The Noncommutative Geometry of Graph $C^{\ast }$-Algebras I: The Index Theorem

Abstract

We investigate conditions on a graph $C^{\ast }$-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth $(1,\infty )$-summable semfinite spectral triple. The local index theorem allows us to compute the pairing with $K$-theory. This produces invariants in the $K$-theory of the fixed point algebra, and these are invariants for a finer structure than the isomorphism class of $C^{\ast }(E)$.

33 pages