I do not believe the above is correct. Again, the idea of leaving property in the care of another is nothing new, it is covered under the section of law known as bailments. That law defines the duties of holders of property relative to costs and abandonment of property.
The owner of a machine never had the ownership of your personal property when you're using his machine or even if you abandon it on his machine.
I want to point out there is no such thing as "finders keepers". If you lose something, you have not lost ownership in it, and someone finding it does not gain ownership in it. For example, if the doors of an armoured truck break open and the money flies out (which has happened several times), it is theft to keep any such money you find and people have gone to jail taking such money. One person who found a large sum of money in the street and kept it went to jail because he failed to make a reasonable effort -- such as reporting the find to the police. The police had a report of the missing money and would've matched it to the owner.
As to intellectual property, this isn't anything new either. Say I go to a library and write up notes on paper, then leave the papers in a book: Those papers remain my property.
JT mentioned internal temporary files -- qsuch things probably never were a person's property in the first place because of their temporary nature. However files I create would be my property.
Now if the computer use was intended to be temporary, the owner would have no obligation to maintain files after I've gone -- they could run an automatic sweep. But if someone comes in asking for a file and it happens to be still available on the computer, they are entitled to get their file back.
I do not believe the owner of any machine ever has ownership of your property that you're running on it, any more than the landlord of your apt has any ownership interests in your furniture kept in an apt you're renting from him.
If I leave an expensive coat at a restaurant, that coat still belongs to me -- the restaurant owner cannot simply give it away and must hold it for a period of time for me to return to claim. Bailment law defines what the hold period would be and the proper care the owner must exercise while holding the coat. Obviously I can't show up five years later and claim it. If I checked the coat in their checkroom, they shouldn't store it next to the stove where grease will splatter on it, but on the other hand, they don't need to store it in a climate controlled vault either.
As to seizing _anything_ for a criminal investigation, property ownership is irrlevent except to state who receives the search warrant. If the cops think there's evidence in a restaurant to catch a criminal who frequents that restaurant, they will search the restaurant and its fixtures.
As to finding confidential information, it is theft to use such information, in some cases a very serious felony. As to a computer rental place, it may be necessary for the store to automatically clean the file space (as my public library does between users) or alert customers to do so. A public service has a duty to take _reasonable_ precautions to protect its customers, just like the restaurant must make a reasonable effort to protect an expensive overcoat left behind, and just as a checkroom won't hang up checked coats in the back alley in the rain. Generally, when such reasonable precautions are taken there is no further liability. Some rental agreements expressly limit or even deny liability -- you see signs in restaurants "not responsible for lost or stolen items".