Correct spacing for Hodge star operator

Question

In my textbook, the Hodge star operator is represented as an asterisk character with the same spacing as in the following image:

But when I use the * character in the obvious way:

\[
\ast d \ast \phi = \frac{\partial A}{\partial x}
\]

It ends up with spacing like an infix operator:

How do I get rid of the extraneous space between *, d and \phi?

Answer 1

You can define

\newcommand{\hodge}{{\star}}

or * if you prefer an asterisk. The additional braces will hide the nature of either \star or * as binary operation symbols.

\documentclass{article}
\usepackage{amsmath} % not needed here, but...
\newcommand{\hodge}{{\star}}
\begin{document}
[
\hodge d \hodge \phi = \frac{\partial A}{\partial x}
]
\end{document}

If you replace \star with * like in

\newcommand{\hodge}{{*}}

you get

Answer 2

An alternative is to use a newcommand putting \mathord\star or \mathord\ast without the curly brackets {}, where you can see that you have the same result of @egreg's answer.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\newcommand{\starr}{\mathord\star}
\newcommand{\astt}{\mathord\ast}
\begin{document}
[
\starr d \starr \phi = \frac{\partial A}{\partial x}
]
[
\astt d \astt\phi = \frac{\partial A}{\partial x}
]
\end{document} 

PS: A \mathord object is kind of neutral to its left and right neighbours where an unary operator binds stronger to its right neighbour than to its left one (expressed be different spacing and potential break points). Difference between \mathop and \mathord