Segal Fréchet algebras (Q1715031)

A Fréchet algebra $\left(\mathcal{B},\tau_{\mathcal{B}}\right) $ is called a Segal Fréchet algebra in a Fréchet algebra $\left(\mathcal{A},\tau_{\mathcal{A}}\right)$ if (a) $\mathcal{B}$ is a dense left ideal in $\mathcal{A}$, (b) $\left(\mathcal{B},\tau_{\mathcal{B}}\right) \hookrightarrow\left(\mathcal{A},\tau_{\mathcal{A}}\right) ,$ and (c) the map $\left(\mathcal{B},\tau_{\mathcal{A}}\left\vert_{\mathcal{B}}\right. \right)$ $\times \left(\mathcal{B},\tau_{\mathcal{B}}\right)\rightarrow\left(\mathcal{B},\tau_{\mathcal{B}}\right),$ $\left(a,b\right) \mapsto ab$ is jointly continuous. If $\mathcal{B}$ and $\mathcal{A}$ are both Banach algebras, this coincides with the concept of abstract Segal algebras defined and investigated by \textit{J. T. Burnham} [Proc. Amer. Math. Soc. 32, 551--555 (1972; Zbl 0234.46050)].\par The main results of the paper under review are the following: (1) If $\left(e_{\alpha}\right) $ is an approximate identity in $\mathcal{B}$, where $\mathcal{B}$ is a proper Segal Fréchet algebra in $\mathcal{A}$, then $\left(e_{\alpha}\right) $ is unbounded in $\mathcal{B}$. (2) If $\mathcal{B}$ is a symmetric Segal Fréchet algebra (i.e., it is a two-sided ideal) in $\mathcal{A}$, then the following statements hold: (i) If $J$ is a left ideal in $\mathcal{A}$, then cl$_{\mathcal{A}}\left(J\right) ,$ the closure of $J$ in $\mathcal{A}$, is a closed left ideal in $\mathcal{A}$. (ii) If $J$ is a left ideal in $\mathcal{A}$, then cl$\left(J\right) \cap\mathcal{B}$ is a closed left ideal in $\mathcal{B}$. (iii) If $I$ is a left ideal in $\mathcal{B}$, then cl$_{\mathcal{A}}\left(I\right) $ is a closed left ideal in $\mathcal{A}$. (iv) If $I$ is a left ideal in $\mathcal{B}$ and $\mathcal{B}$ has left approximate units, then $I=$cl$_{\mathcal{A}}\left(I\right) \cap\mathcal{B}$.