Primitive, factor and prime ideals

Every primitive ideal of a C*-algebra is closed and prime. A long-standing problem of Dixmier asked whether the converse holds, and this was settled by Weaver in 2003, when he gave the first example [1] of a prime C*-algebra that is not primitive. Between primitive ideals and closed prime ideals lies the class of factor ideals, and Weaver’s example also shows that there exist closed prime ideals that are not factor ideals. This leads to the following question:

Is every factor ideal in a C*-algebra primitive?

Background/Motivation: Given a C*-algebra $A$ , a factor representation is a representation $\pi\colon A \to B(H)$ such that the generated von Neumann algebra $\pi(A)''$ is a factor, that is, $Z(\pi(A)'')=\mathbb{C}1$ . One can further distinguish the type of the factor. In particular, an irreducible representation is a type I factor representation. A (two-sided) ideal $I \subseteq A$ is:

primitive if $I = \ker(\pi)$ for some irreducible representation $\pi\colon A \to B(H)$ ;
factor if $I = \ker(\pi)$ for some factor representation $\pi \colon A \to B(H)$ ;
prime if for any ideals $J,K \subseteq A$ with $JK \subseteq I$ we have $J \subseteq I$ or $K \subseteq I$ .

Since irreducible representations are factor representations, we see that every primitive ideal is a factor ideal. Let us see that every factor ideal is a closed prime ideal.

First, if $p,q$ are nonzero projections in a factor $M$ , then $pMq \neq {0}$ . Indeed, projections in a factor are totally ordered with respect to Murray–von Neumann subequivalence, so we may assume that $p \precsim q$ . Then there exists $v \in M$ such that $p = vv^*$ and $v^*v \leq q$ , which implies that $0 \neq v = pvq \in pMq$ . Next, if $x,y$ are nonzereo elements in a factor $M$ , then using Borel functional calculus for $xx^*$ we can find $a \in M$ such that $p:=xx^*a$ is a nonzero projection, and similarly there is $b \in M$ such that $q := by^*y$ is a nonzero projection. It follows that $\{0\} \neq pMq = xx^*aMby^*y \subseteq xMy$ .

Now let $I$ be a factor ideal, that is, $I = \ker(\pi)$ for a representation $\pi \colon A \to B(H)$ such that $M := \pi(A)''$ is a factor, and let $J,K \subseteq A$ be ideals with $JK \subseteq I$ . Then $\pi(J)\pi(A)\pi(K) = \{0\}$ , which implies that $\pi(J) M \pi(K) = \{0\}$ . Then $\pi(J)=\{0\}$ or $\pi(K)=\{0\}$ , and so $J \subseteq I$ or $K \subseteq I$ , showing that $I$ is a prime ideal.

Letting $\mathrm{Prim}(A)$ , $\mathrm{Fac}(A)$ and $\mathrm{Prime}(A)$ denote the set of primitive, of factor, and of closed prime ideals of $A$ , respectively, we thus have the following inclusions:

$\mathrm{Prim}(A) \subseteq \mathrm{Fac}(A) \subseteq \mathrm{Prime}(A).$

There are three important cases in which closed prime ideals are automatically primitive, and hence $\mathrm{Prim}(A) = \mathrm{Fac}(A) = \mathrm{Prime}(A)$ .

Separable C*-algebras. Dixmier [2] showed that every closed prime ideal of a separable C*-algebra is primitive, see also Corollary II.6.5.15 in [3]. To see this, let $A$ be a separable C*-algebras, and let $I \in \mathrm{Prime}(A)$ . Passing to the quotient $A/I$ , we may assume that $A$ is prime (i.e. $\{0\}$ is a prime ideal) and we need to show that $A$ is primitive (i.e. admits a faithful, irreducible representation). We consider the primitive ideal space $X := \mathrm{Prim}(A)$ equipped with the hull-kernel topology, that is, a subset $F \subseteq X$ is closed if and only if $F=\{J \in X : K \subseteq J\}$ for some closed ideal $K$ (namely $K = \bigcap F$ ). Then $X$ is irreducible in the sense that if $F_1,F_2 \subseteq X$ are closed subsets with $X \subseteq F_1 \cup F_2$ , then $X \subseteq F_1$ or $X \subseteq F_2$ . Indeed, the closed ideals $K_1 := \bigcap F_1$ and $K_2 := \bigcap F_2$ satisfy

$K_1K_2 \subseteq K_1 \cap K_2 \subseteq \bigcap X = \{0\}$

since $X \subseteq F_1 \cup F_2$

. Since $\{0\}$

is prime, we get $K_1\subseteq \{0\}$

or $K_2 \subseteq \{0\}$

, which gives $X \subseteq F_1$

or $X \subseteq F_2$

Now let $U \subseteq X$ be a nonempty open subset. Then $U$ is dense, since $X \subseteq (X \setminus U) \cup \overline{U}$ and $X \nsubseteq X \setminus U$ , hence $X \subseteq \overline{U}$ . In general, the primitive ideal space of a C*-algebra is a Baire space (the intersection of a countable family of dense open sets is dense). Now, if $A$ is separable, then $X = \mathrm{Prim}(A)$ is second-countable. Let $U_1,U_2,\dots$ be a basis of nonempty open subsets of $X$ . Using that $X$ is a Baire space, the intersection $G:=\bigcap_n U_n$ is dense in $X$ , and in particular it is not empty. Pick $J \in G$ , which is a primitive ideal of $A$ . Then $J$ is contained in every nonempty open subset of $X$ , and hence $\{J\}$ is dense in $X$ . Then

$X = \overline{\{J\}} = \{P\in X : J \subseteq P\},$

which means that every primitive ideal of $A$

contains $J$

. But the intersection of all primitive ideals of $A$

is $\{0\}$

, and thus $J=\{0\}$

, showing that $\{0\}$

is a primitive ideal.

Von Neumann algebras. Every closed prime ideal in a von Neumann algebra (and more generally, in every AW*-algebra) is primitive. To see this, let $P$ be a closed, prime ideal in an AW*-algebra $A$ . Let $Z$ denote the center and consider $P \cap Z$ . Then $M := P \cap Z$ is a closed, prime ideal of $Z$ . Since closed, prime ideals in commutative C*-algebras are maximal, it follows that $M$ is a maximal ideal of $Z$ . Using that $M$ belongs to the center and applying Cohen’s factorization theorem, it follows that the closed ideal of $A$ generated by $M$ is equal to $AM$ . Then the quotient $A/AM$ is an AW* factor, and we have $AM \subseteq P$ . It was shown in Theorem 6 in [4] that the closed ideals of an AW* factor are well-ordered by inclusion. Since every closed ideal of a C*-algebra is an intersection of primitive ideals, it follows that every (proper) closed ideal of an AW* factor is primitive (see Corollary 7 in [4]). In particular, the (proper) closed ideal $P/AM$ of $A/AM$ is primitive, and it follows that $P$ is a primitive ideal of $A$ .

Type I C*-algebras. Kaplansky showed in Lemma 7.4 in [5] that every prime C*-algebra of type I is primitive. Since type I passes to quotients, we get that closed prime ideals in type I C*-algebras are primitive.

A C*-algebra has type I if and only if its bidual $A^{**}$ is a type I von Neumann algebra. Further, a C*-algebra is type I if and only if every of its factor representations is type I. Thus, a C*-algebra is not type I if and only if it has factor representations of type II or type III. Sakai showed that type III is always realized, that is, a C*-algebra is not type I if and only if it has a factor representation of type III. See IV.1.1.3 and IV.1.5.8 in [3]. For separable C*-algebras, the analog for type II is also known: A separable C*-algebra is not of type I if and only if it admits a factor representation of type II. The nonseparable case seems to be still open:

Does every C*-algebra that is not of type I admit a factor representation of type II?

One can further distinguish type $\mathrm{II}_1$ and type $\mathrm{II}_\infty$ . A non-type I C*-algebra need not have factor representations of type $\mathrm{II}_1$ . For example, simple purely infinite C*-algebras do not admit factor representations of type $\mathrm{II}_1$ , since this would entail the existence of a tracial state. In fact, a unital, simple, non-elementary C*-algebra admits a factor representations of type $\mathrm{II}_1$ if and only it has a tracial state. Simple, purely infinite C*-algebras can have factor representations of type $\mathrm{II}_\infty$ . For the Calkin algebra, this was shown in [6,7].

[1] Weaver. A prime C*-algebra that is not primitive. J. Funct. Anal. 203 (2003), 356–361.

[2] Dixmier. C*-algebras. North-Holland Math. Library, Vol. 15, 1977.

[3] Blackadar. Operator algebras. Theory of C*-algebras and von Neumann algebras. Encyclopaedia Math. Sci., 122, Springer 2006.

[4] Saitô, Wright. Ideals of factors. Rend. Circ. Mat. Palermo (2) 55 (2006), 360–368.

[5] Kaplansky. The structure of certain operator algebras. Trans. Amer. Math. Soc. 70 (1951), 219–255.

[6] Anderson, Bunce. A type II-infty factor representation of the Calkin algebra [Amer. J. Math. 99]

[7] Anderson. Extreme points in sets of positive linear maps on B(H). J. Functional Analysis 31 (1979), 195–217.

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