On the strict topology of the multipliers of a JB∗-algebra

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 117; no. 4
• Main Authors: Fernández-Polo, Francisco J., Garcés, Jorge J., Li, Lei, Peralta, Antonio M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-10-2023; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Continuity (mathematics); Homomorphisms; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Multipliers; Operators (mathematics); Original Paper; Orthogonality; Theoretical; Topology; Multipliers; epimorphisms; Primary 46L05; unital; algebra; Triple homomorphism; Orthogonality preserver; Jordan homomorphism; 17C65; 47B49; 46L70; 47B48; Secondary 47B47; J-strict topology; JB; Extension of Jordan; 17A40
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-023-01476-w

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
)
.