With communication at the speed of Jump, how do financial transactions work across the Imperium?

The most common day-to-day transaction are with coins, backed by complex cryptography and electronics. The most common types are given below.

The above are designed for use in automatic payment machines, and are in common usage, especially under Cr100. They are minted by a major bank normally located at the Sector capital, and are accepted all across the Imperium. The advantage over 'electronic' money is that they are often accepted on worlds which don't have the means to perform electronic funds transfer.

Up to Cr50 are coins. Larger denominations are notes, a sort of plastic material with electronics embedded inside. Cr1,000 and larger are 'large' notes and are very rare, and generally not accepted by automatic machines which is a deliberate choice to prevent them being in common use except for transfer of funds between worlds.

KCr1, KCr2, KCr5, KCr10, KCr20, KCr50

Under KCr10 are sometimes seen in expensive restaurants or bars, but the larger KCr10+ denominations are generally used for transferring funds between systems. If you need to buy a ship in a remote system, and your bank doesn't have a branch in that system, sometimes it's easiest just to take coins. Using these coins in normal day to day transactions thought will cause difficulties.

Finally, there are coins larger than even the above, of KCr100, KCr200, KCr500 and MCr1. These are only ever used for transferring money directly between banks, and are almost never used by 'normal people'. Only banks ever accept them, and will be very suspicious of anyone not authorized to use them trying to.

All A and B class star systems, and many C class systems, will have banks hooked into the galactic financial grid. If you have an account in the system, you can spend and receive money directly via the local network.

It is generally easy to setup a new account with cash, so you don't have to carry coins around with you.

Sometimes it's useful to be able to draw an allowance, such as for expenses, from multiple systems. Rather than moving the entire account, the user has a card with a cryptographically signed script on it, which controls how much can be withdrawn from a generic account. When the user enters the system, they sign on and the bank updates the card to record that it is registered with that system.

They can then use the card according to the rules on it. For example, they might be allowed to withdraw Cr1,000 a month, so that's what they will be limited to.

A history of transactions is stored on the card, probably using some form of blockchain. This means you can't withdraw in one system, then go to a second system and withdraw there to avoid any limits - the card knows what you've done and where, and will upload that information to the new bank.