Abstract. We consider Fréchet algebras which are subalgebras of the algebra F = C [[X]] of formal power series in one variable and of Fn = C [[X1, . . . , Xn]] of formal power series in n variables, where n ∈ N. In each case, these algebras are taken with the topology of coordinatewise convergence.We begin with some basic definitions about Fréchet algebras, (F )-algebras, and other topological algebras, and recall some of their properties; we discuss Michael's problem from 1952 on the continuity of characters on these algebras and some results on uniqueness of topology.A 'test algebra' U for Michael's problem for commutative Fréchet algebras has been described by Clayton and by Dixon and Esterle. We prove that there is an embedding of U into F, and so there is a Fréchet algebra of power series which is a test case for Michael's problem.We also discuss homomorphisms from Fréchet algebras into F. We prove that such a homomorphism is either continuous or a surjection, so answering a question of Dales and McClure from 1977. As corollaries, we note that a subalgebra A of F containing C[X] that is a Banach algebra is already a Banach algebra of power series, in the sense that the embedding of A into F is automatically continuous, and that each (F )-algebra of power series has a unique (F )-algebra 2010 Mathematics Subject Classification: Primary 46H40; Secondary 1325, 13J05, 46J99.

Abstract. We describe all those commutative Fréchet algebras which may be continuously embedded in the algebra C [[X]] in such a way that they contain the polynomials. It is shown that these algebras (except C[[X]] itself) always satisfy a certain equicontinuity condition due to Loy. Using this result, some applications to the theory of automatic continuity are given; in particular, the uniqueness of the Fréchet algebra topology for such algebras is established.

If the indeterminate X in a Frechet algebra A of power series is a power series generator for A, then either A is the algebra of all formal power series or is the Beurling-FWchet algebra on non-negative integers defined by a sequence of weights. Let the topology of A be defined by a sequence of norms. Then A is an inverse limit of a sequence of Banach algebras of power series if and only if each norm in the defining sequence satisfies certain closability condition and an equicontinuity condition due to Loy. A non-Banach uniform Frechet algebra with a power series generator is a nuclear space. A number of examples are discussed; and a functional analytic description of the holomorphic function algebra on a simply connected planar domain is obtained.

Abstract. In 1978, Dales posed a question about the uniqueness of the (F )-algebra topology for (F )-algebras of power series in k indeterminates. We settle this in the affirmative for Fréchet algebras of power series in k indeterminates. The proof goes via first completely characterizing these algebras; in particular, it is shown that the Beurling-Fréchet algebras of semiweight type do not satisfy a certain equicontinuity condition due to Loy. Some applications to the theory of automatic continuity are also given, in particular the case of Fréchet algebras of power series in infinitely many indeterminates.

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