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I am looking for a textbook to cover the following tensor product (and, of course, the theory around it):

  • Let $\otimes_1$ denote the tensor product on Hilbert spaces.
  • Let $\otimes_2$ denote the tensor product on bounded operators defined by $(A\otimes_2 B)(x\otimes_1 y):=Ax\otimes_1 By$.
  • Let $\otimes_3$ denote the tensor product on linear maps on trace-class operators (possibly with additional restrictions such as complete positivity etc.) defined by $(\cal E\otimes_3\cal F)(A\otimes_2 B):=\cal E(A)\otimes_2\cal F(B)$.

The tensor product $\otimes_3$ is the one I am specifically interested in. (For $\otimes_1,\otimes_2$ I have found several textbooks.) For motivation: this is the composition of quantum channels which are usually modeled as completely positive trace-preserving maps.

Specifically, I am interested in things like the existence and well-definedness of $\cal E\otimes_3\cal F$, as well as basic algebraic and topological properties.

The text should be a mathematically rigorous textbook, and not limited only to finite or separable Hilbert spaces.

I need it as a reference for citing in a research paper. Therefore I am looking for a book that spells this tensor product out as explicitly as possible.

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    $\begingroup$ Is Chapter IV of Takesaki's Theory of Operator Algebras I to your liking? $\endgroup$
    – Robert Furber
    Commented May 26, 2019 at 20:57
  • $\begingroup$ @RobertFurber Almost. Unfortunately, I cannot find anything that would cover tensor products of superoperators (functions from trace-class operators to trace-class operators). Proposition IV.5.13 is almost right, except that it works on von-Neumann algebras only (and the trace-class operators are not a von-Neumann algebra). $\endgroup$
    – Dominique Unruh
    Commented May 26, 2019 at 22:27
  • $\begingroup$ The Banach space of trace-class operators, $B_*(H)$, is the predual of the von Neumann algebra $B(H)$. By a standard theorem about weak-* topologies, linear maps $f : B(H_1) \rightarrow B(H_2)$ that are $\sigma$-weakly continuous have a unique "pre-adjoint" $g : B_*(H_2) \rightarrow B_*(H_1)$ such that $\mathrm{tr}(g(\rho)A) = \mathrm{tr}(\rho f(A))$ for all trace-class operators $\rho$ and $A \in B(H)$. It is then possible to show that $g$ is completely positive trace-preserving iff $f$ is completely positive and unital. $\endgroup$
    – Robert Furber
    Commented May 26, 2019 at 22:40
  • $\begingroup$ Hmm... This would probably work, but I would have to additionally show that the tensor product resulting from this is still the one I had in mind, etc. which would need some work. That would make it a bit tricky as a reference in, e.g., the preliminaries of a paper without supporting it with extra proofs in the appendix. But thanks anyway, if I don't find a book that is a better match, then I will do that. (I am editing the question a bit to make clearer what I want.) $\endgroup$
    – Dominique Unruh
    Commented May 27, 2019 at 8:09
  • $\begingroup$ @RobertFurber I could not find a reference for the "standard theorem" you mentioned, so I opened a new question for that. (mathoverflow.net/questions/395480/…) If you happen to know where I can find this theorem, maybe you can chime in there. $\endgroup$
    – Dominique Unruh
    Commented Jun 17, 2021 at 9:39

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