2020/11/16 by Antonio M. Peralta, Peralta, Antonio M.
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2011.07762
openalex publication_date 2020/11/16 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let M and N be unital Jordan-Banach algebras, and let M-1 and\nN-1 denote the sets of invertible elements in M and N, respectively.\nSuppose that mathfrakM\⊆ M-1 and mathfrakN\⊆ N-1\nare clopen subsets of M-1 and N-1, respectively, which are closed for\npowers, inverses and products of the form Ua (b). In this paper we prove\nthat for each surjective isometry \Δ : mathfrakM\→ mathfrakN there\nexists a surjective real-linear isometry T0: M\→ N and an element u0 in\nthe McCrimmon radical of N such that \Δ (a) = T0(a) +u0 for all a\∈\n mathfrakM. smallskip\n Assuming that M and N are unital JB^*-algebras we establish that for\neach surjective isometry \Δ : mathfrakM\→ mathfrakN the element\n\Δ(\1) =u is a unitary element in N and there exist a central\nprojection p\∈ M and a complex-linear Jordan ^*-isomorphism J from M\nonto the u^*-homotope Nu^* such that
Delta (a) = J(p
circ a) + J\n((
textbf1-p)
circ a^*), for all a\∈ mathfrakM. Under the additional\nhypothesis that there is a unitary element \ω0 in N satisfying\nU\ω0 (\Δ(\1)) = \1, we show the existence of a\ncentral projection p\∈ M and a complex-linear Jordan ^*-isomorphism \Φ\nfrom M onto N such that
Delta (a) = Uw0*
left(
Phi (p
circ a) +\n
Phi ((
textbf1-p)
circ a^*)
right), for all a\∈ mathfrakM.\n