Let $A$ be a (unital) $C^\*$-algebra and $X,Y$ right Hilbert $A$-modules which are finitely generated and projective. It seems to be well-known that if $T: X \to Y$ is an $A$-linear map, then $T$ is necessarily adjointable. I could not find a proof though. Can someone give a reference or proof of this little fact?



Linear map between projective finitely generated Hilbert modules is adjointable
Looking at the proof of Lemma 6.21 in the notes of de Commer that you're reading (<https://arxiv.org/pdf/1604.00159.pdf>, per the comments), it seems like the relevant property of the modules is that of having compact identity operator. Suppose that a Hilbert $A$-module $X$ has this property. Since finite sums of the form $\sum\_{i} |x\_i\rangle\,\langle y\_i|$ are dense in the compact operators, some such sum must be very close to the identity, and hence invertible. Let $K=\left( \sum\_i |x\_i\rangle\,\langle y\_i|\right)^{-1}$. Then for every Hilbert module $Y$ and every $A$-linear map $T:X\to Y$ we have
\begin{equation\*}
T = T\circ \operatorname{id}\_{X} = \sum\_i |TKx\_i\rangle\,\langle y\_i|,
\end{equation\*}
which is compact and therefore adjointable.


To tie this to the actual question as asked: for Hilbert modules over unital $C^\*$-algebras, the property of having compact identity operator is equivalent to being finitely generated and projective in the purely algebraic sense, and it is also equivalent to being the image of an orthogonal projection on the Hilbert module $A^n$ for some $n$. These equivalences are explained in Wegge-Olsen's book on $K$-theory.


