Lightning network without main chain? I've been reading about the working of LN. I had this interesting thought, what if we dont need the main blockchain and the miners to have a p2p cash system. Let me explain.

The difference between the current LN and what I thought is, people have channels only with the people they know and trust. There wont be any multisig wallet(may be set a limit for each channel based on their trust of the other person). And when they wanna close the channel, they transact in real life either with bitcoin or fiat (whatever unit they are using). Since it the people they trust there wont be any cheating problem. 

This will be a completely p2p system. Would such system work? Am I missing anything?
You've basically described Ripple (aka XRP). You may want to read up on it - and specifically, criticism of it that can help illuminate why this system does not offer the same benefits that Bitcoin's does.