The tensor algebra of power series spaces
The linear isomorphism type of the tensor algebra T (E) of Frechet spaces and, in particular, of power series spaces is studied. While for nuclear power series spaces of infinite type it is always s the situation for finite type power series spaces is more complicated. The linear isomorphism T (s) ∼= s can be used to define a multiplication on s which makes it a Frechet m-algebra s•. This may be used to give an algebra analogue to the structure theory of s, that is, characterize Frechet m-algebras with (Ω) as quotient algebras of s• and Frechet m-algebras with (DN) and (Ω) as quotient algebras of s• with respect to a complemented ideal. In [8] there was calculated the linear isomorphism type of the space s of rapidly decreasing sequences and all of its complemented subspaces E. It was shown that T (E) ∼= s whenever dim E ≥ 2. This includes all of socalled power series spaces of infinite type, including the space H(C) of entire functions for any dimension d. In the present work we study the tensor algebra of Frechet spaces in a more general context. We use these results to give another proof for the isomorphism theorem in the case of infinite type power series spaces. The isomorphism T (s) ∼= s defines a multiplication on s which turns it into a Frechet m-algebra which we call s•. Its quotient algebras are all nuclear Frechet m-algebras with property (Ω) or, equivalently, which are linearly isomorphic to a quotient of s, and its quotient algebras with respect to a complemented ideal are the nuclear Frechet m-algebras with properties (DN) and (Ω). This gives an “algebra equivalent” to the structure theory of s as developed in [9]. In the last section we extend our study to the finite type power series spaces, including e. g. the spaces s0 of very slowly increasing functions and the spaces H(D) of holomorphic functions on the d-dimensional polydisc. There the situation is much more complicated. But it is interesting to observe that for E = s0 and E = H(D) the results are different, however 2000 Mathematics Subject Classification. Primary: 46H20 Secondary: 46A45, 46A63