In this paper, We define JORDAN algebra bundles of finite type and we give one-one correspondence between JORDAN algebra bundles of finite type and JORDAN rings.

In this paper, we investigate the generalized Hyers-Ulam stability of JORDAN homomorphisms in JORDAN Banach algebras for the functional equation Snk=2 Ski1=2 Sk+1i2=i1+1 · Sn in-k+1=in-k+1 ¦ (Sni=1, i ¹i1, ···, in-k+1 ci -Sn-k+1r=1 cir) + ¦ (Sni=1 ci) -2n-1 ¦ (c1)=0, where n is an integer greater than 1.

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Let A be a unital R-algebra and M be a unital A- bimodule. It is shown that every JORDAN derivation of the trivial extension of A by M, under some conditions, is the sum of a derivation and an antiderivation.

In this ‎paper, ‎we ‎investigate ‎the ‎structure ‎of ‎JORDAN ‎centrali‎zer ‎maps‎ from a unital ‎alg‎ebra ‎$‎A‎$ ‎into a ‎unital‎ ‎‎$‎A‎$‎-bimodule ‎$‎M‎$‎.‎ We applied our results to triangular algebras. In particular, we prove that every ‎ JORDAN centralizer map on a triangular algebra is a centralizer map. Keywords: Centralizer map, JORDAN centralizer map, Triangular algebra. Highlights B.E. Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, vol. 127, 1972 O.T. Mewomo, Various notions of amenability in Banach algebras, Expositiones Mathematicae 29 (2011) 283–299 V. Runde, Lectures on amenability, Springer-Verlag, Berlin, Heidelberg, 2002. V. Runde, Amenable Banach algebras, Springer Monographs in Mathematics, Berlin, 202

Let n nÎN. An additive map h: A®B between algebras A and B is called n-JORDAN homomorphism if h (an) = (h (a)) n for all aÎA. We show that every n-JORDAN homomorphism between commutative Banach algebras is an-ring homomorphism when n<8.For these cases, every involutiven -JORDAN homomorphism between commutative C*-algebras is norm continuous.