Swap definition of starred and non-starred command

You can redefine \abs to call the opposite version of the original command:

\makeatletter
\let\oldabs\abs
\def\abs{\@ifstar{\oldabs}{\oldabs*}}
\makeatother

(The reason that your commented try doesn't work is that technically the * is not part of the macro name, but is read by the macro itself using \@ifstar, \@ifnextchar or similar commands.)

simply define \abs with \def if you do not want to use the star version.

\def\abs#1{\left\lvert#1\right\rvert}

I've had the same problem and wrote a work-around for this: a macro \DeclarePairedDelimiterY, see this question. But note you have to replace the line

{\csname#1Temp*\endcsname{##2}}%

with

{\csname#1Temp\endcsname*{##2}}%

Then you can define \abs by

\DeclarePairedDelimiterY{abs}{\lvert}{\rvert}

where the backslash of \abs is omitted. Then:

• \abs{x} expands to \left\lvert x\right\rvert
• \abs[normal]{x} expands to \lvert x\rvert
• \abs[big]{x} expands to \bigl\lvert x\bigr\rvert
• \abs[Big]{x} expands to \Bigl\lvert x\Bigr\rvert
• \abs[bigg]{x} expands to \biggl\lvert x\biggr\rvert
• \abs[Bigg]{x} expands to \Biggl\lvert x\Biggr\rvert

In contrast to the original \DeclarePairedDelimiter command, you leave out the backslash of \big, \Big etc.

(Unluckily I didn't find your question prior to posting my own one...)