Local facial structure and norm-exposed faces of the unit ball in a JB⁎JB⁎-triple

Each tripotent u   in the bidual A⁎⁎A⁎⁎ of a JB⁎JB⁎-triple A, compact relative to A  , gives rise to a complex Banach space P2(u)AP2(u)A with bidual the JBW⁎JBW⁎-algebra A2⁎⁎(u) that is a weak⁎weak⁎-closed subtriple of A⁎⁎A⁎⁎. Using the fact that the family of tripotents in the JBW⁎JBW⁎-algebra A2⁎⁎(u), compact relative to P2(u)AP2(u)A, coincides with the family of tripotents in A2⁎⁎(u), compact relative to A  , the facial structure of the unit balls in P2(u)AP2(u)A and its dual A2⁎⁎(u)⁎, the predual of A2⁎⁎(u), is revealed. The JBW-algebra Bu⁎⁎, that is the self-adjoint part of the JBW⁎JBW⁎-algebra A2⁎⁎(u), is the bidual of the unital GM-space BuBu that consists of the space of all real-valued weak⁎weak⁎-continuous affine functions on the normal state space of A2⁎⁎(u). It is therefore possible to investigate the ‘real’ local facial structure of the unit balls in BuBu, Bu⁎, and Bu⁎⁎. Some of the results obtained are applied to the problem of identifying the norm-exposed faces of the unit ball A1A1 in A  . It is shown that both locally and globally the norm-exposed faces of A1A1 are those corresponding to compact support tripotents of elements of norm one in the dual space A⁎A⁎ of A.