I know of no "black star" unless you are talking about a black hole. A black hole is a star that runs out of fuel and collapses in on itself. A star is bigger than it normally would be due to the outward force created during the fusion process. (nuclear reaction) Once the star runs out of fuel, that outward force no longer exists and it collapses in on itself. If it does not contain enough mass per volume after collapsing to become a black hole, then it becomes a neutron star or white dwarf.
Can a star with fuel exists yet still be massive enough to create an event horizon? I don't know. If it could, it sure wouldn't last very long as the fusion would be come so extreme that it would burn all of it's fuel rather quickly and turn into a black hole anyway.
VY Canis Majoris which I believe is the largest known star is estimated (with some disagreements) between 600 and 2,100 times the size of our sun. Our sun is over 4 billion years old, yet VY Canis Majoris probably has a life span of only 1 million years due to it's immense size. So, even if a black star could exist, it would be short lived and become a black hole anyhow.
...whatever a black hole actually is.
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