Tensor product of operator algebras $\mathcal B(H) \otimes \mathcal B(K)$

Given Hilbert spaces $$H$$ and $$K$$, denote $$H \otimes K$$ the Hilbert space tensor product with inner product $$\langle x_1\otimes y_1, x_2 \otimes y_2\rangle = \langle x_1, x_2\rangle_H \langle y_1 , y_2 \rangle_K$$, $$\mathcal B(H)$$ the bounded linear operators on $$H$$ and $$\mathcal B(H)\odot \mathcal B(K)$$ the algebraic tensor product.

I saw in some other questions in this page that the closure of $$\mathcal B(H) \odot \mathcal B(K)$$ with respect to the SOT or WOT is precisely $$\mathcal B(H \otimes K)$$. Thus, the tensor product as von Neumann algebras is unique. What about the closure with respect to the operator norm (in infinite dimension)?

On the other hand, the minimal tensor norm on $$A \otimes B$$ for operator spaces $$\rho : A \to \mathcal B(H)$$ and $$\sigma: B \to \mathcal B(K)$$ is defined as $$\lVert x\otimes y\rVert = \lVert \rho\otimes\sigma(x\otimes y)\rVert_{\mathcal B(H \otimes K)}$$, which does not depend on the choice of the *-homomorphisms $$\rho$$ and $$\sigma$$. In particular, when $$A = \mathcal B(H)$$ and $$B=\mathcal B(K)$$ we have an isometric identification $$\mathcal B(H) \otimes_{min} \mathcal B(K) \subset \mathcal B(H \otimes K)$$, where $$\otimes_{min}$$ denotes the completion with respect to the minimal tensor norm. If I understood correctly, the inclusion can be proper in infinite dimension.

I was checking several references about C*-algebras (Murphy, Blackadar, Pisier, Dixmier & Rudin) but I could not find an explicit discussion. I would like to ask for references where these two points are discussed.

1 Answer

Yeah, there a lot of C$${}^*$$-norms on the algebraic tensor product. See for example Ozawa + Pisier A continuum of C$${}^*$$-norms on $$\mathcal{B(H)} \otimes \mathcal{B(H)}$$. Moreover, there is of course the book of Brown and Ozawa that you can refer to, as well as a recent book of Pisier on C*-tensor products.

For the min tensor product (which is the operator norm closure in $$\mathcal{B(H} \otimes \mathcal{K)}$$), its sort of clear that you can't get all of $$\mathcal{B(H} \otimes \mathcal{K)}$$ for the following reason. If you take the min-tensor product of C*-algebras $$A$$ and $$B$$, then $$I \otimes_{\min} J$$ will be an ideal whenever $$I$$ is an ideal of $$A$$ and $$J$$ is an ideal of $$B$$. So think about how many ideals $$\mathcal{B(H)} \otimes_{\min} \mathcal{B(K)}$$ would have if they're both infinite dimensional.

• For the sake of recording the history of this problem, the result that $B(H) \otimes B(K)$ admits at least two tensor norms was first established by Marius Junge and Gilles Pisier in “Bilinear forms on exact operator spaces and $B(H) \otimes B(H)$”. Commented Jun 11 at 17:05