On the strict topology of the multipliers of a JB$^*$-algebra

Abstract

We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB$^*$-algebras admit J-strict continuous extensions to the corresponding type of operators between the multipliers algebras. We characterize J-strict continuous functionals on the multipliers algebra of a JB$^*$-algebra $\mathfrak{A}$, and we establish that the dual of $M(\mathfrak{A})$ with respect to the J-strict topology is isometrically isomorphic to $\mathfrak{A}^*$. We also present a first applications of the J-strict topology of the multipliers algebra, by showing that under the extra hypothesis that $\mathfrak{A}$ and $\mathfrak{B}$ are $\sigma$-unital JB$^*$-algebras, every surjective Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $\mathfrak{A}$ onto $\mathfrak{B}$ admits an extension to a surjective J-strict continuous Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $M(\mathfrak{A})$ onto $M(\mathfrak{B})$.