Finite sums of commutators in C*-algebras

Question

I was looking at Lemma 1.4 below, and I couldn't figure out how Lemma 1.3, being about finite matrices, is used to prove Lemma 1.4. I would guess that there is some convergence involved but I am not sure.

Answer 1

You apply Lemma 1.3 to blocks.

In the first matrix, you take the upper left $3\times 3$ block and apply Lemma 1.3 with $1,-1/2,-1/2$. Next you consider the $12\times 12$ block with coefficients $1/4,1/4, 1/4, 1/4, -1/8,\ldots,-1/8$. And so on.

In the second matrix, after using the minus sign and ignoring the first zero $1\times 1$ block, you apply Lemma 1.3 to the $6\times 6$ block with coefficients $1/2, 1/2, -1/4, -1/4, -1/4, -1/4$. Then the $24\times 24$ block with coefficients $1/8,\ldots,1/8,-1/16,\ldots,-1/16$. Etc.

The choice of coefficients guarantees that the matrices are norm limits of matrices in the algebraic tensor product (I'm assuming it's $A\otimes K$, where $K$ are the compact operators; i.e. the stabilization of $A$).