Graded pseudo-H-rings

Abstract

Consider a pseudo-$H$-space $E$ endowed with a separately continuous biadditive associative multiplication which induces a grading on $E$ with respect to an abelian group $G$. We call such a space a graded pseudo-$H$-ring and we show that it has the form $E=cl(U+{\sum }_j{I}_j)$ with $U$ a closed subspace of $E_1$ (the summand associated to the unit element in $G$), and any $I_j$ runs over a well described closed graded ideal of $E$, satisfying $I_j{I}_k=0$ if $j\ne k$. We also give a context in which graded simplicity of $E$ is characterized. Moreover, the second Wedderburn-type theorem is given for certain graded pseudo-$H$-rings.