{
  "id": "2010.00978",
  "version": "v1",
  "published": "2020-10-02T13:16:59.000Z",
  "updated": "2020-10-02T13:16:59.000Z",
  "title": "Orthogonality in Banach Spaces via projective tensor product",
  "authors": [
    "Kousik Dhara",
    "Narayan Rakshit",
    "Jaydeb Sarkar",
    "Aryaman Sensarma"
  ],
  "comment": "8 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "Let $X$ be a complex Banach space and $x,y\\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \\|x+\\lambda y\\|_{X} \\geq \\|x\\|_{X}$ for all $\\lambda \\in \\mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal to $y$ if and only if there exists a semi-inner product $\\varphi$ on $X$ such that $\\|\\varphi\\| = 1$, $\\varphi(x,x)=\\|x\\|^2$ and $\\varphi(x,y)=0$. A similar result holds for $C^*$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-02T13:16:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B28",
      "47A30",
      "46B20",
      "46B22",
      "47B01",
      "47L05"
    ],
    "keywords": [
      "orthogonality",
      "birkhoff-james orthogonal",
      "projective tensor product spaces",
      "complex banach space",
      "similar result holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}