Isomorphism and Morita equivalence of graph algebras

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