The fixed point property in \(JB^*\)-triples and preduals of \(JBW^*\)-triples
We recall that a complex \(JB^*\)-triple is a complex Banach space \(X\) equipped with a continuous triple product \(\{\cdot,\cdot,\cdot\}: X\times X\times X\to X\), \((x,y,z)\mapsto\{x,y,z\}\), which is billinear and symmetric in the outer variables and conjugate linear in the middle one and satisfies: (a) (Jordan identity) \(L(x, y)\{a,b,c\}= \{L(x,y)a, b, c\}- \{a,L(x, y)b, c\}+ \{a, b, L(x,y)c\}\) for all \(x,y,a,b, c\in X\), where \(L(x, y): X\to Y\) is the linear mapping given by \(L(x, y)z= \{x, y, z\}\); (b) the map \(L(x, y)\) is a Hermitian operator \((\|\exp(irL(x, x))\|=1\) for every \(r\in\mathbb{R}\)) with nonnegative spectrum for all \(x\in X\); (c) \(\|\{x,x,x\}\|=\| x\|^3\) for all \(x\in X\). By a real or complex \(JBW^*\)-triple, we mean a real or complex \(JB^*\)-triple which is also a dual Banach space whose triple product is separately weak\(^*\)-continuous. Every real Hilbert space, the Banach space of all bounded linear operators between real Hilbert spaces are examples of real \(JB^*\)-triples [cf.\ \textit{J.\,M.\thinspace Isidro}, \textit{W.\,Kaup} and \textit{A.\,Rodríguez Palacios}, ``On the real forms of \(JB^*\)-triples'', Manuscr.\ Math.\ 86, No.\,3, 311--335 (1995; Zbl 0834.17047)]. In the present paper, the authors prove that, for every member \(X\) in the class of real or complex \(JB^*\)-triples or preduals of \(JBW^*\)-triples, the following assertions are equivalent: {\parindent7mm \begin{itemize}\item[(1)] \(X\) has the fixed point property, \item[(2)] \(X\) has the super fixed point property, \item[(3)] \(X\) has normal structure, \item[(4)] \(X\) has uniform normal structure, \item[(5)] the Banach space of \(X\) is reflexive. \end{itemize}} As a consequence, a real or complex \(C^*\)-algebra or the predual of a real or complex \(W^*\)-algebra having the fixed point property must be finite-dimensional.
scientific article; zbMATH DE number 5602711
scientific article