Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}_k$, where $\mathcal{A}_k = \mathcal{A} \, \, \forall k$. I am interested in operators like $O_i = \otimes_{k=1}^{i-1} \mathbb{I}_k \otimes O\otimes_{k=i+1}^{N} \mathbb{I}_k $, where $\mathbb{I}$ is the identity operator. I will list at the bottom few references where the algebra above is better introduced.
Let me also introduce the operator $\widehat{O}_N = \sum_{i=1}^{N} O_i$ which is the subject of this post. My doubts are the following:
- Under which conditions the limit $\lim_{N \to +\infty} \widehat{O}_N$ converges either in norm, or strongly or weakly? Can you please link me a good reference where to study it?
- Given $\omega_N$ a state on $\mathcal{A}^{N}$, and given two operators $O,P \, \in \mathcal{A}$ such that the commutator $[O,P] = T \in \mathcal{A}$, I am interested in computing $\lim_{N \to +\infty} \omega_N ([\widehat{O}_N,\widehat{P}_N])$. My doubt here is how to proceed. For example, should I firstly compute the commutator $[\widehat{O}_N,\widehat{P}_N]$, then perform the limit $\lim_{N \to +\infty}$ or should I firstly compute the limit $\lim_{N \to +\infty} \widehat{O}_N$ and $\lim_{N \to +\infty} \widehat{P}_N$ and then compute $[\lim_{N \to +\infty} \widehat{O}_N, \lim_{N \to +\infty} \widehat{P}_N]$ ?
Reference:
- "Symmetric states of infinite tensor products of C* algebras", by E. Stormer, Journal of Functional Analysis 3, 48-68 (1969);
- "Certain factors constructed as infinite tensor products", by D. J. C. Bures, Composition Mathematica 15, 169-191, (1962-1964)