The easy way out

Most countries don't have a 'central bank' in the way that some people in this thread are talking about.

In the US, there is one entity that is allowed to create money, but all the banks do their own thing and interact with each other without involving this central entity, except in unusual situations. There is a Government entity that does audits of Banks to see if they are being run in what are considered 'sane' ways by the current policies (and a different Government entity that does audits of Credit Unions)

Backups and records that the other entities keep can do a lot to track down fraudulent transactions, but that is a matter of enlightend self interest (if an entity can't prove what happened, they are liable)

As for bitcoins maintaining a history of transactions forever, that seems to me like a very bad thing.

Imagine that a bitcoin that you get paid was used at some point in the past by a terrorist or druglord. The "asset forfeiture" laws say that governments (not just in the US) can grab any assets that they had. Now you can loose the bitcoin because it can be proved that it went through their hands X years and Y transactions ago.