Dual Manifolds of 𝐽⁢𝐵∗-Triples of the Form C(X, U)

Symmetric manifolds which are biholomorphically equivalent to a bounded domain in a Banach space are called bounded symmetric domains. It has been shown that every bounded symmetric domain is biholomorphically equivalent to the open unit ball in a Banach space endowed with a triple product {.,. ${}^{\ast}$,.} and called a $J^{\ast}$-triple system. This is the closest we get to a Riemann Mapping type theorem in higher dimensions. Kaup has shown that the category of simply connected symmetric manifolds is equivalent to the category of $J^{\ast}$-triple systems. A $J^{\ast}$-triple is called positive (negative) if a certain class of operators have positive (negative) spectrum. The open unit ball of a positive $J^{\ast}$-triple (U, {.,.${}^{\ast}$,.}) is a bounded symmetric domain and the simply connected symmetric manifold associated to the negative triple (U, --{.,.${}^{\ast}$,.}) is called the dual manifold of U. Let X be a compact Hausdorff space and U a $JB^{\ast}$-triple system. If the dual manifold of U is M, we show that the corresponding dual manifold of C(X, U) is the universal covering of $\scr{F}_{x}(M)=\{f\in C(X,M)\colon f\text{is homotopic to a constant map}\}$. We also show that the dual manifold of the $JB^{\ast}$-triple C(X) admits no non-constant ℂ-valued holomorphic mappings and examine concretely the case X ⊂ ℝ.