centering / aligning arbitrary amsmath objects

Question

See how the little ^\frown ^⌢ symbols are aligned (on the last 3 lines) ?
That's what I want.

But see how it's just a concatenation symbol for (C_\alpha) (C_α) and (C_\beta) (C_β) ?
I want to make that obvious, so I want to push both halves out to the sides (see the second image).

\documentclass[preview,border=2mm]{standalone}
\usepackage{amsmath}
\providecommand{\abs}[1]{\lvert#1\rvert}

\begin{document}

    $$C=E_k(m)$$
    \begin{align*}
        C_\alpha = & (E_k(m) \oplus m) & C_\beta = & (E_k(m) \oplus 1^{\abs{C}}) \\
        C_\alpha = & (C      \oplus m) & C_\beta = & (C      \oplus 1^{\abs{C}}) %%
    \end{align*}

    $$C_\omega = (C_\alpha) ^\frown (C_\beta)\ \ $$
    \begin{align*}
        C_\omega = (E_k(m) \oplus m) & ^\frown (E_k(m) \oplus 1^{|C|}) \\
        C_\omega = (C      \oplus m) & ^\frown (C      \oplus 1^{|C|}) %%
    \end{align*}

\end{document}

---

Like this, but with the ^⌢ symbol back in it's original spot in the middle.

\documentclass[preview,border=2mm]{standalone}
\usepackage{amsmath}
\providecommand{\abs}[1]{\lvert#1\rvert}

\begin{document}

    $$C=E_k(m)$$
    \begin{align*}
        C_\alpha = & (E_k(m) \oplus m) & C_\beta = & (E_k(m) \oplus 1^{\abs{C}}) \\
        C_\alpha = & (C      \oplus m) & C_\beta = & (C      \oplus 1^{\abs{C}}) %%
    \end{align*}

    $$C_\Omega = (C_\alpha) ^\frown (C_\beta)\ \ $$
    \begin{align*}
        C_\Omega = & (E_k(m) \oplus m) & ^\frown & (E_k(m) \oplus 1^{|C|}) \\
        C_\Omega = & (C      \oplus m) & ^\frown & (C      \oplus 1^{|C|}) %%
    \end{align*}

\end{document}

---

If I could keep it all in the same environment that would also be good, rather than calling \begin{align*} ... \end{align*} and $$ ... $$ twice separately.

Answer 1

Although I would not recommend the desired output, one way to do this is to use alignat. A slightly better (IMO) is the second version and third version:

Notes

• Don't use $$ for display math. See Why is \[ … \] preferable to $$ … $$?.
• I think it is a good idea to define \FrownOp (perhaps with a more meaningful name) as a binary operator

Code

\documentclass[preview,border=2mm]{standalone}
\usepackage{mathtools}% <-- includes amsmath
\providecommand{\abs}[1]{\lvert#1\rvert}
\newcommand*{\FrownOp}{\mathbin{^\frown}}%
\begin{document}
\begin{alignat}{4}
             &       &      C &= E_k(m) \
    C_\alpha &=  (E_k(m) \oplus m) &\qquad &          &C_\beta &=  (E_k(m) \oplus 1^{\abs{C}}) \
    C_\alpha &=  (C      \oplus m) &\qquad &          &C_\beta &=  (C      \oplus 1^{\abs{C}}) \
             &     &C_\Omega = (C_\alpha)  &\FrownOp (C_\beta) \
    C_\Omega &=  (E_k(m) \oplus m) &\qquad &\FrownOp  &&(E_k(m) \oplus 1^{|C|}) \
    C_\Omega &=  (C      \oplus m) &\qquad &\FrownOp  &&(C      \oplus 1^{|C|}) %%
\end{alignat}
\textbf{Alternative suggestion 1:}
\begin{alignat}{4}
             &                     & \mathllap{C} &= E_k(m) \
    C_\alpha &=  (E_k(m) \oplus m) & & &  C_\beta &= (E_k(m) \oplus 1^{\abs{C}}) \
             &=  (C      \oplus m) & & &          &= (C      \oplus 1^{\abs{C}}) \
    \shortintertext{Thus,}
    C_\Omega &= (C_\alpha)        &&\FrownOp (C_\beta) \
             &= (E_k(m) \oplus m) &&\FrownOp (E_k(m) \oplus 1^{|C|}) \
             &= (C      \oplus m) &&\FrownOp (C      \oplus 1^{|C|}) %%
\end{alignat}
\textbf{Alternative suggestion 2:}
\begin{alignat}{4}
     C &= E_k(m) \
    \shortintertext{Thus,}
    C_\Omega &= (C_\alpha) \FrownOp (C_\beta), \
    \shortintertext{where}
    C_\alpha &= (E_k(m) \oplus m)           &&=  (C \oplus m) \
    C_\beta  &= (E_k(m) \oplus 1^{\abs{C}}) &&=  (C \oplus 1^{\abs{C}}).
\end{alignat}
\end{document}