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

We introduce the Jordan-strict topology on the multiplier algebra of a JB ^* -algebra, a notion which was missing despite the forty 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 𝔄 is J-strict dense in its multiplier algebra M(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 multiplier algebras. We characterize J-strict continuous functionals on the multiplier algebra of a JB ^* -algebra 𝔄 , and we establish that the dual of M(𝔄) with respect to the J-strict topology is isometrically isomorphic to 𝔄^* . We also present a first application of the J-strict topology of the multiplier algebra, by showing that under the extra hypothesis that 𝔄 and 𝔅 are σ -unital JB ^* -algebras, every surjective Jordan ^* -homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from 𝔄 onto 𝔅 admits an extension to a surjective J-strict continuous Jordan ^* -homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from M(𝔄) onto M(𝔅) .