Classical, but there’s a set of CDs called “Beethoven for Relaxation,” and ditto for Brahms, Mozart, and Bach. I find them very VERY relaxing.
Here’s a piano cover of a deadmau5 song. It might be slightly more upbeat/energetic than some of the suggestions in this thread, but I still find it soothing.
As for Oscar Peterson, I’m not so sure – his ballads, yes, but a lot of his stuff is intricate, fast, be-bop or even swing-style jazz. Really fast, awesome, intricate fingerwork is not what I call relaxing, personally.
Add me to the list of fans of Chopin and Satie.
Also, again absolutely not jazzy, but I love Philip Glass.
Jazz pianists to consider or to check out at YouTube:
George Shearing
Bill Evans
Ahmad Jamal
Keith Jarrett
Lyle Mays
Tommy Flanagan
Peter Nero
Whisperings, er, rocks. Or tinkles. Thank for the link!
A couple at random:
Schumann, Romance op. 28, no. 2 Schumann: Romance op. 28 #2, Stephen Ham, Piano - YouTube
Schubert, Impromptu in G-flat major Schubert: Impromptu in G flat Op. 90 No. 3 - YouTube
He sings and plays trumpet in a small group with piano but your mention of soothing standards makes me think that Chet Baker ( Misty, Someone to watch over me ) should be mentioned. Try on pandora . He perfectly fits your request.
George Winston - very soothing, nice background music
Yes. You will find him on the Wyndham Hill label I linked to near the start of this thread. 
Anything by John Thulin. His stuff is great.
Phillip Glass - Solo Piano Link
Also, from his Glassworks album, the song “Opening” is wonderful, but that album is not strictly piano.
I also enjoy
Music for Airports by Brian Eno, but it’s not strictly piano. Mostly, but not all. It will put you to sleep most definitely though. I absolutely love that album.
Music For Airports
Suzanne Ciani, a New Age artist, has 3 non-electronic albums featuring simple piano arrangements of her songs - “Pianissimo” volumes 1-3. They’re all available on CD and to download from itunes (for some reason amazon only has volumes 1 & 2 available to download.) Lovely melodies, very soothing.
Soothing? That’s “get up and jump around like a spastic cat” music!
Well, whatever works for you. 
I always enjoyed George Winston. http://www.amazon.com/Seasons-Piano-Solos-George-Winston/dp/B0009LDNCW
My roommate and I used to fall asleep to his music in college.
Carl Doy - any of his CD’s heres one example
Most of Bill Evans’ *Conversations with Myself *would fill the bill.
Raul di Blasio:
Both of these are from the album ‘Solo’ but I also highly recommend ‘Grand Exitos’. It’s the most interesting and varied mix of sad, sleepy, jazzy, fluty and modern piano I’ve heard.