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

We introduce the Jordan-strict topology on the multipliers algebra of a JB*-algebra. 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 A is J-strict dense in its multipliers algebra $M(A)$, and that the 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 show that, under some additional hypothesis this extension is surjective.