If you are selling a lightning channel (selling a lease for a set time duration) your liquidity would be stored in that channel (controlled by your lightning node) on the lightning network. Your lightning node is usually operating through your BTC node. All of these are your nodes and your wallets. So yes, sort of. It's indirectly on chain. Its just on layer two, the lightning network, fully under your control. You are only subject to your honor in keeping the channel open for the agreed upon duration for the agreed upon fees. If you fail, your reputation and ability to do such business on Amboss is all that's at stake. This assumes you can keep your lightning node up and running and have the knowledge/skills to recover your funds in case of a node failure.

In short, you have control but the risk is your LN node failing and your loss of funds due to failing to have proper backups/redundancies in case your node goes down. With proper operation knowledge and backup/restoration plans the risk is low. But there is risk.

That said, if you fire up a LN with say 1BTC and sell one channel for .5btc, this leaves you with .5btc to open other channels that will make your leased channel valuable to the person leasing it. For instance, if the only channel you have is that one .5btc channel then it is useless to the person you leased it to as no one can route a payment to them through it. They'll likely do their own due diligence and not lease from you in that case. However if they do, that's on them. But, you would only get the leasing fee and you wouldn't make any extra money routing payments through that channel to them.

If you didn't know this already, you probably don't have a lightning node. Perhaps a wallet but not a node. If that's the case, I'd suggest you begin by looking into node operation and see if the time and effort is worth it to you. If you are familiar with lightning nodes enough to know it's not worth the time and effort for you, then this isn't for you either.

No judgment, just the facts from assumptions based on your question.

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Thank you for the comprehensive sum up… i am running my own node for some time now but never played around with liquidity offers hence i am trying to understand the topic better.

But then this would imply i have 0.5 ₿ onchain which i can offer to be utilized and when successful this will be claimed for the channel to be opened which means at the end they will get outbound and i will get inbound capacity through the lease, if that understanding is correct ?

When you sell/lease a channel, you will open the channel to the buyer who has paid you for the channel. Since they are buying from you this means they want inbound liquidity for themselves. When you open to them, you will have the outbound from the start. However they wouldn't be buying it if they didn't plan on using it so the channel should move more liquidity to your inbound side over time. Depending on the buyer this could be very quickly. You'll have to decide on a fee rate and if/how you want to keep the channel balanced based on it's profitability to you and the terms of your agreement with them.

They want to receive payments through your channel. They're paying you for their inbound capacity to them.