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 each couple of elements $a,b$ in a unital complex Jordan--Banach algebra $\mathfrak{A}$ the identities $$\underset{n\to \infty }{\lim }{\left(e^{\frac{a}{n}}\circ e^{\frac{b}{n}}\right)}^n=e^{a+b},\underset{n\to \infty }{\lim }{\left(U_{e^{\frac{a}{n}}}\left(e^{\frac{b}{n}}\right)\right)}^n=e^{2a+b},and$$ $$\underset{n\to \infty }{\lim }{\left(U_{e^{\frac{a}{n}},e^{\frac{c}{n}}}\left(e^{\frac{b}{n}}\right)\right)}^n=e^{a+b+c}$$ 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 $f:\mathfrak{A}\to \mathbb{C}$ 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 $\psi :\mathfrak{A}\to \mathbb{C}$ such that $f(x)=\psi (x),$ for every $x$ in the connected component of set of all invertible elements of $\mathfrak{A}$ containing the unit element. If we additionally assume that $\mathfrak{A}$ is a JB${}^{\ast }$-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, Shulz, and Tour{é}.