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Smoothness Properties of the Unit Ball in a JB * ‐Triple
Author(s) -
Edwards C. Martin,
Rüttimann Gottfried T.
Publication year - 1996
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/28.2.156
Subject(s) - unit sphere , mathematics , norm (philosophy) , ball (mathematics) , sequence (biology) , element (criminal law) , combinatorics , pure mathematics , smoothness , mathematical analysis , chemistry , biochemistry , political science , law
An element a of norm one in a JB * ‐triple A is said to be smooth if there exists a unique element x in the unit ball A 1 * of the dual A * of A at which a attains its norm, and is said to be Fréchet‐smooth if, in addition, any sequence ( x n ) of elements in A 1 * for which ( x n ( a )) converges to one necessarily converges in norm to x . The sequence ( a 2 n +1 ) of odd powers of a converges in the weak * ‐topology to a tripotent u ( a ) in the JBW * ‐envelope A ** of A . It is shown that a is smooth if and only if u ( a ) is a minimal tripotent in A ** and a is Fréchet‐smooth if and only if, in addition, u ( a ) lies in A .