Hermitian geometry on the resolvent set (II)

For an element A in a unital C*-algebra ℬ, the operator-valued 1-form ωa(z) = (z − A)−1dz is analytic on the resolvent set ρ(A), which plays an important role in the functional calculus of A. This paper defines a class of Hermitian metrics on ρ(A) through the coupling of the operator-valued (1, 1)-form ΩA = −ω*A Λ ωa with tracial and vector states. Its main goal is to study the connection between A and the properties of the metric concerning curvature, arc length, completeness and singularity. A particular example is when A is quasi-nilpotent, in which case the metric lives on the punctured complex plane ℂ {0}. The notion of the power set is defined to gauge the “blow-up” rate of the metric at 0, and examples are given to indicate a likely link with A’s hyper-invariant subspaces.