Projective spectrum in Banach algebras

TL;DR

This paper explores the properties of the projective spectrum in Banach algebras, analyzing its structure, especially in infinite-dimensional and special algebra cases, and investigates related differential forms and cohomology.

Contribution

It introduces the concept of projective spectrum in Banach algebras, studies its geometric properties, and links it to differential forms and cohomology in the context of $C^*$-algebras.

Findings

In finite dimensions, projective spectrum forms a hypersurface.

In infinite dimensions, projective spectrum can be complex but retains some hypersurface-like properties.

For $C^*$-algebras, the projective resolvent set decomposes into domains of holomorphy.

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 $H^1_d(P^c(A), \cc)$.