We prove the Kummer duality arising from norm algebraic tori over a certain family of fields. Our family contains relatively large subfields of prime cyclotomic fields.

We prove that, if A is an absolute-valued *-algebra in the sense of [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247–258], then the normed space of A becomes a trigonometric algebra (in the meaning of [P.A. Terekhin, Trigonometric algebras, J. Math. Sci. (New York) 95 (1999) 2156–2160]) under the product defined by . Moreover, we show that, “essentially,” all infinite-dimensional complete trigonometric algebras derive from absolute-valued *-algebras by the above construction method.