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On the Facial Structure of the Unit Balls in a JBW ∗ ‐Triple and its Predual
Author(s) -
Edwards C. M.,
Rüttimann G. T.
Publication year - 1988
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-38.2.317
Subject(s) - mathematics , unit sphere , lattice (music) , combinatorics , norm (philosophy) , unit (ring theory) , pure mathematics , partially ordered set , physics , philosophy , mathematics education , epistemology , acoustics
The set of tripotents in a JBW ∗ ‐triple U with its natural ordering and with a largest element adjoined is shown to be a complete lattice, order isomorphic to the lattice of norm closed faces in the unit ball U ∗1 of the predual U ∗ of U and anti‐order isomorphic to the lattice of weak ∗ closed faces of the unit ball U 1 in U . As a consequence, the set of partial isometries in a W ∗ ‐algebra with its natural ordering and again with a largest element adjoined forms a complete lattice and every non‐empty weak ∗ closed face of its unit ball is of the form u+(1−uu∗) U (1−u∗u) 1 for some unique partial isometry u .