# States and left ideals

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.

• 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). – David Handelman Nov 14 '15 at 19:31

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).

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

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.