Absolute-valued algebras with involution, and infinite-dimensional Terekhin's trigonometric algebras

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 x∧y:=(x∗y−y∗x)2. Moreover, we show that, “essentially,” all infinite-dimensional complete trigonometric algebras derive from absolute-valued ∗-algebras by the above construction method.