{
  "id": "0804.0387",
  "version": "v1",
  "published": "2008-04-02T16:00:25.000Z",
  "updated": "2008-04-02T16:00:25.000Z",
  "title": "Projective spectrum in Banach algebras",
  "authors": [
    "Rongwei Yang"
  ],
  "categories": [
    "math.FA"
  ],
  "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 $H^1_d(P^c(A), \\cc)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-02T16:00:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "projective spectrum",
      "paper studies maurer-cartan type",
      "infinite dimensional cases",
      "unital banach algebra",
      "rham cohomology space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.0387Y"
    }
  }
}