Maps on idempotents

Tom 169 / 2005

Peter Šemrl Studia Mathematica 169 (2005), 21-44 MSC: 06A06, 47B49. DOI: 10.4064/sm169-1-2


Let $X$ be an infinite-dimensional real or complex Banach space, $B(X)$ the algebra of all bounded linear operators on $X$, and $P(X)\subset B(X)$ the subset of all idempotents. We characterize bijective maps on $P(X)$ preserving commutativity in both directions. This unifies and extends the characterizations of two types of automorphisms of $P(X)$, with respect to the orthogonality relation and with respect to the usual partial order; the latter have been previously characterized by Ovchinnikov. We also describe bijective orthogonality preserving maps on the set of idempotents of a fixed finite rank. As an application we present a nonlinear extension of the structural result for bijective linear biseparating maps on $B(X)$.


  • Peter ŠemrlDepartment of Mathematics
    University of Ljubljana
    Jadranska 19
    SI-1000 Ljubljana, Slovenia

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