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

Abstract

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 $${\mathfrak {A}}$$ A is J-strict dense in its multiplier algebra $$M({\mathfrak {A}})$$ 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 $${\mathfrak {A}}$$ A , and we establish that the dual of $$M({\mathfrak {A}})$$ M ( A ) with respect to the J-strict topology is isometrically isomorphic to $${\mathfrak {A}}^*$$ A ∗ . We also present a first application of the J-strict topology of the multiplier algebra, by showing that under the extra hypothesis that $${\mathfrak {A}}$$ A and $${\mathfrak {B}}$$ B are $$\sigma $$ σ -unital JB$$^*$$ ∗ -algebras, every surjective Jordan $$^*$$ ∗ -homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $${\mathfrak {A}}$$ A onto $${\mathfrak {B}}$$ B admits an extension to a surjective J-strict continuous Jordan $$^*$$ ∗ -homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $$M({\mathfrak {A}})$$ M ( A ) onto $$M({\mathfrak {B}})$$ M ( B ) .