Two notions of bundle of C* algebras

Question

One can find in the literature two notions of $C^*$-algebra over a topological space $X$.

The first is as the data of an open surjection $ \pi: B \rightarrow X$ together with the structure of a $C^*$-algebra on each fiber $B_x = \pi^{-1} \{ x \}$ such that:

• all the algebraic operations are continuous on the fiber products over $X$.

• The norm functions $b \mapsto \Vert b \Vert_{B_x}$ on $B$ is upper semicontinuous.

• There is one last condition that roughly says that the topology of $B$ is determined by the topology of $X$ and the norm functions.

One can eventually add additional condition: like that the norm function is continuous, or that there is enough of continuous local sections/global sections of the map $B \rightarrow X$.

One can also find an apparently completely different notion in the litterature:

A bundle of $C^*$-algebras over $X$ is the data of a $C^*$-algebra $A$ together with a continuous map $Prim(A) \rightarrow X$ where $Prim(A)$ is the space of primitive ideal of $A$ with the Jacobson topology.

When $X$ is nice (like locally compact Hausdorff , but completely regular and paracompact should be enough) it is well known that these definitions are equivalent (it follow from the Dauns-hoffmann theorem).

But what happen when $X$ is more general ?

I roughly see that it should works (one should not expect the two notions to be equivalent, but there is way to go from one to the other eventually under some assumptions and this should give a close relationship between the two notions. For example the second notion can only corresponds to bundle where there is enough global sections...) but I haven't look at the details yet.

I wanted to know first if there is some literature about this question ?

(If there is then I don't need to figure it out myself, and if there is none maybe I should figure it out and write something by myself)

Answer 1

While I don't know of any reference that answers this question explicitly, I have a few observations that might be helpful. For an upper-semicontinuous C$^*$-bundle $\pi : \mathcal{A} \to X$ and an open subset $U \subseteq X$, $\Gamma^b (U, \mathcal{A})$ denotes the C$^*$-algebras of continuous sections $s:U \to \mathcal{A}$ that are norm bounded. $\Gamma_0 (U, \mathcal{A} )$ denotes the C$^*$-subalgebra of $\Gamma^b(U,\mathcal{A})$ consisting of those sections $s$ that `vanish at infinity' on $U$ in the sense that $$ \{ x \in U : \| s(x) \| \geq \alpha \} $$ is compact for all $\alpha>0$.

1. Suppose that $A$ is a C*-algebra and $\phi: \mathrm{Prim}(A) \to X$ is continuous. Combining Remark 3.7.3 and Theorem 5.6 of the paper "Sheaves of C$^*$-algebras" by Ara and Mathieu, we do get an upper-semicontinuous C$^*$-bundle $\pi: \mathcal{B} \to X$ over $X$ together with a canonical embedding of the multiplier algebra $M(A)$ into the C$^*$-algebra $\Gamma^b (X, \mathcal{B} )$. If we then regard $A \subseteq \Gamma^b(X , \mathcal{B})$, one could then consider in each fibre $\mathcal{B}_x$ of $\mathcal{B}$ the subalgebra $$ \mathcal{A}_x:=\{ b \in \mathcal{B}_x : b=a(x)\mbox{ for some }a \in A \}. $$ It looks plausible that $A$ would then be isomorphic to (a subalgebra of) $\Gamma_0 (X, \mathcal{A} )$ where $\mathcal{A}$ is a sub-bundle of $\mathcal{B}$ with fibres $\mathcal{A}_x$, though I haven't checked the details.

2. As for the converse, suppose that $A = \Gamma_0 (X , \mathcal{A} )$ for some bundle $\pi : \mathcal{A} \to X$. I think that the existence of a map $\phi : \mathrm{Prim} (A) \to X$ can possibly be deduced from Lemma 2.25 of the paper "C*-algebras over topological spaces : the bootstrap class," by Meyer and Nest (here we need the assumption that $X$ is a sober space):

   Let $\mathcal{O}(X)$ and $\mathcal{I} (A)$ denote the directed sets of open subsets of $X$ and closed-two-sided ideals of $A$ respectively (both ordered with respect to inclusion). Then there is a bijective correspondance between continuous maps $\mathrm{Prim}(A) \to X$ and maps $\mathcal{O} (X) \to \mathcal{I} (A)$ that commute with arbitrary suprema and finite infima.

   Certainly, if $A$ is isomorphic to $\Gamma_0 (X, \mathcal{A} )$ for some u.s.c. C*-bundle $\pi:\mathcal{A} \to X$, then there is a natural map $\mathcal{O}(X) \to \mathcal{I} (A)$ where $U \subseteq X$ is identified with the ideal $$\Gamma_0(U,\mathcal{A})=\{ a \in A: a(x) = 0 \mbox{ for all } x \in X \backslash U \} $$ of $A$, which appears to have the required properties.

3. It is worth pointing out that when constructing the bundle in point 1. one cannot simply adapt the proof from the case of locally compact Hausdorff $X$ in general (the one used in Appendix C of "Crossed products of C$^*$-algebras" by Dana Williams for example).

   Indeed, for $x \in X$ define the ideal $$ I_x : = \bigcap \{ P \in \mathrm{Prim} (A) : \phi (P) = x \} $$ and the quotient C*-algebra $A_x = A/I_x$. Then there is a natural way to regard each $a \in A$ as a cross section $X \to \coprod_{x \in X} A_x$, letting $a(x)$ be the image of $a$ under the quotient mapping $A \to A/I_x$. Indeed, for locally compact Hausdorff $X$, this identification gives the required $*$-isomorphism $A \to \Gamma_0(X,\mathcal{A})$.

   Returning to the general case, if we take $X= \mathrm{Prim} (A)$ and $\phi$ to be the identity map, this construction yields an upper-semicontinuous C*-bundle with fibres $A_x$ and section algebra $\Gamma_0(X,\mathcal{A})\cong A$ if and only if $\mathrm{Prim} (A)$ is Hausdorff, in which case the bundle is in fact continuous. Thus it looks like the fibres of the bundle constructed in point 1. will not coincide with $A_x$ in general.