PROJECTIVE SPECTRUM IN BANACH ALGEBRAS

Abstract

For a tuple A = (A 1 , A 2 , …, A n ) of elements in a unital algebra [Formula: see text] over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂ n , or respectively z ∈ ℙ n-1 such that A(z) = z 1 A 1 + z 2 A 2 + ⋯ + z n A n is not invertible in [Formula: see text]. In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case [Formula: see text] is reflexive or is a C*-algebra, the projective resolvent setP c (A) := ℂ n \ P(A) is shown to be a disjoint union of domains of holomorphy. [Formula: see text]-valued 1-form A -1 (z)dA(z) reveals the topology of P c (A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology [Formula: see text] is established.