If I understand you correctly, the answer is yes. You could use a Bitcoin-like system simply to track the movement of 'shares' or 'units'. All the 'coins' could start out pre-mined, and the system's sole purpose could be to track the movement of those units.
However, there would be a lot of reasons why this would be a bad idea. The primary one is this: the Bitcoin system makes a lot of trade-offs to get the benefit of decentralization (nobody needs to trust anyone else at all ever). If you want to have one central authority anyway, you are trading off all those things for nothing.
Securing the block chain requires sufficient hashing power such that a malicious organization can't easily muster 51% of it. So to keep the system secure on a day-to-day basis, you'd need some way to fund that massive amount of computing power. It's hard to see what benefit you'd get back for that cost.
There are systems, even cryptographic ones with similar properties, that are more suitable for a centralized currency. The basic design change would be that instead of having a block chain, you'd simply have the central authority (or authorities) sign all transactions.
For example, if I had 50 units of this currency, I could request a signed and dated statement from the central authority saying that I had those coins and showing my public key. If I want to transfer 10 units to you, I make a signed statement saying so and get the central authority to sign it. You can now prove you have those 10 coins with that signed statement. I can prove I have 40 coins by showing the signed statements and the central authority can't argue I made a transaction I didn't make because I have to sign them. So you get the same security properties and don't need all the CPU power.