BE350 pilot reported landing safely after the left engine was shut down due to fuel starvation related to fuel pump failure.

Narrative

Descending into ZZZ from ZZZ1. I started my descent checklist 12/15 minutes out from ZZZ; and noticed I had a 200LBS fuel imbalance on the left side. This was not an issue; because I was still landing with 700LBS on the left side; and 900LBS on the right. ATC started giving vectors for traffic; and for the R-NAV GPS Runway XX approach. After a few minutes; I was cleared for the approach. Upon joining the final approach course; the wings rolled level; and the left engine failed. I then reported to tower the problem; and was asked if I wanted to [request priority handling]. I replied that it was not a [priority]. That decision was made because I was by myself; without passengers. The aircraft was performing great on one engine; and I didn't want to cause problems with numerous aircraft waiting to take off. I landed the aircraft; and taxied to the ramp under my own power. Before shutting down; I had a look at the fuel gauges; and the left fuel gauge was empty. The next day; local mechanics; told me that the low pressure fuel pump gasket had completely failed. The fuel pump poured out about 110 gallons of fuel in 12/15 minutes; therefore starving the left engine of fuel. Looking back; maybe declaring an emergency would have been the right thing to do; but that's not the decision I made. The tower did not respond and I've yet to hear from the FAA. If a problem like this occurs again; I will declare an emergency; just so I have priority and everyone including tower; and aircraft wanting to take off and land are all alert and aware of the situation.