Most die sets are in fact Star, with only the sizer die being from LifeTyme of RLS. Star started selling reloaders with carbide sizing dies as an option in the 50's, and by the mid 70's no longer offered a steel sizing die as an option. (based on the price lists I have accumulated).
Bill Cunningham who bought the Star Reloader assets in 1993 sold parts from the Star inventory, but at some point had CH Tool and Die make some dies for him. I believe he also had Lee make some dies for him. That 40 S&W set that I use personally was purchased from Bill Cunningham five or six years ago. It has a carbide sizing die marked Lee. I have had two other 40 S&W sets pass through may hands, and both had the same unmarked carbide sizing die. The one that I have for sale now came directly from Star San Diego, but I don't know who made the sizing die. The rest of the dies in the set are obviously Star.
I have purchased several CH tool and Die Star type dies and they appear to be well made. The only Lee that I have is the afore mentioned 40 S&W sizer die, that came via Bill Cunningham and I have no complaints.
Does anyone know when Lifetyme went out of business? I have never seen a Lifetyme carbide sizing die in 40 S&W. Did Star offer only the Lifetyme carbide sizing dies as and OEM option, or die they also sell the RLS Perfection dies as an OEM option?