# Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 389-402.

Certain subalgebras of the tensor product of graph algebras**Authors**: Amy B. Chambers

**Author institution:**Department of Mathematics, Tennessee Technological University, Cookeville, TN, 38501, U.S.A.

**Summary:**Using an action of the unit circle, we construct a conditional expectation from the tensor product of two graph algebras, $C^*(E_1)\otimes C^*(E_2)$, onto a defined subalgebra $\mathcal{B}$. In addition, we make precise the required hypotheses for this subalgebra $\mathcal{B}$ to be isomorphic to the graph algebra $C^*( \mathcal{E})$ for the graph $\mathcal{E}$ defined using the Cartesian products of the vertex and edge sets of the graphs $E_1$ and $E_2$. We study two concrete examples of the conditional expectation constructed for the general case, and we discuss the ideas of index and Paschke crossed product by an endomorphism.

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