Entire functions and {$m$}-convex structure in commutative Baire algebras

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We show that a unitary commutative locally convex algebra, with a continuous product which is a Baire space and in which entire functions operate is actually m-convex. Whence, as a consequence, the same result of Mitiagin, Rolewicz and Zelazko, in commutative B0-algebras. It is known that entire functions operate in complete m-convex algebras [1]. In [3] Mitiagin, Rolewicz and Zelazko show that a unitary commutative B0-algebra in which all entire functions operate is necessarily m-convex. Their proof is quite long and more or less technical. They use particular properties of B0-algebras, a Baire argument and the polarisation formula. Here we show that any unitary commutative locally convex algebra, with a continuous product which is a Baire space and in which all entire functions operate is actually m-convex. The proof is short, direct and selfcontained. A locally convex algebra (A; ), l. c. a. in brief, is an algebra over a eld K (K = R or C) with a Hausdor locally-convex topology for which the product is separately continuous. If the product is continuous in two variables, (A; ) is said to be with continuous product. A l. c. a. (A; ) is said to bem-convex (l. m. c. a.) if the origin 0 admits a fundamental system of idempotent neighbourhoods ([2]). An