Hermitian geometry on resolvent set(I)

TL;DR

This paper explores the geometric properties of the joint resolvent set of tuples in Banach algebras using Hermitian metrics derived from the fundamental form, linking metric properties to algebraic features like commutativity.

Contribution

It introduces a geometric framework for analyzing joint spectra in Banach algebras using Hermitian metrics and establishes connections between metric properties and algebraic structures.

Findings

Kählerness of the metric is equivalent to the tuple's commutativity.

Completeness of the metric relates to the Fuglede-Kadison determinant.

The fundamental form encodes topological information about the joint resolvent set.

Abstract

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ 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 $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ 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 {\em fundamental form} $\Omega_A=-\omega^*_A\wedge \omega_A$ and its coupling with faithful states $\phi$ on ${\mathcal B}$, i.e. $\phi(\Omega_A)$. The connection between the tuple $A$ and the metric is the main subject of this paper. In particular, it shows that the K\"{a}hlerness of the metric is tied with the commutativity of the tuple, and its completeness is related to the Fuglede-Kadison determinant.