When I had just turned SIX, my father was an exhibitor in a home-computer show in a medium-sized mall. He brought me along. While he was working in his booth, I was allowed to wander that end of the mall (about 8 stores), ride on the nearby escalators, or sit by the fountain around the corner. I was not allowed to go into the department stores, stay upstairs (having ridden the escalator up, I was to come straight back down), or leave the building. If any adult told me to quit doing something (touching merchandise, etc.) I was to obey. The restroom was far away, so when I had to go I came to get my dad to take me into the men's room.
This worked out fine. Nobody made the slightest attempt to kidnap me, molest me, push LSD on me, or sell me into slavery. Nobody even "bothered" me. I had a lot of fun browsing the stores, watching the people, and drawing or reading on a bench by the fountain. I misbehaved slightly a few times but stopped when someone told me to. Only two bad things happened:
1. I left my bag unattended by the fountain when I went to show my dad a drawing I had made, and it was stolen.
2. Standing on the ledge next to the Up escalator and pushing my hands on the railing (pretending I was making it go), I forgot to let go and found myself riding up the outside of the escalator. When I did let go, I fell and skinned my knee.
Both of these were learning experiences. #1 led my dad to apologize for not telling me to keep my bag with me and to demonstrate how we make a police report. I never recovered the bag, which contained a favorite book, and felt the loss acutely. #2 brought a few strangers running to see if I was okay and walking me back to my dad.
It all depends on the child and the place, but in general the idea of an 8-year-old being unsupervised in public is fine with me. I see kids that young walking around alone or in small groups in my neighborhood sometimes. I wouldn't let a kid violate any posted policy against unsupervised children, though. In a mall, specifically, I think it's important to have a specific time and place to meet, and the younger the child the more often you check in. I wouldn't let a child younger than about 13 wander a mall if I wasn't in the same mall myself.