You set aside some funds on the global ledger and then you can use those funds to transact in a much more efficient, truly p2p manner, without touching the global ledger. Eventually, you net out everything and settle it all at once on-chain, utilizing its features to resolve any conflicts.

.. between the two parties that started the link, up to the initial value. It's so limited as to be useless. It's like a bar tab where you have to put down a deposit of the maximum amount you might spend first.

https://docs.lightning.engineering/the-lightning-network/pay...

But to make that work, you have to be online in real time to receive funds. It's not like on-chain transactions that complete without needing your direct involvement. If you're not online, then somebody has to do that on your behalf and now you're back to having someone else custody your funds. Here's a good article explaining it:

https://github.com/moneybutton/yours-channels/blob/master/do...

(You also need someone online monitoring for fraud, even without intermediates, but at least that part can be outsourced without trusting someone with custody.)

Assuming of course that there's a path between the sender and receiver, and that there's enough liquidity at each step of the way.

Otherwise, you need to setup a new channel, paying dearly for it once more.