Some characterizations of dualizing complexes in terms of $G_C$-dimension

Let $(R,\text{fm})$ be a local ring and $C$ be a homologically bounded and finitely generated $R$-complex. Then, we prove that $C$ is a dualizing complex of $R$ if and only if $C$ is a Cohen-Macaulay semidualizing complex of type one or ${\mu }_R^{\inf C+{\dim }_R(C)}(\text{fm},R)={\beta }_{\inf C}^R(C)$. Also, we show that a semidualizing complex $C$ is dualizing if and only if there exists a type one Cohen-Macaulay $R$-module of finite $G_C$-dimension or there exists a type one Cohen-Macaulay $R$-complex of finite $G_C$-dimension such that ${\dim }_R(X)={\dim }_R(C)-{\text{gr}}_C(X)$. Furthermore, for a semidualizing $R$-complex $C$, we prove that $C\sim R$ if and only if there exists a type one Cohen-Macaulay $R$-module $M$ which belongs to the Auslander class ${\mathcal{A}}_C(R)$.