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 A defined by x boolean AND 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. (c) 2005 Elsevier Inc. All rights reserved.
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