Wish this wasn't the highest-rated comment, as it's not correct.
If it needed a "56kHz channel", it would not have worked in the phone-over-copper-wire days either.
If we assume the parent meant 56kbps, that's still incorrect. The fastest fax machines use V.34bis modems (ignoring ISDN), which give you 33.6kbps, though many (most?) still support the oldest standard, V.27, at 2400bps!
The real issue sending a (T.30) fax over a cell phone line would be jitter and packet loss, since most cell networks are IP-based nowadays. T.30 is not very resilient to noise or timing issues. There's the T.38 standard, which was designed for sending faxes over a packet-switched network, but it's still sensitive to timing issues (esp. since often you'd be talking to a real fax machine on the other end, after going through a T.38/T.30 gateway).
Cell phones certainly don't give you the bandwidth that a copper landline does, but it'd certainly be enough to transmit a fax... if you could keep the error rate low enough... which I doubt.
(Source: I built the initial production version of Twilio's fax product.)