Analyticity and naturality of the multi-variable functional calculus

Mackey-complete complex commutative continuous inverse algebras generalize complex commutative Banach algebras. After constructing the Gelfand transform for these algebras, we develop the functional calculus for holomorphic functions on neighbourhoods of the joint spectra of finitely many elements and for holomorphic functions on neighbourhoods of the Gelfand spectrum. To this end, we study the algebra of holomorphic germs in weak^*${\mathrm{weak}}^{*}$-compact subsets of the dual. We emphasize the simultaneous analyticity of the functional calculus in both the function and its arguments and its naturality. Finally, we treat systems of analytic equations in these algebras.