Some classes of Banach analytic spaces (Q2821811)

The paper is motivated by results of \textit{V. Mascioni} and \textit{L. Molnár} [Can. Math. Bull. 41, No. 4, 434--441 (1998; Zbl 0919.47024)] with conclusive answers about linear maps between von Neumann factors preserving extreme points. The authors present counterexamples to show that similar conclusions need not be true in general for unital \(C^*\)-algebras, and in order to find appropriate extensions, they revisit the topics when maximal isometries are replaced with a natural generalization of Brown-Pedersen quasi-invertible elements in the setting of JB\(^*\)-triples. Recall that JB\(^*\)-triples are Banach spaces with holomorphically symmetric unit ball, or equivalently, Banach spaces equipped with a 3-variable product \(\{ x,y,z\}\) (like \(2\{ x,y,z\} = xy^*z+zy^*x\) for \(C^*\)-algebras) satisfying some familiar metric Jordan-triple \(^*\)-algebra axioms established by \textit{W. Kaup} [Math. Z. 183, 503--529 (1983; Zbl 0519.32024)]. An element \(x\) in a JB\(^*\)-triple \((E, \{ \cdot, \cdot, \cdot\})\) is Brown-Pedersen quasi-invertible if \(B(x, y) = 0\) for some (not necessarily unique) \(y \in E\) in terms of the Bergman operator \(B(x,y)= \text{Id}_E - 2L(x,y) + Q(x)Q(y)\) with the linear and quadratic representations \(L(a,b):z\mapsto \{ a,b,z\}\), \(Q(a):z\mapsto \{ a,z,a\}\). In particular, the von Neumann regular elements (defined as those \(a\in E\) admitting a necessarily unique generalized inverse \(a^\wedge\) satisfying \(Q(a)(a^\wedge) = a, Q(a^\wedge)(a) = a^\wedge\) and \(Q(a)Q(a^\wedge) = Q(a^\wedge)Q(a)\)) are all Brown-Pedersen quasi-invertible. NEWLINENEWLINENEWLINENEWLINEAccording to the main results, given two JB\(^*\)-triples \(E\) and \(F\) where the unit ball of \(E\) admits extreme points, a linear map \(T:E\to F\) between two JB\(^*\)-triples preserving Braun-Pedersen quasi-invertibility and being a \(^\wedge\)-homomorphism for von Neumann regular elements (``strongly preserving Braun-Pedersen quasi-invertible elements'' in the terminology of the authors) is a homomorphism for the triple products. If \(E\) is weakly compact, all bounded linear maps strongly preserving von Neumann regularity are triple homomorphisms. In the case of unital \(C^*\)-algebras, a linear map \(T:A\to B\) strongly preserving Braun-Pedersen quasi-invertibility can be written in the form \(T(x)=T(1)S(x)\) \((x\in A)\) by means of a suitable unital Jordan \(^*\)-homomorphism \(S:A\to B\).