Decomposition and Integrability of Jordan Higher *-Derivations of Incidence Algebras

Let I(X,R) be the incidence algebra determined by a unital commutative coefficient ring R and a poset X all of whose intervals are finite. We show that each Jordan higher α-derivation of I(X,R), for any R-linear automorphism or anti-automorphism α, is obtained by twisting a transposed one with an invertible power series whose coefficients lie in I(X,R). For an involution built from an order-reversing involution λ of X and a multiplicative function, with R reduced, we prove that the odd components tk of a transposed Jordan higher *-derivation of order N are zero at the matrix unit eλ(y)y whenever λ(y)<y and 2k≤N. As a consequence, for algebras of upper triangular matrices over a field of characteristic other than 2 with such an involution, a Jordan *-derivation is inner exactly when it occurs as the first component of a Jordan higher *-derivation of order 2. We also determine all transposed Jordan higher *-derivations of the triangular 2×2 matrices over a field. In particular, the bound 2k≤N is optimal.