+1 to the above.
Yes, check the arrangement first. If you listen to/watch jazz piano players, you'll hear/see that they keep their left hand well out of the way of the bass player ... rarely going below the E below middle C and mostly don't play the root note of the chord.
(Not your boogiewoogie and the likes type of piano though, cos that's a particular style)
If the piano part exactly double the bass part, that can work too depending on the song, but you want to avoid the piano playing low chord notes in the bass region.
With the 'live' playing and the 'recording' thingy: you'll see Elton John hammering away on 'live' with his left hand, but most of it will be equed out of the way and the bright end of the piano focused on.
You want to avoid the piano area (kind of the octave below the G below middle C) being too chordal down there as it doesn't sit very well with bass. If you eq out a lot of the piano though and make the piano quite bright, that could work too.
It does depend a lot on the song though.
The thing is I suppose that pianists playing solo are used to covering all the parts (bass, chords and melody), but when playing piano in a band, you've got to adjust the playing style a lot or you run the risk of annoying the bass player because the piano is treading on his/her bass note choice and sometimes clashing badly. So they've got to be exactly together on the same bass note or the bottom notes of the piano removed.
I've seen myself sometimes recording a midi piano part and then going into the piano roll and taking a lot of the low piano notes back out!