## A local algebra structure for $$H^ p$$ of the polydisc.(English)Zbl 0766.46040

N. M. Wigley [Can. Math. Bull. 18, 597-603 (1975; Zbl 0324.46051)] showed that, for $$p\geq 1$$, $$H^ p(\Delta)$$ can be given a Banach algebra structure under the Duhamel product given by $f*g(z)={d\over dz}\int^ z_ 0f(z-t)g(t)dt,$ where $$\Delta$$ is the open unit disc. The authors define on $$H^ p(\Delta^ n)$$ a natural extension of the Dunhamel product as follows: $f*g(z_ 1,\dots,z_ n)={\partial^ n\over\partial z_ 1,\dots,\partial z_ n}\int^{z_ n}_ 0\cdots\int^{z_ 1}_ 0f(z_ 1-t_ 1,\dots,z_ n-t_ n)g(t_ 1,\dots,t_ n)dt_ 1, \cdots,dt_ n.$ And they show that $$H^ p(\Delta^ n)$$, $$0<p\leq\infty$$, is a local (Banach or $$F$$-) algebra. This result follows from a more general study on vector-valued analytic functions: For $$p\geq 1$$ let $$B$$ be a Banach algebra and for $$0<p<1$$ let $$B$$ be a $$p$$-normed $$F$$-algebra. Let $$H^ p(\Delta,B)$$ be the space of $$B$$-valued analytic functions for which the usual $$H^ p$$ norm is finite. The authors define the Duhamel product on $$H^ p(\Delta,B)$$ and show that $$H^ p(\Delta,B)$$ is a Banach algebra for $$p\geq 1$$ and an $$F$$- algebra for $$0<p<1$$. They also show that there is a natural isomorphism between the maximal ideal space of $$H^ p(\Delta,B)$$ and that of $$B$$.
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables