As in do they no longer have the aura of the deceased owner?

From an SR3 perspective:

SR3 Core, Page 190
"Only one person may bond a focus at a time, and only that person can use it. If you find or take someone else’s focus, you have to bond it to yourself before you can use it, which severs the bond with the previous owner. A focus always holds the astral signature of its owner (see Astral Signatures, p. 172). The connection between a focus and its owner can be tracked through astral space (see Astral Tracking, p. 176)."

Doesn't say anything about the link/signature dropping on death. We have always played it that it remains, which means taking a foci from a corporate guard / initiate group member etc is a dangerous thing. If they're working solo - that's fine, but if they have friends that might follow up on their death, they could run a trace on the signature, and potentially find you - and take retribution, making it a risky venture. Normally that kind of stuff is saved for when you have a decent force warded box that you can put the foci in, to make tracking it down so much harder - to reduce that risk.

My judgement call would be yes they do, but only until/unless someone else bonds to it.