For a finite integer $N$, let $A_n = \bigotimes^n M_N(\mathbb{C})$. $A_n$ embeds in $A_{n+1}$. Let $A_\infty = \cup A_n$. Are the (complex) irreducible representations of $A_\infty$ known? It is usually seen in the context of C*-algebras where $A_\infty$ is embedded in a GNS triple.
2
-
$\begingroup$ Is there a unique way to embed $A_n$ in $A_{n+1}$? Is it obvious that this union does not depend on how they are embedded? (I guess technically the union is a direct limit, and this is why it does not seem obvious that it should be independent of this choice). $\endgroup$– Tobias KildetoftCommented Sep 18, 2013 at 6:14
-
$\begingroup$ Thanks Tobias. I should have said, assuming the embedding: For $a_n \in A_n$, $a_n \mapsto a_n \otimes \mathbb{I}_{n+1}$, where $\mathbb{I}_{n+1}$ is the identity on factor indexed $n+1$. $\endgroup$– magya_bloomCommented Sep 18, 2013 at 16:08
Add a comment
|