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Computation of the λ u-function in JB ∗-algebras

Motivated by the work of Gert K. Pedersen on a geometric function, which is defined on the unit ball of a C∗-algebra and called the λu-function, the present author recently initiated a study of the λufunction in the more general setting of JB∗-algebras. He used his earlier results on the geometry of the unit ball to investigate certain convex combinations of elements in a JB∗-algebra and to obtain analogues of some related C∗-algebra results, including a formula to compute λufunction on invertible elements in a JB∗-algebra. The main purpose in this article is to investigate the computation of the λu-function on noninvertible elements in the unit ball of a JB∗-algebra. Additional results that relate the λu-function to convex combinations, unitary rank, and distance to the invertibles in the C∗-algebra setting are generalized to the JB∗-algebra context. Results of G. K. Pedersen and M. Rørdam are generalized. An open problem is presented.