{
  "id": "1312.6569",
  "version": "v1",
  "published": "2013-12-23T15:23:22.000Z",
  "updated": "2013-12-23T15:23:22.000Z",
  "title": "Projective Spectrum and Cyclic Cohomolgy",
  "authors": [
    "Patrick Cade",
    "Rongwei Yang"
  ],
  "journal": "Projective Spectrum and Cyclic Cohomology (with R. Yang), Journal of Functional Analysis, 265 (2013), pp. 1916-1933",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a tuple $A=(A_1,\\ A_2,\\ ...,\\ A_n)$ of elements in a unital algebra ${\\mathcal B}$ over $\\mathbb{C}$, its {\\em projective spectrum} $P(A)$ or $p(A)$ is the collection of $z\\in \\mathbb{C}^n$, or respectively $z\\in \\mathbb{P}^{n-1}$ such that the multi-parameter pencil $A(z)=z_1A_1+z_2A_2+\\cdots +z_nA_n$ is not invertible in ${\\mathcal B}$. ${\\mathcal B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological information about $P^c(A):=\\mathbb{C}^n\\setminus P(A)$. In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of ${\\mathcal B}$ does a similar job. In fact, a Chen-Weil type map $\\kappa$ from the cyclic cohomology of ${\\mathcal B}$ to the de Rham cohomology $H^*_d(P^c(A),\\ \\mathbb{C})$ is established. As an example, we prove a closed high-order form of the classical Jacobi's formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-23T15:23:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13"
    ],
    "keywords": [
      "projective spectrum",
      "cyclic cohomolgy",
      "cyclic cohomology",
      "invariant multi-linear functionals",
      "chen-weil type map"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.6569C"
    }
  }
}