In this sequel to arXiv1407.4089 by the second author, we extend to multi-dimensional (or infinite-dimensional) settings the Goldie equation arising in the general regular variation of `General regular variation, Popa groups and quantifier weakening, J. Math. Anal. Appl. 483 (2020) 123610, 31 pp. (arXiv1901.05996). The theory focusses on extension of the treatment there of Popa groups, permitting a characterization of Popa homomorphisms (in two complementary theorems, 4A and 4B below). This in turn enables a characterization of the (real-valued) solutions of the multivariate Goldie equation, to be presented in the further sequel arXiv1910.05817. The Popa groups here contribute to a structure theorem describing Banach-algebra value
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