Is entering a closed but unlocked door without force considered trespass or break and enter?the incident was at a shopping complex when the shop...

As stated above, it does depend on the state, but let me
pose the situation in another way.  Suppose it was a private home, an the owner left the
door unlocked at night when he went to bed.  Can I walk into the house and only be
considered trespassing?  The homeowner's failure to lock the door is not legal
justification for entry, as we have a right to privacy the courts have consistently
upheld.

When we talk in a legal sense about private
property, and especially in the case of a business where property is maintained and a
target for thieves, "breaking" the plane of the doorway outside of business hours would
most likely be considered breaking and entering as well, and/or burglary.  If someone is
inside, then most likely robbery would be the charge.