Projective spectrum in Banach algebras

For a tuple $A=(A_0,A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n)$ of elements in a unital Banach algebra $\mathcal{B}$, its {\em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0,z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n]\in \text{pn}$ such that $A(z)=z_0A_0+z_1A_1+… +z_nA_n$isnotinvertiblein${\mathcal B}$\mathrm{.}Thepre-imageof$p(A)$in${\cc}^{n+1}$isdenotedby$P(A)$\mathrm{.}When${\mathcal B}$isthe$k\times k$matrixalgebra$M_k(\cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When $\mathcal{B}$ is reflexive or is a $C^{\ast }$-algebra, the {\em projective resolvent set} $P^c(A):={\text{cc}}^{n+1}\setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${\mathcal B}$-valued1-form$A^{-1}(z)dA(z)$on$P^c(A)$. As a consequence, we show that if $\mathcal{B}$ is a $C^{\ast }$-algebra with a trace $\varphi$, then $\varphi (A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H_d^1(P^c(A),\text{cc})$.