Theorem of the month – Dauns-Hofmann theorem

In this installment of THEOREM OF THE MONTH! we will look at another big theorem in operator theory, namely the Dauns-Hofmann theorem. To set the scene, we will first introduce the notion of the primitive ideal spectrum (or space), its topology and list without proofs some of its most important properties. The theorem extends beautifully the spectral theory for commutative C*-algebras by treating the resulting algebra as sections of certain bundles of C*-algebras over the primitive ideal spectrum rather than the more familiar spectrum of a commutative C*-algebra.

Primitive Ideals

There are various candidates for the spectrum of a noncommutative C*-algebra. We denote by $\hat{\mathcal{A}}$ the collection of all unitary equivalence classes of irreducible representations of $\mathcal{A}$ . Another candidate is the primitive ideal spectrum.

A primitive ideal $I\subset \mathcal{A}$ of a C*-algebra $\mathcal{A}$ is the kernel of an irreducible representation, that is, we can find an irreducible representation $\pi: \mathcal{A} \to B(H)$ such that $I = \pi^{-1}(0)$ . Primitive ideals are always closed two sided prime ideals and the converse holds if $\mathcal{A}$ is assumed to be separable. Any maximal ideal is automatically primitive (why? Spoiler – compose the quotient map with the universal representation of the quotient space, noting that $\mathcal{A}/I$ simple if $I$ is maximal). The term is borrowed from ring theory, where a primitive ideal of a ring $R$ is an ideal $I\subset R$ for which there is a simple left $R$ -module $M$ such that $I \cdot M = 0$ .

It turns out that ring theoretic primitive ideals of a C*-algebra $\mathcal{A}$ when treated simply as a ring agrees with the C*-algebraic primitive ideals defined above. This is one of the many consequences of Kaddisons transitivity theorem (the C*-algebraic equivalent of Jacobsons density theorem) which often ties the knot between the algebraic and topological structure of C*-algebras.

The primitive ideal space, denoted $Prim(\mathcal{A})$ is the collection of all primitive ideal endowed with the so called Jacobson topology or sometimes the hull-kernel topology, defined as follows: Let $I\subset \mathcal{A}$ be any subset and define

$hull(I) = \{ x\in Prim(\mathcal{A}) ~|~ I\subset x \} = Prim(\mathcal{A}/I)$

The collection of all $hull(I)$ where $I$ runs over all subsets of $\mathcal{A}$ satisfy the axioms of the closed sets of a topology, meaning their complements are precisely the open sets in the Jacobson topology. In this topology, given a collection of primitive ideals $J_\alpha$ their closure is

$\overline{J_\alpha} = \{J~|~ \cap J_\alpha \subset J \}.$

This assignment determines the topology uniquely (we have previously blogged about how topologies may be determined through by means of a closure function in a previous post).
Similarly one endows $\hat{\mathcal{A}}$ with the topology making the canonical map

$\pi \mapsto ker(\pi)$

a quotient map. This topology is not in general that nice as it satisfies very few our favorite separation axioms. Nonetheless, let’s list some properties of the primitive ideal space to get a feel for what we are working with:

$Prim(\mathcal{A})$ is always $T_0$ (while $\hat{\mathcal{A}}$ is $T_0$ if and only if $\hat{\mathcal{A}} = Prim(\mathcal{A})$ !).
$Prim(\mathcal{A})$ is compact if $\mathcal{A}$ is unital (just like the commutative case!).
$Prim(\mathcal{A})$ is invariant under strong Morita equivalence (hence also under stabilization).
$I\in Prim(\mathcal{A})$ is closed if and only if it is maximal (just like the Zariki topology on $Spec(R)$ )
Any closed subset $X\subset Prim(\mathcal{A})$ corresponds to an ideal $J_X := \cap_{I \not\in X} I$ .
If $\mathcal{A}$ is separable, then $Prim(\mathcal{A})$ is second countable Baire space.
We have a continuous open mapping $\Theta: P(\mathcal{A}) \to Prim(\mathcal{A})$ given by $\phi \mapsto ker(\pi_\phi)$ where $P(\mathcal{A})$ denotes the pure states on $\mathcal{A}$ with the weak*-topology (also called ultraweak topology) and $\pi_\phi$ denotes the GNS representation associated to $\phi$ (recall that the GNS representation is irreducible if and only if $\phi$ is a (scalar multiple of a) pure state).
For any type I C*-algebra the canonical quotient map $\hat{\mathcal{A}}\to Prim(\mathcal{A})$ is a homeomorphism and if $\mathcal{A}$ is separable the converse holds in which case $\mathcal{A}$ is liminal (or CCR) if and only if $Prim(\mathcal{A})$ is $T_1$ .
Morita equivalent C*-algebras have homeomorphic primitive ideal spaces.
Continuous trace C*-algebras have Hausdorff primitive ideal spaces.
If $\mathcal{A} = C_0(X)$ is a commutative C*-algebra, then $\hat{\mathcal{A}} = Prim(\mathcal{A}) = X$ .
For any $a\in \mathcal{A}$ we have $||a|| = \sup_{I\in Prim(\mathcal{A})}||a||_{\mathcal{A}/I}$ (By the GNS-theorem, knowing that $||x||= \sup{\phi \in P(\mathcal{A})} \phi(x)$ )
For any $a\in \mathcal{A}$ the map $a\mapsto ||a||_{\mathcal{A}/I}$ ( $I\in Prim(\mathcal{A})$ is lower semi-continuous. If $Prim(\mathcal{A})$ is Hausdorff, or $a\in Z(\mathcal{A})$ (the center of $\mathcal{A}$ ) then it is continuous.

The list could be made much much longer, but let’s stop and turn our focus back on the theorem

Dauns-Hofmann theorem

For the proof of the Dauns-Hofmann theorem we will need the following lemma (Lemma 1 ) which we restate word for word without proof here to shorten the exposition. We will also follow the proof from that article later on.

Lemma 1 Let $A$ be a linear space. Let $J_0, …, J_n$ be closed subspaces of $A$ such that for each $k = 1, …, n$ the canonical isomorphism

$\frac{J_0+ … + J_k}{(J_0 + … + J_k)\cap J_{k + 1}} \to \frac{J_0+ … + J_k + J_{k + 1}}{J_{k + 1}}$

is an isometry. Let $x\in J_0+ … + J_n$ . Then there exist $x_0\in J_0$ , …, $x_n \in J_n$ such that

$x = x_0+ … + x_n$

and for each $i = 0, …, n$ we have $||x_i||\leq 3||x||$

There are many theorems that go under the name Dauns-Hofmann theorem so the first order of business is to fix which one we are going consider in this post. The theorem we have in mind is the following –

Theorem 1 (Dauns-Hofmann) Let $\mathcal{A}$ be C*-algebra, $x\in \mathcal{A}$ and $f\in C_b(Prim(\mathcal{A}))$ any bounded continuous function on $Prim(\mathcal{A})$ . Then there exists a unique element $y \in \mathcal{A}$ such that $[f(I)x] = [y]$ in $\mathcal{A}/I$ for all $I\in Prime(\mathcal{A})$ .

Proof

What we are going to do is similar to the approximation of an $L^2$ -function by simple (or “step”) functions. We will partition the prime spectrum into pieces where the function $f$ is very close to a constant, then approximate $f(t)x(t)$ thereon, then letting $y$ be the limit as the approximation gets more and more refined.

By taking the real part of $f$ and scaling it, we may assume $f(Prim(\mathcal{A}))\subset [-1,1]$ . Writing $f = f^+ - f^-$ , where $f^\pm$ are positive functions, we can further assume $f(Prim(\mathcal{A})) \subset [0,1]$ .

For each fixed choice of positive integer $n = 1,2,...$ we get an open cover of $Prim(\mathcal{A})$ consisting of $n$ elements of the form

$U_k = f^{-1}\left(\frac{k-1}{n},\frac{k+1}{n}\right) \qquad 1\leq k \leq n.$

Note that on each $U_k$ we have $\left| f(t) - \frac{k}{n}\right| \leq \frac{1}{n}$ . For each $1\leq k \leq n$ , let $J_k$ be the ideal

$J_k = \bigcap_{I\in X\backslash U_k} I$

Since $U_i$ cover $X$ , we can readily check that

$\sum^n_{k = 1} J_k = A.$

Now using the previous lemma, we may write $x$ as

(1) $\begin{equation*} x = \sum_{k = 1}^n x_k \quad x_k\in J_k \quad \text{and} \quad ||x_k|| \leq 3||x|| \end{equation*}$

Since $f$ is “close” to the constant function $\frac{k}{n}$ on $U_k$ , the sensible candidate for aproximating $f(t)x$ would be

$y_n := \sum_{k = 1}^n \frac{k}{n}x_k.$

This actually does the trick, as we get the following chain of (in)equalities

$\begin{align*} ||f(t)x(t) - y_n(t)|| & = ||\sum_{k = 1}^n (f(t) - \frac{k}{n})x_k(t)|| \\ &\leq \sum_{k = 1}^n |(f(t) - \frac{k}{n})| || x_k(t)|| \\ & \leq \frac{1}{n} \sum_{k = 1}^n || x_k(t)|| \\ & = \frac{1}{n} \sum_{k = i_0}^{i_0 + 1} || x_k(t)|| \leq \frac{6}{n}||x|| \end{align*}$

The last equality follows from the fact that $x_k(t) \neq 0$ only if $t\in U_k$ , which happens for at most two values of $k$ , here denoted by $i_0$ and $i_{0}+1$ . The last inequality follows from the preceding lemma.

Finally since

$||y_n|| = sup_{I} ||y_n(I)||$

where $I$ runs over all the primitive ideal the sequence turns out to be Cauchy (hence convergent in $\mathcal{A}$ ). To see this, assume $m = n + s$ for some $n,s \in \N$ , then if

$x_1 + ... + x_m = \tilde{x}_1 + ... +\tilde{x}_n = x$

are the the sequence of equation (1) we have that if $I\in Prim(\mathcal{A})$ is such that $\frac{t_0}{m} \leq f(I) \leq \frac{t_0+1}{m}$ , then

$|| \sum^n_{i} \tilde{x}_i(I) - \sum^m_{x_j}(I)|| 0$

so, after a little computation one can show that

$||y_n(I) - y_m(I)|| \leq \frac{3||x||}{n}.$

Letting $y = \lim_{n}y_n \in \in A$ be the limit, one can easily verify that this has the desired properties.

It may be slightly counter intuitive that we associate the ideal $J_k$ to the open $U_k$ in the above theorem. Heuristically, one could think of the element $x\in \mathcal{A}$ as represented by a function $x: Prim(\mathcal{A}) \to \prod_{I\in Prim(\mathcal{A})}A/I$ given by $I\mapsto [x]\in \mathcal{A}/I$ , so its values on an open $U_k$ could be thought of as the “component” of (the function) $x$ that are non-zero on $U_k$ or equivalently, the component of (the element) $x$ that are contained in

$J_k = \bigcap_{I\in Prim(\mathcal{A} ) \backslash U_k } I$

Another formulation of the above theorem is given in and and yes Hoffman in the first reference is indeed Hofmann.

Corollary 1 (Dauns-Hofmann) Let $\mathcal{A}$ be a unital (resp. non-unital) C*-algebra and let $Z : = Z(\mathcal{A})$ (resp. $Z = Z(M(\mathcal{A}))$ ) its center. Then, for each $x\in Z_+$ , the function $x\mapsto \hat{x}$ where $\hat{x}(I) = ||x||_{\mathcal{A}/I}$ is continuousand extends to an isomorphism $Z \simeq C_0(Prim(\mathcal{A}))$ (rep. $Z \simeq C_b(Prim(\mathcal{A}))$ ).

Proof

We will repeat the argument of , which in turn seems to follow that of with small changes, namely we tackle the non-unital case.
The isomorphism in question is defined by the map which assigns to an $x\in Z$

$\hat{x}: Prim(\mathcal{A}) \to \R \qquad \hat{x}(I) = \sup{|x(J)|~|~ J \not\in I } = ||x||_{\mathcal{A}/I}.$

If $J \in Prim(\mathcal{A})$ let $\pi$ be an irreducible representation of $\mathcal{A}$ with kernel $J$ . Then $\pi$ restricts to a representation of $Z$ . Recall that for a representation $\pi$ being irreducible is equivalent to having trivial commutant, that is $\pi(M(\mathcal{A}))' = \C$ . It follows that we must have $\pi(Z) \subset \C$ , hence it is also irreducible (a character).
The kernel of this restricted representation in $Z$ is $Z\cap J$ . We can thus define a map

$\alpha : Prim(\mathcal{A})\to Prim(Z) \qquad J\mapsto J\cap Z.$

To see that $\alpha$ is continuous it suffices to show for any closed subset $I\subset Z$ the set $\alpha^{-1}(I)$ is closed in $Prim(\mathcal{A})$ . This can be shown as follows

$\begin{align*} \alpha^{-1}(I) & = \{ J\in Prim(\mathcal{A}) ~|~ \alpha^{-1}(I) \subset J\} \\ & = \overline{\alpha^{-1}(I)}. \end{align*}$

Now since $Z$ is commutative, given an irreducible representation $\pi$ of $Z$ there is a pure state $\pi$ on $Z$ such that $\pi = \pi_\phi$ (the GNS representation of $\phi$ ). It is always possible to extend pure states from $Z$ to $M(\mathcal{A})$ (Murphy Th. 5.1.13) and it is not that much work to check the corresponding GNS representation of $M(\mathcal{A})$ by this extended pure state indeed extends the representation $\pi$ , hence the map $\alpha$ is surjective.

It follows that the induced map

$\alpha_\star : C_0(\hat{Z}) \to C_b(Prim(\mathcal{A})) \qquad f\mapsto f\circ \alpha$

on the function algebras is injective.

Surjectivity of $\alpha_\star$ can be proved by putting $a = 1$ in Theorem 1 above. Explicitly this yields that for every $f\in C_b(Prim(\mathcal{A}))$ there is a $y\in M(\mathcal{A})$ such that

$|| f(t)1 - y(t)||_{M(\mathcal{A})/t} = 0 \qquad \text{for all }t\in Prim(\mathcal{A}).$

Note that $y\in Z(M(\mathcal{A}))$ , since for any $z\in M(\mathcal{A})$ we have

$||yz - zy|| = \sup_{t}||y(t)z(t) - z(t)y(t)|| = \sup_t|| f(t)z(t) - f(t)z(t)|| = 0.$

The map in the theorem is often referred to as the Dauns-Hofmann isomorphism. As we have seen, one can describe it either as the map on the spectrum (with $Z = Z(M(\mathcal{A}))$ )

$\sigma_{\mathcal{A}}: Prim(Z)\to Prim(\mathcal{A})\qquad I \mapsto I\cap Z$

or on the function algebras by

$\tilde{\sigma_\mathcal{A}}: C_b(Prim(\mathcal{A}))\to Z(M(\mathcal{A}))$

sending $f$ to the “multiplication by $f$ ” operator as the following lemma (left unproved) indicates:

Lemma 2 Keeping the notation of Theorem 1 above, isomorphism in Corollary 1 is the inverse of the assignment $\tilde{\sigma_\mathcal{A}}: C_b(Prim(\mathcal{A}))\to Z(M(\mathcal{A}))$ given by

$\tilde{\sigma_\mathcal{A}}(f)(x) = y, \qquad \text{for all }x\in \mathcal{A}$

Note that $\tilde{\sigma_\mathcal{A}}(f)$ is indeed in $Z(M(\mathcal{A}))$ since it is easy to check that

(2) $\begin{align*} \tilde{\sigma_\mathcal{A}}(f)(ab)= a\tilde{\sigma_\mathcal{A}}(f)(b) = \tilde{\sigma_\mathcal{A}}(f)(a) b \end{align*}$

and the fact that $T\in Z(M(\mathcal{A}))$ if and only if

$T(ab) = T(a)b = a T(b)$

for all $a, b\in \mathcal{A}$ .

**Continuous fields of C*-algebras and $C_0(X)$ -algebras**

Maybe the most common incarnation of the Dauns-Hofmann theorem is the one mentioned in the introduction to this blog post, which relates a C*-algebra to the C*-algebra of sections of a bundle over its primitive ideal space. This seems however to be a result of Fell. To make sense of the theorem we need to introduce the notion of C*-algebraic bundles and, for the sake of completeness, the closely related (but weaker) notion of a $C_0(X)$ algebra. The theory of bundles of C*-algebras is popular in physics, where it pops up in applications with zazzy names like (strict) deformation quantization. The consequence of this popularity is the existence of a plethora of (not always) equivalent definitions under the same name.

Definition 1 ( $C(X)$ -algebra) Given a locally compact Huasdorff space $X$ , a $C^*$ -algebra $\mathcal{A}$ is called a $C_0(X)$ -algebra if it comes equipped with a $\star$ -homomorphism $\phi: C_0(X)\to Z(M(\mathcal{A}))$ that is non-degenerate in the sense that $\phi( C_0(X))A = A$ .

Be warned that most authors use their own flavour of the above definition: some requiring that the map $\phi$ be injective (see Ex. 3 oc Appendix C in for a pathological example with non-injective maps), some requiring that the space $X$ be compact, $\sigma$ -compact or paracompact, some do not make any assumptions on the non-degeneracy of $\phi$ . We have opted for the definition found in and which is close to the one given in the original paper of Kapsarov .

Another definition found in the literature is the following

Definition 2 ( $C(X)$ -algebra) Given a locally compact Huasdorff space $X$ , a $C^*$ -algebra $\mathcal{A}$ is called a $C_0(X)$ -algebra if there is a continuous map

$\sigma_\mathcal{A}: Prim(\mathcal{A}) \to X$

These two definitions are indeed equivalent, see for instance Prop. C.5 for the proof. A third definition (which seems rather strange) is the following (see p. 135 of )

Definition 3 ( $C(X)$ -algebra) Let $X$ be a locally compact Hausdorff space. A $C_0(X)$ -algebra is a central $C(X)$ -module $\mathcal{A}$ with an involution satisfying

$(f\cdot a)^\star = a^\star \cdot \overline{f}$

I suspect the author intended to add a norm and a C*-identity to the definition, so use it with caution. There is a related notion of bundles of C*-algebras or continuous field of C*-algebras which seem to be used interchangeably. The following definition is taken from Dixmier –

Definition 4 (Continuous fields of C*-algebras) Let $X$ be a topological space and $(\mathcal{A}x){x\in X}$ a collection of C*-algebras parametrized by the points in $X$ . Let $\Gamma$ be collection of (some) functions $f: X\to \bigsqcup \mathcal{A}x$ such that $f(x)\in \mathcal{A}_x$ for all $x\in X$ (that is, $\Gamma \subset \prod \mathcal{A}_x$ ). The pair $(X, \Gamma)$ is said to be a continuous field of C*-algebras if the following conditions are satisfied:

$\Gamma$ is a complex subalgebra of $\prod \mathcal{A}x$ , that is closed under pointwise involution, addition and multiplication.
For every $x\in X$ , the set $f(x)$ for $f\in \Gamma$ is dense in $\mathcal{A}_x$ .
For every $f\in \Gamma$ , the map $x\mapsto ||f(x)||$ is continuous.
(Local approximation) If $f\in \prod\mathcal{A}_x$ is an arbitrary element which for each point $x\in X$ and every $\epsilon >0$ , there is a neighborhood $x\in U \subset X$ and a function $f_{U, \epsilon}\in \Gamma$ such that
$||f(y) - f_{U, \epsilon}(y)||< \epsilon$
for all $y\in U$ , then $f\in \Gamma$ .

In summary, a continuous field of C*-algebras is not a C*-algebra in its own right, but rather a collection of sections defined to be continuous. There is however a way to associate a C*-algebra to any continuous field of C*-algebras. This associated C*-algebra consists of the subset $\mathcal{A}\subset \Gamma$ of continuous sections that vanish at infinity with the supremum norm. We call this the C*-algebra associated with the continuous field or just the algebra of continuous sections.

Similarly we define an upper semi-continuous field of C*-algebras (or upper semi-continuous bundle) by only requiring that the map
$f\in \Gamma$ , the map $x\mapsto ||f(x)||$ in the above definition be upper semi-continuous.

The following theorem summarizes the basic facts about such bundles and $C_0(X)$ -algebras (Theorem C.28 ) –

Theorem 2 For a C*-algebra $\mathcal{A}$ , the following are equivalent

$\mathcal{A}$ is a $C_0(X)$ -algebra
The induced map $\sigma_{\mathcal{A}}: Prim(\mathcal{A})\to X$ is continuous
$\mathcal{A}$ is the upper semi-continuous sections vanishing at infinity of an upper semi-continuous field of C*-algebras over $X$ .

The section algebra in the third point above is a continuous field if and only if $\sigma_\mathcal{A}$ is an open and continuous map.

In particular when $Prim(\mathcal{A})$ is Hausdorff we can set $X = Prim(\mathcal{A})$ and $\sigma_\mathcal{A} = id$ and get the following corollary, often called the Dauns-Hofmann theorem:

Corollary 2 (Dauns-Hofmann) Let $\mathcal{A}$ be C*-algebra with $Prim(\mathcal{A})$ Hausdoff. Then $\mathcal{A}$ is isomorphic to the sections of a continuous field over $Prim(\mathcal{A})$ whose fibers at $I\in Prim(\mathcal{A})$ are isomorphic to $\mathcal{A}/I$ .

Note that taking $\sigma_\mathcal{A}$ to be any other open map produces a different realization of $\mathcal{A}$ as as continuous sections over $Prim(\mathcal{A})$ .
It should be emphasised also that this indeed is a direct extension of the characterization of commutative C*-algebras as continuous functions over its (maximal ideal spectrum = primitive ideal spectrum = character space). If $\mathcal{A} = C_0(X)$ then $Prim(\mathcal{A}) = X$ , the points in $X$ correspond to the maximal ideal

$I_x = \{f \in C_0(X) ~|~ f(x) = 0\}$

of functions vanishing at $x$ . It is easy to verify that $C(X)/I_x = \C$ for all points hence the trivial bundle

$X\times \C$

has fibers $C(X)/I_x$ at each point and the algebra $C_0(X)$ can be thought of as sections of this bundle vanishing at infinity. The above theorem then directly generalizes this characterization in the case where $Prim(\mathcal{A})$ is Hausdorff.

Example Before wrapping things up, let’s look at some trivial examples. Taking $q: X\to Y$ to be a quotient map, $C_0(X)$ is a $C_0(Y)$ -algebra. The algebra $C_0(X)$ is obtained as the continuous sections of a bundle over $Y$ if and only if $q$ is an open map. Note that all quotients by group actions are open maps, so for a locally compact $G$ -space $X$ the algebra $C(X)$ is a bundle of C*-algebras over $C(X/G).$ Be warned that the same does not always hold for the crossed product $C(X)\rtimes G$ however, which is always a $C(X/G)$ -algebra, but not a bundle in general (unless the group $G$ is amenable (see the Gootman–Rosenberg–Sauvageot theorem in ).

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Primitive Ideals

Dauns-Hofmann theorem

Continuous fields of C*-algebras and -algebras

References

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**Continuous fields of C*-algebras and $C_0(X)$ -algebras**