Dual Manifolds of 𝐽𝐵∗-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. ,. {^,. } and called a J^-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^-triple systems. A J^-triple is called positive (negative) if a certain class of operators have positive (negative) spectrum. The open unit ball of a positive J^-triple (U,. ,. {^,. }) is a bounded symmetric domain and the simply connected symmetric manifold associated to the negative triple (U, --. ,. {^,. }) is called the dual manifold of U. Let X be a compact Hausdorff space and U a JB^-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 Fₗ (M) =\f C (X, M) fis homotopic to a constant map\. We also show that the dual manifold of the JB^-triple C (X) admits no non-constant ℂ-valued holomorphic mappings and examine concretely the case X ⊂ ℝ.