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Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) \overline{\otimes} L^{2}(M)$ with commuting left and right actions of $M$ given by $x(\xi \otimes \eta)y=(x \xi) \otimes (\eta y)$ for any $x,y \in M$, where $x \xi$ and $\eta y$ refer, of course, to the standard left and right actions on $L^{2}(M)$.

Intuitively, this bimodule is regarded as "coarse" since there is no "connection" between the two copies of $L^{2}(M)$ making it up. If this is the case, what if we consider $L^{2}(M) \overline{\otimes} L^{2}(M) \overline{\otimes} L^{2}(M)$ with elements of $M$ acting only on the left on the left copy and the right on the right copy of $L^{2}(M)$? There still should be no "connection".

Question: As correspondences, is $L^{2}(M) \overline{\otimes} L^{2}(M) \overline{\otimes} L^{2}(M)$ always isomorphic to the coarse correspondence $L^{2}(M) \overline{\otimes} L^{2}(M)$? If so, what is an isomorphism?

As far as I can see, the former is isomorphic to a direct sum of the latter. The intuition suggests that more may be true. (I imagine an answer to this question can be seen by viewing the bimodules as morphisms of M into amplifications of M, but I haven't worked it out.)

EDIT/ADDENDUM: Since this is already pretty silly, why not make it sillier? Is it possible for $M$ to be nonamenable and yet have $L^{2}(M)$ weakly contained in $L^{2}(M) \overline{\otimes} L^{2}(M) \overline{\otimes} L^{2}(M)$?

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The above is quite a silly question, actually...but that's research. Many thanks to Andreas for taking the time to type an answer!!! – Jon Bannon Mar 2 '11 at 18:27
up vote 1 down vote accepted

The commutant of the action of $M \otimes M^{op}$ on $L^2(M) \otimes_2 L^2(M)$ is $M^{op} \bar \otimes M$ with the obvious action, whereas the commutant of the action on $L^2(M) \otimes_2 L^2(M) \otimes_2 L^2(M)$ is $M^{op} \bar \otimes B(L^2(M)) \bar \otimes M$ (again with the obvious action). The first algebra is finite, the second infinite. Hence, these bi-modules cannot be isomorphic.

(I denote the Hilbert space tensor product by $\otimes_2$ and the von Neumann algebra tensor product by $\bar \otimes$.)

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$L^2(M)\otimes L^2(M)\otimes L^2(M)$ is not isomorphic to the coarse correspondence, but it is an infinite direct sum of coarse correspondences. In that sense it is clear that it does not present a non-trivial relation between the left and right actions. – Steven Deprez Mar 8 '11 at 12:29

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