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

Abstract

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 $\mathfrak {A}$ the identities

$$\begin{aligned}{} & {} \lim _{n\rightarrow \infty } \left( e^{\frac{a}{n}}\circ e^{\frac{b}{n}} \right) ^{n} = e^{a+b},\ \lim _{n\rightarrow \infty } \left( U_{e^{\frac{a}{n}}} \left( e^{\frac{b}{n}}\right) \right) ^{n} = e^{2 a+b}, \hbox { and }\\{} & {} \lim _{n\rightarrow \infty } \left( U_{e^{\frac{a}{n}},e^{\frac{c}{n}}} \left( e^{\frac{b}{n}}\right) \right) ^{n} = e^{a+b + c} \end{aligned}$$

hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in $\mathfrak {A}$. These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals $f:\mathfrak {A}\rightarrow \mathbb {C}$ satisfying $f(U_x (y))=U_{f(x)}f(y),$ for all $x,y\in \mathfrak {A}$. We prove that for any such a functional f,  there exists a unique continuous (Jordan-) multiplicative linear functional $\psi :\mathfrak {A}\rightarrow \mathbb {C}$ such that $f(x)=\psi (x),$ for every x in the connected component of the set of all invertible elements of $\mathfrak {A}$ containing the unit element. If we additionally assume that $\mathfrak {A}$ is a JB$^*$-algebra and f is continuous, then f is a linear multiplicative functional on $\mathfrak {A}$. The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Schulz, and Touré.