Given a nontrivial left ideal $J$ of a unital $C^\*$ algebra $A$, is there a state on $A$ which vanishes on all elements of $J$? (Left or right doesn't matter, just not 2-sided.) 


The problem came from the idea of a state as evaluation at a 'point' of a noncommutative space. If an ideal corresponds to a vanishing 'set', then can we look at 'points' of that 'set'? I must admit that this problem as I stated it sounds unlikely to be true, but is there another version which might work?


I would also be interested in any related references which people could suggest about the theory of 1-sided ideals in operator algebras.



States and left ideals
Yes if you restrict to positive elements. This is because a closed subset of a unital C\*-algebra is a left ideal if and only if it is a left kernel of a certain state. This was first observed by Prosser:



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> R. T. Prosser, On the ideal structure of operator algebras, *Mem. Amer. Math. Soc.* **45** (1963).
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See also Theorem 10.2.10 in



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> R. V. Kadison and J. R. Ringrose, *Fundamentals of the theory of operator algebras II: Advanced theory*, Pure Appl. Math. **100**, Academic Press, Orlando, Florida, 1986.
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and Theorem 5.3.5 in 



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> G. J. Murphy, *$C^\ast$-algebras and operator theory*, Academic Press, Boston, MA, 1990.
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You will find the most comprehensive exposition of one-sided ideals of $C^\ast$-algebras in



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> T. Palmer, *Banach Algebras and the General Theory of ${}^\ast$-Algebras: Volume 2, ${}^\ast$-Algebras*
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