They can only prosecute for trespassing if the owner of the establishment asks you to leave after seeing it, and you refuse. But that is all, and the fine for trespassing is minute.
At least in Texas.
OK, I looked up the law, and Minnesota is similar.
Subd. 17.Posting; trespass.(a) A person carrying a firearm on or about his or her person or clothes under a permit or otherwise who remains at a private establishment knowing that the operator of the establishment or its agent has made a reasonable request that firearms not be brought into the establishment may be ordered to leave the premises. A person who fails to leave when so requested is guilty of a petty misdemeanor. The fine for a first offense must not exceed $25. Notwithstanding section 609.531, a firearm carried in violation of this subdivision is not subject to forfeiture.
(b) As used in this subdivision, the terms in this paragraph have the meanings given.
(1) "Reasonable request" means a request made under the following circumstances:
(i) the requester has prominently posted a conspicuous sign at every entrance to the establishment containing the following language: "(INDICATE IDENTITY OF OPERATOR) BANS GUNS IN THESE PREMISES."; or
(ii) the requester or the requester's agent personally informs the person that guns are prohibited in the premises and demands compliance.
(2) "Prominently" means readily visible and within four feet laterally of the entrance with the bottom of the sign at a height of four to six feet above the floor.
(3) "Conspicuous" means lettering in black arial typeface at least 1-1/2 inches in height against a bright contrasting background that is at least 187 square inches in area.
(4) "Private establishment" means a building, structure, or portion thereof that is owned, leased, controlled, or operated by a nongovernmental entity for a nongovernmental purpose.
(c) The owner or operator of a private establishment may not prohibit the lawful carry or possession of firearms in a parking facility or parking area.
(d) This subdivision does not apply to private residences. The lawful possessor of a private residence may prohibit firearms, and provide notice thereof, in any lawful manner.
(e) A landlord may not restrict the lawful carry or possession of firearms by tenants or their guests.
(f) Notwithstanding any inconsistent provisions in section 609.605, this subdivision sets forth the exclusive criteria to notify a permit holder when otherwise lawful firearm possession is not allowed in a private establishment and sets forth the exclusive penalty for such activity.
(g) This subdivision does not apply to:
(1) an active licensed peace officer; or
(2) a security guard acting in the course and scope of employment.
Of course, I can't carry at work, as my employer has excercised its right to prohibit me.