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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.

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  • $\begingroup$ Classification results for closed right (or left!) ideals exist, and are closely related to the Cuntz group, K${}_0^*$; see Blackadar & Handelman, "Dimension functions,..." (ca 1982), and recent developments in the past few years in the use of the Cuntz group (many references). $\endgroup$ – David Handelman Nov 14 '15 at 19:31
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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:

R. T. Prosser, On the ideal structure of operator algebras, Mem. Amer. Math. Soc. 45 (1963).

See also Theorem 10.2.10 in

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.

and Theorem 5.3.5 in

G. J. Murphy, $C^\ast$-algebras and operator theory, Academic Press, Boston, MA, 1990.

You will find the most comprehensive exposition of one-sided ideals of $C^\ast$-algebras in

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

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Just a couple of minor comments on Tomek's answer.

1) I don't think you need to restrict to positive elements. Indeed, if $I$ is a closed left ideal then $I=AI_+$ where $I_+=I\cap A_+=$ the positive elements of $I$. And for any $a\in A$, $b\in I_+$ and state $\phi$ vanishing on $I_+$, Cauchy-Schwarz yields $$|\phi(ab)|^2\leq\phi(aba^*)\phi(b)=0,$$ so $\phi$ also vanishes on $I$.

2) For separable C*-algebras, each proper closed left ideal is the left kernel of a single state, but not in general. For example, there is no faithful state on the Calkin algebra $\mathcal{B}(H)/\mathcal{K}(H)$ or even its commutative C*-subalgebra $C(\beta\mathbb{N}\setminus\mathbb{N})$, i.e. $\{0\}$ is a proper closed (left) ideal but not the left kernel of a single state. Rather, in general the closed left ideals correspond to (weak*-)closed faces of states ($F$ is a face of a convex subset $C$ of a vector space if, for all $x,y\in C$ and $t\in(0,1)$, $x,y\in F\Leftrightarrow tx+(1-t)y\in F$). Specifically, for any $F\subseteq A^*$ and $I\subseteq A$ define $$\begin{align*} {}^\perp F &=\{a\in A:\forall \phi\in F(a\phi=0)\}\text{ and}\\ I^\perp &=\{\phi\in A^*:\forall a\in I(a\phi=0)\text{ and }\phi(1)=1=||\phi||\}, \end{align*}$$ where $a\phi\in A^*$ is defined by $a\phi(b)=\phi(ba)$. Then the maps $F\mapsto{}^\perp F$ and $I\mapsto I^\perp$ are mutually inverse bijections between closed faces of states and closed left ideals.

Personally, I think the best reference for this kind of thing is the following.

Pedersen, Gert K. $C^∗$-algebras and their automorphism groups. London Mathematical Society Monographs, 14. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. ix+416 pp. ISBN: 0-12-549450-5.

It sounds like you might be interested in the non-commutative topological theory of C*-algebras, in which case you could also take a look at various papers by Akemann et al.

But if you don't want to go through the references, here's an outline of a proof answering your original question:

Take a proper left ideal $I$ of $A$. Note no $a\in I$ can be (left) invertible as then we would have $1=a^{-1}a\in I$ and hence $I=A$. This means $0\in\sigma(a)$ and hence $\lambda\in\sigma(a+\lambda1)$ so $|\lambda|\leq||a+\lambda1||$, for all $a\in I$ and $\lambda\in\mathbb{C}$. Thus the linear functional $\phi$ defined on $I+\mathbb{C}1$ by $\phi(a+\lambda1)=\lambda$ has norm $1$ and, by Hahn-Banach, has a norm $1$ extension to $A$. Thus this extension is a state on $A$ vanishing on $I$.

The same is true even for non-unital $A$ so long as you are talking about proper closed left ideals.

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