Symmetric tensor algebras and integral decompositions
Summary: Let \(\lambda\) be a perfect sequence space closed under the Cauchy convolution product, with respect to which it is a topological algebra with jointly continuous product. Suppose, in addition, that \(\lambda\) is type (h) and \(\lambda\subset h\). Let \(E[t]\) be a nuclear Fréchet space, with the topology t defined by the seminorms \(\{p_ r^{(1)}:\) \(r\geq 0\}\). The corresponding regular tensor algebra is \[ T(E,\lambda)=\{x\in \prod_{n\geq 0}{\hat \otimes}^ n E:\;[p_ r^{(n)}(x_ n)]\in \lambda,\quad r\geq 0\}, \] equipped with the topology \(t(E,\lambda^{\times})\) determined by the seminorms \[ p_{u,r}(x)=\sum_{n\geq 0}| u_ n| p_ r^{(n)}(x_ n),\quad (u\in \lambda^{\times}). \] The algebra product is \[ (xy)_ n=\sum_{0\leq k\leq n}x_{n-k}\otimes y_ k. \] If \(S_ n\) is the symmetrizing operator on \({\hat \otimes}^ n E\) for all \(n\geq 0\), we consider the symmetric subalgebra \(S[T(E,\lambda)]\), where \(S(x_ n)=(S_ nx_ n).\) We show that functionals on \(T(E,\lambda)\) and \(S[T(e,\lambda)]\) can be represented by complete finite complex Borel measures if they are continuous with respect to a certain locally convex topology \(\hat t(E,\lambda^{\times})\) on \(T(E,\lambda)\) and its restriction \(\hat t_ S(E,\lambda^{\times})\) to \(S[T(E,\lambda^{\times})]\) respectively. The topology \(\hat t(E,\lambda^{\times})\) is coarser than \(t(E,\lambda^{\times})\) and non-Hausdorff, whereas \(\hat t_ S(E,\lambda^{\times})\) is Hausdorff. If the algebraic positive cone in \(\lambda\) is normal, then so is the positive cone on \(S[T(E,\lambda)]\). Using the isomorphism \(\lambda\cong T({\mathbb{C}},\lambda)\), we show that the cone in \(s[T(E,\lambda)]\) is normal for all nuclear Fréchet spaces \(E[t]\) if and only if \(t({\mathbb{C}},\lambda^{\times})= \hat t({\mathbb{C}},\lambda^{\times})\).
perfect sequence space closed under the Cauchy convolution product
topological algebra with jointly continuous product
scientific article; zbMATH DE number 4153238