# Isomorphism between $\mathcal{L}_{M_n(A)}(E^n)$ and $\mathcal{L}_{A}(E^n)$

The following doubt came after reading the book "Hilbert C*-modules" by E.C. Lance. Let $$A$$ be a C*-algebra and $$E$$ a Hilbert $$A$$-module, there's a natural structure of Hilbert $$A$$-module on $$E^n$$ given by $$\langle (x_1,\dots,x_n),(y_1,\dots,y_n)\rangle_A = \sum \langle x_i,y_i\rangle$$ but also on page 39 of the book we're introduced a Hilbert $$M_n(A)$$-module structure on $$E^n$$ given by $$(x_1,\dots,x_n)\cdot (a_{ij})=\left(\sum x_ia_{i1},\dots ,\sum x_ia_{in}\right)$$ and $$\langle (x_1,\dots,x_n),(y_1,\dots,y_n)\rangle_{M_n(A)}=(\langle x_i,y_j\rangle)$$ Later on, on page 58, there's a result that states $$\mathcal{L}_{M_n(A)}(E^n)\simeq \mathcal{L}_{A}(E^n)$$. The *-homomorphism that establishes this isomorphism according to results on the previous pages seems to be $$T\mapsto T$$. I'm pretty sure this isn't exactly the isomorphism since $$T$$ being adjointable in the $$M_n(A)$$ sense doesn't seem to imply it being adjointable in the $$A$$ sense.

My question is: explicitly what would be the isomorphism between these two algebras? If $$\varphi:\mathcal{L}_{M_n(A)}(E^n)\rightarrow \mathcal{L}_{A}(E^n)$$ is the isomorphism then what would $$\varphi(T)(x_1,\dots,x_n)$$ be?

Realized the morphism $$T\mapsto T$$ actually works. Take $$x,y\in E^n$$, $$T\in \mathcal{L}_{M_n(A)}(E^n)$$ and write $$T=(T_1,\dots,T_n)$$ and $$T^*=(S_1,\dots,S_n)$$ where $$T_i,S_i:E^n\rightarrow E$$ are linear maps. Then $$(\langle T_i(x),y_j \rangle)_{i,j=1}^n=\langle T(x),y \rangle_{M_n(A)}=\langle x , S(y) \rangle_{M_n(A)}=(\langle x_i,S_j(y) \rangle)_{i,j=1}^n$$ meaning $$\langle T_i(x),y_j \rangle= \langle x_i,S_j(y) \rangle$$ for any $$i,j\in \{1,\dots, n \}$$. But now $$\langle T(x),y \rangle_{A}=\sum_{i=1}^n \langle T_i(x), y_i\rangle=\sum_{i=1}^n \langle x_i, S(y)\rangle=\langle x , T^*(y) \rangle$$ this means not only that $$T\in \mathcal{L}_A(E^n)$$ but that it has the same adjoint as in the $$M_n(A)$$ case and therefore $$T\mapsto T$$ is a *-isomorphism between $$\mathcal{L}_{M_n(A)}(E^n)$$ and $$\mathcal{L}_A(E^n)$$.