This question has a subjective component but I would like answers that try to stick to concrete observable facts, such as which papers use which terminology. However, the informed impressions of those working with operator spaces or operator algebras is welcome.

The projective and Haagerup tensor products of operator spaces can be seen as the universal constructions that linearize certain kinds of bilinear maps on operator spaces. Here is what I can gather from trying to survey the literature:

  • The book of Effros+Ruan calls the first kind of bilinear maps "completely bounded" and the second kind "multiplicatively bounded".

  • The early papers of Blecher+Paulsen call the first kind of bilinear maps "jointly completely bounded" and the second kind "completely bounded".

  • The book of Pisier seems, if I have read it correctly, to duck the issue wherever possible by constantly referring to c.b. maps $E\to F^*$ (which correspond to linear functionals on the projective tensor product of $E$ and $F$ in the operator-space category).

My question is this: is there a current consensus on which terminology to use?

  • $\begingroup$ As an aside, I am aware that Helemskii has used the terminology "weakly cb" and "strongly cb" for these two cases, but I suspect that this is not commonly used. $\endgroup$ – Yemon Choi May 18 '16 at 18:24

I would say that there is now a consensus to use the terminology of jointly completely bounded for the cb maps $E \to F^*$ (and to use "completely bounded" for the one that correspond to the Haagerup tensor product, but I know less recent work on this notion).

At least all the papers from the current millenium that I know on the subject use the terminology jointly completely bounded. To quote Pisier and Shlyakhtenko in the introduction of their Inventiones paper The second notion came first, was then called simply "complete boundedness" and has proved extremely fruitful. To (reluctantly) conform with the already established tradition, we will call the first ones "jointly c.b." although calling them c.b. would definitely be more natural from the viewpoint of operator space theory, it would have been more natural to call "completely bounded" the ones that are now called "jointly completely bounded".

Other examples of recent papers using the terminology of jointly completely bounded bilinear maps include The Effros-Ruan conjecture for bilinear forms on $C^*$-algebras by Haagerup and Musat, Elementary Proofs of Grothendieck Theorems for Completely Bounded Norms by Regev and Vidick, Grothendieck's Theorem, past and present by Pisier.

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  • $\begingroup$ Merci Mikael. It seems like I will need to do some "search and replace" tonight on something I have been typing... $\endgroup$ – Yemon Choi May 18 '16 at 20:39

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