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