I think that yes, I email my T to keep the connection and also because I usually have a lot of reactions to my session and I want her to know what they are. Since our rule is that she doesn't email me back except for emails about scheduling (or my birthday when she sent me a surprise birthday greeting

) I feel like it's not a burden on her for me to email her. All I ask is for her to read them.
I like to process my sessions via emails to her, at least the first day or two afterwards. Then I seem to be okay until my next session.
When my T used to respond to emails, I found her to be just as warm as she was in our sessions. She was briefer, though. She used to tell me that she agonized over what to write!

It was important for her to say the "right" thing. She doesn't like emailing because, like others say, words can get misinterpreted. I used to be disappointed when she didn't address all of my concerns, so now I just write and write, knowing she won't address anything. It works for me.