{
  "id": "1711.05652",
  "version": "v1",
  "published": "2017-11-15T16:29:29.000Z",
  "updated": "2017-11-15T16:29:29.000Z",
  "title": "On the unit sphere of positive operators",
  "authors": [
    "Antonio M. Peralta"
  ],
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "Given a C$^*$-algebra $A$, let $S(A^+)$ denote the set of those positive elements in the unit sphere of $A$. Let $H_1$, $H_2,$ $H_3$ and $H_4$ be complex Hilbert spaces, where $H_3$ and $H_4$ are infinite dimensional and separable. Let $E$ and $P$ be subsets of a Banach space $X$. The unit sphere around $E$ in $P$ is defined as the set $$Sph(E;P) :=\\left\\{ x\\in P : \\|x-b\\|=1 \\hbox{ for all } b\\in E \\right\\}.$$ In a first result we establish a geometric characterization of the projections in $B(H)$ by showing that an element $a\\in S(B(H_1)^+)$ is a projection if and only if $$Sph^+_{B(H)} \\left( Sph^+_{B(H)}(a) \\right) =\\{a\\}.$$ The same characterization holds when $B(H_1)$ is replaced with $K(H_3)$. This characterization is applied to establish a positive variant to Tingley's problem by showing that every surjective isometry $\\Delta : S(B(H_1)^+)\\to S(B(H_2)^+)$ or (respectively, $\\Delta : S(K(H_3)^+)\\to S(K(H_4)^+)$) admits a unique extension to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$ (respectively, from $K(H_3)$ onto $B(H_4)$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-15T16:29:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit sphere",
      "positive operators",
      "surjective complex linear isometry",
      "complex hilbert spaces",
      "unique extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}