Some Metric and Homotopy Properties of Partial Isometries
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Publication:5267513
DOI10.3318/PRIA.2016.116.08zbMath1379.46040arXiv1603.07002OpenAlexW2963879158MaRDI QIDQ5267513
Publication date: 13 June 2017
Published in: Mathematical Proceedings of the Royal Irish Academy (Search for Journal in Brave)
Abstract: We show that ||u*u - v*v|| leq ||u - v|| for partial isometries u and v. There is a stronger inequality if both u and v are extreme points of the unit ball of a C*-algebra, and both inequalities are sharp. If u and v are partial isometries in a C*-algebra A such that ||u - v|| < 1, then u and v are homotopic through partial isometries in A. If both u and v are extremal, then it is sufficient that ||u - v|| < 2. The constants 1 and 2 are both sharp. We also discuss the continuity points of the map which assigns to each closed range element of A the partial isometry in its canonical polar decomposition.
Full work available at URL: https://arxiv.org/abs/1603.07002
General theory of (C^*)-algebras (46L05) Special classes of linear operators (47B99) Linear operators in (C^*)- or von Neumann algebras (47C15)
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