If you have a chance encounter like that, you should always act like you don't know the other person; be discrete.

The adventurous part of me would like to stir up a conversation but given the nature of this business, discretion in public is important.

She could have been headed for a date or coming for a date.

It is an unwritten rule that you do speak to a provider unless she starts the conversation. It is hard enough determining who is L.E. without some stranger talking to her on the street or elevator

I had this happen to me once, we smiled at each other (I had seen her on the way up) and we shared a nice, quiet ride down to the first floor, and never saw each other again.

You should resist the urge to ask her for her fs rate.  Really now, its an unwritten rule that unless you are already engaged as provider/client at the time neither party should initiate conversation or make any gesture or expression as though you are familiar or know the other person.  perhaps a private setting might be an exemption but only if the two of you know each other.  if she does not know who you are, you would make her very very uncomfortable by making any statement which discloses your knowledge of her as a provider.  

I also wouldn't be surprised if she was associated with the provider you had just seen.

Elevator rules......

Look up at the floor numbers-don't make eye contact-don't talk unless absolutely necessary.
 
Most ladies want to come and go from a hotel discreetly and might feel threatened by someone too chatty.  In fact, many are taught not to get into an elevator with a lone male as it can be a place to be attacked!  And what if it is a case of mistaken identity?  You might out yourself!

It is usually best to say nothing and smile to yourself.  Then post it here on TER.

You recognize her, she does not know you at all.  
I would not say anything to her in that situation.
If I had seen her and we were alone I might politely say hello, but nothing else.
I would not ask why she was there or bring up anything hobby related.  
Her visit may not have been work related.   But whatever her reason for being there, it's none of my business.

Hmmm

How many times have we all seen someone we recognize from somewhere? I just smile and move on with my day.

Devilin

Tickle your ass with a feather??

What?! Excuse me??!!

I said it's particularly nasty weather, don't you think?