Throughout let $B$ be a stable C\*-algebra, i.e. $B\cong B\otimes K$, where $K$ is the C\*-algebra of compact operators on an infinite dimensional separable Hilbert space. It is well-known that any countably generated Hilbert $B$-module $X$ is singly generated, i.e. there exists a positive element $b\in B$ such that $X\cong \overline{bB}$.


Assume the following as the definition of c.p.c. order zero maps between C\*-algebras.


**Definition** A completely positive linear map $\phi:A\to B$ between C\*-algebras has order zero if there exist a positive element $h\in\mathcal M(C)\cap C'$ and a $\*$-homomorphism $\pi:A\to\mathcal M(C)\cap\{h\}'$ such that $\Vert h\Vert = 1$ and
$$\phi(a) = h\pi(a)=\pi(a)h$$
for any $a\in A$, where $C = C^\*(\phi(A))\subset B$, i.e. the C\*-algebra generated by the image of $\phi$ and $\mathcal M(C)$ is the multiplier algebra of $C$.


Let $A$ be a separable C\*-algebra and let $\phi:A\to B$ be a c.p.c. order zero map. One can construct a Hilbert $B$-module $X\_\phi$ out of $\phi$ by setting
$$X\_\phi:=\overline{\phi(A)B}.$$
Conversely, given a countably generated Hilbert $B$-module $X$ and a separable C\*-algebra $A$, when is that there exists a c.p.c. order zero map $\phi:A\to B$ such that $X\cong\overline{\phi(A)B}$?


As a special case, suppose that $X$ is the inductive limit of a sequence of isometric inclusions of modules $\overline{\phi\_n(A)B}\hookrightarrow\overline{\phi\_{n+1}(A)B}$, where $\{\phi\_n\}\_{n\in\mathbb N}$ is a sequence of c.p.c. order zero maps. Is there a c.p.c. order zero map $\phi$ such that $X\cong\overline{\phi(A)B}$? I believe this last question has a positive answer when the connecting maps "commute" with the c.p.c. order zero maps, so I would rather be interested in the case where there are no *a priori* connections between the $\phi\_n$s.



When are countably generated Hilbert modules generated by c.p.c. order zero maps?
If $A=\mathbb C$ then the answer to both questions is yes.


If $A=M\_2(\mathbb C)$, then the modules $H=\overline{\phi(A)B}$ are those that have a direct sum decomposition $H\cong E\oplus E$ (where $E= \overline{\phi(e\_{1,1})B}$).
It is clear that not all modules need to have this property. The answer to the second question is also negative in this case. One can arrange for this: a locally compact space $X$ covered by compactly contained open sets $\bigcup\_n U\_n=X$
and a dimension 2 vector bundle over $X$ that is trivial on all the sets $U\_n$
but non-trivial on $X$ (a ``phantom" vector bundle). ($X$ can be obtained by a telescoping construction. See Example 5.6 of <http://arxiv.org/abs/0910.2967>). Viewing the vector bundle as a Hilbert module $H$ over $C\_0(X)$, the Hilbert modules $HC\_0(U\_n)$ have the desired direct sum decomposition but $H$ itself does not.


If $A=\mathcal K$, then the modules in question have the form $\bigoplus\_{n=1}^\infty E$ (a.k.a., a stable module). By Kasparov's stabilization $H$ is isomorphic to $\ell^2(I)$, where $I$ is a closed two-sided ideal of $B$. Again this need not exhaust all possible modules, but the second question has a positive answer in this case. If $H=\overline{\bigcup\_n H\_n}$ and all the modules $H\_n$ are stable (and countably generated) then $H$ is stable.


