Characterization of commutative algebras embedded into the algebra of smooth operators

Abstract

The paper deal with the noncommutative Fréchet * -algebra L(s ′ , s) of the so-called smooth opertors, i.e. linear and continuous operators acting from the space s ′ of slowly increasing sequences to the Fréchet space s of rapidly decreasing sequences. By a canonical identification, this algebra of smooth operators can be also seen as the algebra of the rapidly decreasing matrices. We give a full description of closed commutative * -subalgebras of this algebra and we show that every closed subspace of s with basis is isomorphic (as a Fréchet space) to some closed commutative * -subalgebra of L(s ′ , s). As a consequence, we give some equivalent formulation of the long-standing Quasi-equivalence Conjecture for closed subspaces of s.

Fréchet algebras with a dominating Hilbert algebra norm

“…The algebra K ∞ is isomorphic as the Fréchet * -algebra to the algebra L (s ′ , s) of compact smooth operators. Moreover, it is isomorphic as a Fréchet space to the space s. For further information concerning the algebra K ∞ we refer the reader to [5,6,7]. Moreover, by the Weyl asymptotic formula [4, Note III.15, p. 184], there is a constant C > 0 depending only on n and the choice of a Riemannian metric such that ( 6)…”