How can we replace a math symbol?

Question

I found a way to define a math symbol as follow:

\documentclass{article}

\makeatletter
\begingroup\lccode`~=`*
\lowercase{\endgroup\def~}{\times}
\mathcode`*="8000
\makeatother

\begin{document}
    $a*b$
\end{document}

Theh problem is that I have to assign a letter which appears on keyboard "*" to replace "\times" (that means the ASCII code is in range 0-255). Because function \lccode just accept value in that range.

How can I define an arbitrary math symbol and assign it to the same character? Something like:

\begingroup\lccode`~=`\times         % it doesn't work in this way
\lowercase{\endgroup\def~}{new\ \times}
\mathcode`\times="8000

In other words, we redefine a math symbol which has value of mathcode is out of ASCII range. Because we can do as follow:

\lccode`~=`0
\lccode`~=`\_
\lccode`~=`\=
 ...

but cannot with

\lccode`~=`\times
\lccode`~=`\star
...

Answer 1

I found the answer from @Henri Menke (thank you for that). I realize that we should use \lccode to redefine symbols which are typed directly by single symbol. Regarding operators (such as \times, \div, etc.), which have mathcode's out of the range, we can redefine them by using:

\let <new command> <old command>
\renewcommand <old command> (<new reformation>)