| If E is a directed graph, the graph algebra C*(E) is the universal C*-algebra generated by a family of partial isometrics and projections corresponding to the edges and vertices of the graph E satisfying certain relations that makes it a Cuntz-Krieger E-system. Graph algebras have been much studied in the last ten years by D. Pask, A. Kumjian, and I. Raeburn and have proved useful in the general structure theory of C*-algebras. At the same time, the research area of C*-subalgebras of C*-algebras and the concept of index was introduced by Y. Watatani, motivated by the success of V. Jones for von Neumann algebras. Watatani developed the notions of a conditional expectation of a C*-algebra onto a C*-subalgebra and the index of a conditional expectation.; In this thesis, we find a conditional expectation from the tensor product of two graph algebras, C*(E1) ⊗ C*(E2), onto the subalgebra B = span{lcub}S muSnu* ⊗ S˜ alphaS˜beta* : mu, nu are paths in E1 with the same source and alpha, beta are paths in E2 with the same source and |mu| - |nu| = |alpha| - |beta|{rcub}. Using an action of the unit circle T on C*(E1) ⊗ C*(E2), we prove that there always exists a conditional expectation from C*(E 1) ⊗ C*(E2) onto B. In addition, we make precise the required hypotheses for this subalgebra B to be isomorphic to the graph algebra C*( E ) for the graph E defined using the Cartesian products of the vertex and edge sets of the graphs E1 and E2. Finally, we apply our results to two examples of conditional expectations, one of index-finite type and one not of index-finite type. |
