Karakterizacija jordanskih preslikav na C[zvezdica]-algebrah z ničelnim produktom

Naj bosta ▫$A$▫ in ▫$B$▫ ▫$C^ast$▫-algebri, ▫$X$▫ naj bo bistveni Banachov ▫$A$▫-bimodul in naj bosta ▫$T colon A to B$▫ in ▫$S colon A to X$▫ zvezni linearni preslikavi▫$T$▫ naj bo surjektivna. Denimo, da je ▫$T(a)T(b) + T(b)T(a) = 0$▫ in ▫$S(a)b + bS(a) + aS(b) + S(b)a = 0$▫, kadarkoli ▫$a, b in A$▫ zadoščata ▫$ab = ba = 0$▫. Dokažemo, da je ▫$T = wPhi$▫ in ▫$S = D + wPsi$▫, kjer ▫$w$▫ leži v centru multiplikatorske algebre ▫$B$▫, ▫$Phicolon A to B$▫ je jordanski epimorfizem, ▫$D colon A to X$▫ je odvajanje in ▫$Psi colon A to X$▫ je bimodulski homomorfizem.Let ▫$A$▫ and ▫$B$▫ be ▫$C^ast$▫-algebras, let ▫$X$▫ be an essential Banach ▫$A$▫-bimodule and let ▫$T colon A to B$▫ and ▫$S colon A to X$▫ be continuous linear maps with ▫$T$▫ surjective. Suppose that ▫$T(a)T(b) + T(b)T(a) = 0$▫ and ▫$S(a)b + bS(a) + aS(b) + S(b)a = 0$▫ whenever ▫$a, b in A$▫ are such that ▫$ab =ba = 0$▫. We prove that then ▫$T = wPhi$▫ and ▫$S = D + wPsi$▫, where ▫$w$▫ lies in the centre of the multiplier algebra of ▫$B$▫, ▫$Phicolon A to B$▫ is a Jordan epimorphism, ▫$D colon A to X$▫ is a derivation and ▫$Psi colon A to X$▫ is a bimodule homomorphism