Endomorphisms of Leavitt Path Algebras

Directed graphs have played a prominent role as a tool for encoding information for certain classes of C*-algebras, particularly AF-algebras and Cuntz-Krieger algebras. These constructions have been generalized to a class of C*-algebras known as graph C*-algebras, which have found applications to several areas of C*-algebra theory. One prominent area of investigation has been the application of Elliott’s classification program to the class of graph C*-algebras. Rørdam was able to prove that K-theory invariants classify certain simple Cuntz-Krieger algebras, and this classification has been extended to broader classes of graph C*-algebras, including even certain non-simple cases. Another avenue for extending these classification results is to consider Leavitt path algebras, algebraic analogues of the graph C*-algebras, and ask to what extent K-theory groups can be used to classify them. This dissertation explores a specific, but important, aspect of the classification of Leavitt path algebras. In particular, we investigate the question of whether L(E2) and L(E2−) are *-isomorphic. We do this by examining the diagonal of a Leavitt path algebra, and producing methods to construct endomorphisms of Leavitt path algebras that take a given maximal abelian subalgebra (MASA) to the diagonal.