* obtains a special mathcode inside the AMS math operator setup, but here you want the standard star which is still available as \ast :
\documentclass{article} \usepackage{amsmath} \DeclareMathOperator*{\wslima}{w^{*}-lim} \DeclareMathOperator*{\wslimb}{w^{\ast}-lim} \begin{document} \[\wslima a \quad \wslimb a\] \end{document} Note the reason for the redefinition is to make it possible to just use w* which is the intended markup here although as you note it does not produce the same symbol (or use the same font) as a normal math ^*.
\documentclass{article} \usepackage{amsmath} \DeclareMathOperator*{\wslim}{w*-lim} \DeclareMathOperator*{\wslima}{w^{*}-lim} \DeclareMathOperator*{\wslimb}{w^{\ast}-lim} \begin{document} \[\wslim a \quad \wslima a \quad \wslimb a\] \end{document}