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edited Sep 29, 2013 at 18:55

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I believe the problem here is the notation. Conway uses the notation

$H^{(n)}$ to mean the $n$-fold direct sum of $H$, and
$A^{(n)}$ for the $n$-fold direct sum of the operator $A\in B(H)$.
$\mathcal{A}$ for the $C^*$ or von Neumann algebra.

(I didn't have Conway's book in my office, but I looked at the preview on Google Books, and searched for "cardinal number," and I found Definition 6.3.(a). I don't know if he also uses the notation $\mathcal{A}^{(n)}$ for the $n$-fold direct sum of $\mathcal{A}$, but I can see how this could be confusing!)

In what you've written, $\mathcal{A}^{(n)}$ should be the $n$-fold amplification of $\mathcal{A}$, i.e., the image of the map $$ A\mapsto \begin{pmatrix} A & & \\ & \ddots & \\ & & A \end{pmatrix} \in B(H^{(n)}). $$ So $M_n(A)$$M_n(\mathcal{A})$ is the operators on $H^{(n)}$ commuting with the diagonal action of the algebra $\mathcal{A}'$, as expected. I hope this clears up the confusion.

I think it would be more clear to use lower case letters for operators and upper case letters for algebras, e.g., $a\in \mathcal{A}$.

I believe the problem here is the notation. Conway uses the notation

$H^{(n)}$ to mean the $n$-fold direct sum of $H$, and
$A^{(n)}$ for the $n$-fold direct sum of the operator $A\in B(H)$.
$\mathcal{A}$ for the $C^*$ or von Neumann algebra.

In what you've written, $\mathcal{A}^{(n)}$ should be the $n$-fold amplification of $\mathcal{A}$, i.e., the image of the map $$ A\mapsto \begin{pmatrix} A & & \\ & \ddots & \\ & & A \end{pmatrix} \in B(H^{(n)}). $$ So $M_n(A)$ is the operators on $H^{(n)}$ commuting with the diagonal action of the algebra $\mathcal{A}'$, as expected. I hope this clears up the confusion.

I think it would be more clear to use lower case letters for operators and upper case letters for algebras, e.g., $a\in \mathcal{A}$.

I believe the problem here is the notation. Conway uses the notation

$H^{(n)}$ to mean the $n$-fold direct sum of $H$, and
$A^{(n)}$ for the $n$-fold direct sum of the operator $A\in B(H)$.
$\mathcal{A}$ for the $C^*$ or von Neumann algebra.

In what you've written, $\mathcal{A}^{(n)}$ should be the $n$-fold amplification of $\mathcal{A}$, i.e., the image of the map $$ A\mapsto \begin{pmatrix} A & & \\ & \ddots & \\ & & A \end{pmatrix} \in B(H^{(n)}). $$ So $M_n(\mathcal{A})$ is the operators on $H^{(n)}$ commuting with the diagonal action of the algebra $\mathcal{A}'$, as expected. I hope this clears up the confusion.

I think it would be more clear to use lower case letters for operators and upper case letters for algebras, e.g., $a\in \mathcal{A}$.

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created Sep 29, 2013 at 18:27

Dave Penneys

5.4k
2
27
47

I believe the problem here is the notation. Conway uses the notation

$H^{(n)}$ to mean the $n$-fold direct sum of $H$, and
$A^{(n)}$ for the $n$-fold direct sum of the operator $A\in B(H)$.
$\mathcal{A}$ for the $C^*$ or von Neumann algebra.

In what you've written, $\mathcal{A}^{(n)}$ should be the $n$-fold amplification of $\mathcal{A}$, i.e., the image of the map $$ A\mapsto \begin{pmatrix} A & & \\ & \ddots & \\ & & A \end{pmatrix} \in B(H^{(n)}). $$ So $M_n(A)$ is the operators on $H^{(n)}$ commuting with the diagonal action of the algebra $\mathcal{A}'$, as expected. I hope this clears up the confusion.

I think it would be more clear to use lower case letters for operators and upper case letters for algebras, e.g., $a\in \mathcal{A}$.