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Here's a new look for electronic cash. Instead of having a bank where
people have accounts, have the "banker" be a money changer. He
changes Federal Reserve Notes into crypto dollars. Anybody can send
him dollars, and include an email address and public key, and he will
email them that same amount of electronic cash. Anybody can email him
electronic cash, and include a snail-mail address, and he will send
them that same amount of U.S. dollars.
You don't have a permanent relationship with him, you just use the
service when you need to change between electronic and paper dollars.
With the Chaum cash that was discussed here a while ago, where you
need to "deposit" it right away when you get it, here's what you could
do instead. Send it to the money changer, and ask for fresh crypto
cash back. It's like you combined a "deposit" with an exactly equal
"withdrawal". What you get back is good cash that you can hold or
spend as needed. There are no account numbers involved, no demand
deposits, no bank account at all.
Eric Hughes mentioned demand deposits as defining a bank, so maybe
this would be a way around that.