The only setup I'm aware of that could do what you say is e-cash (blind signatures, part of Open Transactions). But this algorithm requires a server to perform the settlement. Perhaps the servers could evolve to multiple hidden servers where very few trust is needed. If you could easily move money from one e-cash server to another (can we?), and if you could be sure they can't steal from you (multi-signatures!), then we could practically have an e-cash decentralized network. Perhaps it's time to study Open Transactions...
There is a concept in Open Transactions known as "voting pools" where a network of independent but cooperating transaction servers share public keys accept bitcoin deposits to multisig address and then issue BTC-backed OT assets for users to transact with in the OT system.
What this means is that several servers have to cooperate in order to process withdrawals therefore no single person can steal the bitcoins.
Open Transaction's security model is not as strong as Bitcoin's, but transaction servers in a voting pool can be set up to allow realtime public auditing of both their blockchain holding and the OT assets they issue based on them (to check for fractional reserve, etc).
I'm not a fan of off-chain transactions in general, but I think OT has enough additional features that it's worth building on right now, and in the long term we should get the necessary opcodes added to Bitcoin such that we would no longer need the OT transaction servers and could just process smart contracts directly on the blockchain.