Hermitian Geometry on Resolvent Set

Abstract

For a tuple ${A} = ({A}_{1}, {A}_{2}, \ldots, {A}_{n})$ of elements in a unital Banach algebra $\mathcal{B}$, its projective joint spectrum P(A) is the collection of ${z} \in {\mathbb{C}}^{n}$ such that ${A}(z) = {z}_{1}{A}_{1} + {z}_{2}{A}_{2} + \cdots + {z}_{n}{A}_{n}$ is not invertible. It is known that the $\mathcal{B}$-valued 1-form ${\omega}_{A}(z) = {A}^{-1}(z){dA}(z)$ contains much topological information about the joint resolvent set P^c(A). This paper studies geometric properties of P^c(A) with respect to Hermitian metrics defined through the $\mathcal{B}$-valued fundamental form ${\Omega}_{A} = -{\omega}^{\ast}_{A} \wedge {\omega}_{A}$ and its coupling with faithful states φ on $\mathcal{B}$, i.e., φ(Ω_A). The connection between the tuple A and the metric is the main subject of this paper. In particular, it shows that the Kählerness of the metric is tied with the commutativity of the tuple, and its completeness is related to the Fuglede–Kadison determinant.