes and the Baum—Connes conjecture

We associate a noncommutative C ^*-algebra with every locally finite simplicial complex.We determine the K-theory of these algebras and show that they can be used to obtain a conceptual explanation for the Baum—Connes conjecture. Our main technical result determines the algebra generated by the coefficients of a universal projection in an algebraic crossed product by a discrete group , in terms of such a noncommutative algebra associated with a simplicial complex defined by . Submitted: May 2001, Revised: October 2001, Revised: April 2002.