This is possible when you can create commodities out of thin air (like "paper silver").
I have thought of making a proof-of-concept "Casascius Bank" but where I accept no more than about 1000 BTC in total deposits.
The real goal, though, wouldn't be to become a bank. It would be to illustrate a proof-of-concept: provably secure depositing.
My "bank" would create cryptographic signatures on a daily basis proving in fact that I have every single bitcoin I say I have on deposit. Meanwhile, I would break down that 1000 BTC into "ranges" to assert that they are owned by specific depositors (but only in a manner where the depositors themselves can identify their own funds, not the public). For example, if I held 4 deposits for 4 people, and they were 100, 200, 300, and 400, then I would say that coins 1-100 belong to depositor 1, coins 201-400 belong to depositor 2, 401-600 and 601-1000 belong to depositors 3 and 4. I would sign and send messages to each depositor, which would allow them to reconcile, catch, and out me if (for example), I told two people they each own the same range of coins.
The real goal would be to help the public know they really can and should demand proof that their deposits are safe and sound and not fractionally pledged, the same way "provably fair gaming" is teaching us by example that we should demand, well, provably fair games.
In turn, this would cut down the number of "paper bitcoins" in existence (or rather, I should say, "bitcoin-denominated promises", because the "paper bitcoins" I have in mind - like bitaddress.org and bip38 - are actually close to the best kind of bitcoins you can have!)
I actually thought of the same thing recently, great minds think alike
, ahum .
You could publish the holdings of each user by providing the hash of their username and then the balance that corresponds to that user. EDIT: I see piramida posted a similar idea.
Of course, your users still need to trust you not to elope with the funds, but at least you can prove you are still holding the money.
I was actually thinking of this being used by a bitcoin backed central bank, which could do micro-transactions without clogging the blockchain. This bank would produce bitcoin-replacement currency that would be created the moment a user deposited bitcoins in a "shared address" (only possible to withdraw with approval of both parties, I don't know the proper term).
The downside is that this bank would be a single point of failure, so it is probably not a good idea until bitcoin is really mainstream and free of governmental crackdown fear.