However the server has a large amount of ram memory and I suspect all of those indexes are already cached in ram.
Then there may be no benefit to be had.
I have read that tablespaces introduce overhead of maintenance and introduce complication for replication.
Yes, they are a nuisance for the humans who need to document, maintain, configure, etc. And they can induce administrators into making mistakes which can prolong outages or cause data loss.
But on the other hand I have this ssd disk ready for something.
That isn't a good reason. Unless your users are complaining, or you think they will be soon as things scale up, or you think they would be complaining of they weren't too apathetic to, then I would make no change that adds complexity just because the hardware exists.
But I would turn on track_io_timing, and load pg_stat_statements, and probably set up auto_explain. That way when problems do arrive, you will be prepared to tackle them with empirical data.