$\|A\|,\|B\|\leq 1$. How do I prove $\left(\begin{array}{}1&A&AB\\A^*&1&B\\B^*A^*&B^*&1\end{array}\right)$ is positive.

Question

$A,B$ are elements of a unital C*-algebra $\mathscr A$ with $\|A\|\leq 1$ and $\|B\|\leq 1$.

$X=\left(\begin{array}{}1&A&AB\\A^*&1&B\\B^*A^*&B^*&1\end{array}\right)\in M_3(\mathscr A)=\mathscr A\otimes M_3(\mathbb C)$ .

I tried to find a matrix such that $Y^*Y=X$ and failed.

Then I tried to show the invertibility of $X+\lambda$ for $\lambda>0$. This is equivalent to the invertibility of $X$ for $\|A\|,\|B\|<1$. And failed again.

Could you please give me some hint?

Answer 1

Hint $$ \pmatrix{ 1&0&0\\-A^*&1&0\\0&-B^*&1} \pmatrix{1&A&AB\\A^*&1&B\\B^*A^*&B^*&1} \pmatrix{ 1&-A&0\\ 0&1&-B\\0&0&1} = \pmatrix{1&0&0 \\ 0& 1-A^*A &0 \\ 0&0&1-B^*B} $$

Answer 2

Another elegant solution:

$\left(\begin{array}{}A&0&0\\1&0&0\\B^*&0&0\end{array}\right)\left(\begin{array}{}A^*&1&B\\0&0&0\\0&0&0\end{array}\right)=\left(\begin{array}{}AA^*&A&AB\\A^*&1&B\\B^*A^*&B^*&B^*B\end{array}\right)\leq \left(\begin{array}{}1&A&AB\\A^*&1&B\\B^*A^*&B^*&1\end{array}\right)$