Lie-Trotter Formulae in Jordan-Banach Algebras with Applications to the Study of Spectral-Valued Multiplicative Functionals

We establish some Lie-Trotter formulae for unital complex Jordan-Banach algebras, showing that for any elements a, b, c in a unital complex Jordan-Banach algebra U the identities lim(n ->infinity) (e(a/n) circle e(b/n))(n) = e(a+b), lim(n ->infinity) (U-ea/n (e(b/n)))(n) = e(2a+b), and lim(n ->infinity) (U-ea/n,U-ec/n (e(b/n)))(n) = e(a+b+c) hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in U. These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals f : U -> C satisfying f(U-x(y)) = U-f(x)f(y), for all x, y is an element of U. We prove that for any such a functional f, there exists a unique continuous (Jordan-) multiplicative linear functional psi: U -> C such that f(x) = psi(x), for every x in the connected component of the set of all invertible elements of A containing the unit element. If we additionally assume that U is a JB*-algebra and f is continuous, then f is a linear multiplicative functional on U. The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Schulz, and Toure.