Introduction to graph C*-algebras and tensor products of graph algebras

Abstract

This thesis explores some of the basic concepts of C*-algebras in functional analysis. Also a selection from the first few chapters of Iain Raeburn's book “Graph Algebras” are worked out in full detail introducing graph C*-algebras. Graph C*-algebras are explained here in terms of Cuntz-Krieger families of bounded linear operators on a Hilbert space. From this it is shown that graph algebras are linearly generated and have a universality property.

Two important uniqueness theorems in relation to graph C*-algebras are also presented: the Cuntz-Krieger uniqueness theorem and the gauge-invariant uniqueness theorem. Tensor products of graph C*-algebras are introduced briefly with a focus in finding C*-subalgebras of the tensor product C*-algebra formed by tensoring two graph algebras as well as analyzing when these certain subalgebras are isomorphic to the C*-algebra made by the Cartesian product graph E₁ x E₂.