2008/04/02 by Rongwei Yang, Yang, Rongwei · 3 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Holomorphic and Operator Theory #math.FA
paper · pdf · doi:10.48550/arxiv.0804.0387
arxiv created 2008/04/02 · arxiv updated 2009/12/01
For a tuple A=(A0, A1, ..., An) of elements in a unital Banach algebra \mathcal B, its \em projective spectrum p(A) is defined to be the collection of z=[z0, z1, ..., zn]∈ \pn such that A(z)=z0A0+z1A1+... +znAn is not invertible in \mathcal B. The pre-image of p(A) in \ccn+1 is denoted by P(A). When \mathcal B is the k× k matrix algebra Mk(\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 Pc(A):=\ccn+1∖ 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 Pc(A). As a consequence, we show that if \mathcal B is a C^*-algebra with a trace ϕ, then ϕ(A-1(z)dA(z)) is a nontrivial element in the de Rham cohomology space H1d(Pc(A), \cc).