Isomorphism and Morita equivalence of graph algebras

Abstract

For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^{\ast }(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$. We prove that in certain situations isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^{\ast }(E)$ and $C^{\ast }(F)$. Conversely, we show that *-isomorphisms between $C^{\ast }(E)$ and $C^{\ast }(F)$ produce isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ in specific cases. The relationship between Leavitt path algebras and graph C*-algebras is also explored in the context of Morita equivalence.

40 pages, Version II comments: Section 10 updated, typos corrected, and a few proofs rewritten

References in corpus (2)

• $K$-theory of Leavitt path algebras
• Continuity of ring *-homomorphisms between C*-algebras

Cited by in corpus (2)

• The matrix type of purely infinite simple Leavitt path algebras
• The graded Grothendieck group and the classification of Leavitt path algebras