A quick fix without the hassles of renewing the section command is to use addtocounter.
Everytime you use the \section*{} command, right after that you can say \addtocounter{section}{1}
\chapter{}
\section*{Preliminaries}
\addtocounter{section}{1}
\subsection{Banach Algebras}
\section*{Preliminaries}
\addtocounter{section}{1}
\subsection{Banach Algebras}
This will give you a result of,
Chapter 1
Preliminaries
1.1.1 Banach Algebras
Preliminaries
1.2.2 Banach Algebras
Essentially, it is just adding 1 to your section counter whenever you create a section so when the subsection checks the section counter, it has the updated counter.
And the advantage is that if you add another section now which needs to be numbered like,
\chapter{}
\section*{Preliminaries}
\addtocounter{section}{1}
\subsection{Banach Algebras}
\section*{Preliminaries}
\addtocounter{section}{1}
\subsection{Banach Algebras}
\section{Preliminaries}
You will get the correct section number (i.e. 1.3)
Chapter 1
Preliminaries
1.1.1 Banach Algebras
Preliminaries
1.2.2 Banach Algebras
1.3 Preliminaries
The main drawback is that you will have to remember to add to the counter everytime you created a \section*{}
You can add \setcounter{subsection}{0} everytime you create one of those sections to reset the subsection counter... sorry I missed that one.
Thanks for clarifying.