# Thread: Show that two C*-algebras are morita equivalent

1. ## Show that two C*-algebras are morita equivalent

Let $\displaystyle A$ be a C*-algebra, I would like to show that $\displaystyle A$ is Morita equivalent to $\displaystyle M_l(A)$

Here follows a description of my plan to show this:
Define $\displaystyle F(A):=\{(a_1,\dots,a_l):a_i\in A\}$
let $\displaystyle \overline{a}:=(a_1,\dots,a_l)$ be the row matrix, then we can define the following inner products
$\displaystyle _{A}\langle \overline{a},\overline{b}\rangle:=\overline{a}\ove rline{b}^*$
$\displaystyle \langle \overline{a},\overline{b}\rangle_{M_l(A)}:=\overli ne{a}^*\overline{b}$

The above choice of inner product yields the desired result which we need to have an imprimitivity bimodule, namely
$\displaystyle _{A}\langle \overline{a},\overline{b}\rangle \overline{c}=\overline{a}\overline{b}^*\overline{c }=\overline{a}\langle \overline{b},\overline{c}\rangle_{M_l(A)}$

I can show that $\displaystyle F(A)$ in complete in the norm induced by the $\displaystyle A$-inner product. But not for the norm induced by the $\displaystyle M_l(A)$-inner product.

I can also show that $\displaystyle I_{A}:=\text{span}\{_{A}\langle \overline{a},\overline{b}\rangle:\overline{a},\ove rline{b}\in A\}$ is dense in $\displaystyle A$. Is it possible to show that
$\displaystyle I_{M_{l}(A)}:=\text{span}\{\langle \overline{a},\overline{b}\rangle_{M_l(A)}:\overlin e{a},\overline{b}\in A\}$ is dense in $\displaystyle M_l(A)$?

Or is this idea totally wrong?

2. I figured it out, here is an outline for those that might be interested

The idea I had worked perfectly well, we can show that
$\displaystyle I_{M_l(A)}=\{\langle \overline{a},\overline{b}\rangle\}$ is dense in $\displaystyle M_l(A)$ irrespective if $\displaystyle A$ is unital or not. The trick is in noticing that we can create matrices with either 1 (in the unital case) or the approximate identity (non-unital case) at the positions $\displaystyle (i,j)$. The span of these matrices will be dense in $\displaystyle M_l(A)$.

The Cauchy sequences are also easy since we are dealing with a finite dimensional space $\displaystyle F(A)$. The inner products $\displaystyle \langle\cdot,\cdot\rangle_{M_l(A)}$ and $\displaystyle _{A}\langle\cdot,\cdot\rangle$ induce well defined norms. Now there is a theorem which states that every finite dimensional normed space in complete.

All the conditions are then fullfilled and we see that $\displaystyle F(A)$ is in fact an $\displaystyle A-M_{l}(A)$-Imprimitivity bimodule and hence $\displaystyle A$ and $\displaystyle M_l(A)$ are Morita equivalent.