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 $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ 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 $C^*$-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 $C^*$-algebra with a trace $\phi$, then $\phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), \cc)$.