In the first season of startrek the next generation there's an episode called
'11001001' in which Riker creates a jazz club on the holodeck. He plays
'the nearness of you' with the trio. I know the actor who plays the pianist
is Jack Sheldon, my question is: who is really playing there?

I can't prove it, but I suspect that Jack Sheldon played the piano, in
addition to appearing onscreen. Sheldon, in addition to being an
actor, is an accomplished jazz musician. He is best known as a
vocalist and trumpeter, but he does play piano:

"...a chance encounter with Dragnet star and producer Jack Webb led to
a number of guest appearances on Dragnet. 'I was playing piano down at
a little club called The China Trader in Toluca Lake,' Sheldon
recalls. 'I spotted Jack and I told him I couldn?t get a job. He gave
me a whole bunch of Dragnets. He was a great guy and really helped me
out'."

http://www.what-a-character.com/cgi-bin/display.cgi?id=982797537