Some places such as shopping malls have no gun
signs posted that are so small and conspicuous it
would make you wonder why they posted it at all.

The reason is simple. Liability.

The mall owners know that bad people don't care
about signs. They also don't want to be held
responsible if your hurt, because mall security was
unable to defend you from an armed robber after
you left your gun in the car due to the sign.

But if you happen to defend yourself with your
firearm and miss hitting an innocent person or have
an unintentional negligent discharge they also want
to be able to say that they had a sign posted that
guns were not allowed inside the mall.

I am not aware of any Florida state laws that make it
illegal to carry a legally concealed firearm into a
private place of business that prohibits firearms.

However, if you are asked to leave the private
property or secure your firearm in your car and you
refuse you maybe arrested for "trespassing with a
firearm", a much more serious charge than simply
trespassing.

Keep your gun properly secured and concealed in
public.  No one should be able to see that you're
packing.  If they can tell you are carrying a gun, you
give up your tactical advantage of surprise and you
now become the first target of the bad guy or the
nervous shopper who reports you to store security
while the terrorist quietly checks out through the
cashier with 10 propane bottles, 10 teddy bears, and
10 boxes of bb's. Yes, that is a true story. Nobody
thought to ask the man why he was buying the items
until after he left the store.