Hermitian geometry on resolvent set(I)

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.