Projective spectrum in Banach algebras

Abstract

For a tuple $A=(A_0,A_1,...,A_n)$ of elements in a unital Banach algebra $\mathcal{B}$, its {\em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]\in \pn$ such that $A(z)=z_0{A}_0+z_1{A}_1+...+z_n{A}_n$ is not invertible in $\mathcal{B}$. The pre-image of $p(A)$ in ${\cc}^{n+1}$ is denoted by $P(A)$. When $\mathcal{B}$ is the $k\times k$ matrix algebra $M_k(\cc)$, 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} $P^c(A):=\cc^{n+1}\setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type $\mathcal{B}$-valued 1-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 $H^1_d(P^c(A), \cc)$.