Fréchet algebras with a dominating Hilbert algebra norm

Abstract

Let L * (s) be the maximal O * -algebra of unbounded operators on ℓ 2 whose domain is the space s of rapidly decreasing sequences. This is a noncommutative topological algebra with involution which can be identified, for instance, with the algebra L (s) ∩ L (s ′ ) or the algebra of multipliers for the algebra L (s ′ , s) of smooth compact operators. We give a simple characterization of unital commutative Fréchet * -subalgebras of L * (s) isomorphic as a Fréchet spaces to nuclear power series spaces Λ∞(α) of infinite type. It appears that many natural Fréchet * -algebras are closed * -subalgebras of L * (s), for example, the algebras C ∞ (M ) of smooth functions on smooth compact manifolds and the algebra S (R n ) of smooth rapidly decreasing functions on R n .