The Image of Unitary Representation in the Space of Bimodules

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TL;DR: Bimodules over a von Neumann algebra are commonly understood as
a generalization of group representation. Indeed, when the von Neumann algebra $N$ is $mathcalL G$, the unitary representations embed into the
bimodules.



Question: Can we describe the image of this embedding? Which are the
bimodules over $mathcalL G$ that come from a representation?




Let $G$ be a discrete group. The four objects in the diagram bellow are:



  • $mathrmRep_cyc(G)$: its cyclic unitary representations, i.e. unitary representations $pi: G to U(H_pi)$ for which there is a vector $xi_pi in H_pi$ such that $pi(g) xi : g in G$ spans a dense subset of $H_pi$.

  • $mathcalP(G)$: The cone of positive definite functions on $G$.

  • $ _N H _N$: The space of all $N$-$N$-bimodules, i.e: Hilbert spaces with normal commuting left and right actions of $N$. In this case $N = mathcalL G$, the left regular von Neumann algebra of $G$. We ask the bimodules to be cyclic, i.e.: there has to be a $xi in H$ such that $x cdot xi cdot y$ is dense and that $xi$ is left $N$-bounded.

  • $mathcalCP(N to N)$: The cone of completely positive and normal maps.

All those objects are closely related. For instance, there are bijective correspondences between the objects in each of the columns of the diagram bellow. The objects in the right column are interpreted as "generalizations" of those on the right column from the context of groups to that of von Neumann algebras. The arrows from the left column to the right one are injections.





In particular the maps above are given by:




  • (i) Given a cyclic representation $pi$ there is a positive definite function $varphi$ given by $$varphi(g) = langle xi, pi(g) xi rangle.$$


  • (ii) Given a positive definite function $varphi$ we can obtain a rep. by a GNS-type construction. Define a positive form in $ell^1(G)$ given by $langle varphi, zeta^ast ast zeta rangle$. Quotienting out the nullspace and taking completions gives a Hilbert space with a cyclic representation of $G$.


  • (iii) Every unitary representation $pi$ gives rise to a $mathcalL G$-bimodule $ell^2(G) otimes H_pi$ with left and right actions given by:
    $$
    lambda_g cdot ( delta_k otimes xi ) cdot lambda_h = delta_g k h otimes pi(g) xi.
    $$


  • (iv) If $varphi$ is positive definite, the Fourier multiplier given by extension of $lambda_g mapsto varphi(g) lambda_g$ is normal and cp.


  • (v) Given a bimodule $H$ with a left $N$-bounded cyclic vector $xi$ we can construct a normal cp map by $x mapsto L_xi^ast x L_xi$


  • (vi) Given a normal cp map $phi: N to N$ we can construct a bimodule $L^2(N otimes_varphi N)$ by a GNS-type construction. Take $N otimes_alg N$ and complete it with respect to
    $$
    biglangle x_1 otimes y_1, x_2 otimes y_2 bigrangle = tau_N big( x_1^ast , phi(y_1^* y_2) , x_2 big).
    $$
    The actions are given by $x cdot (a otimes b) cdot y = x a otimes b y$. The construction above requires that $N$ is finite since it uses the trace (no problem since $N = mathcal L G$).

More on (iii), (v) and (vi) can be found in [AP: Chapter 13].




Question 1: The image of (iv) can be described as the $Delta$-equivariant cp maps, i.e. maps $T$ such that:
$$ (T otimes id) cdot Delta = Delta cdot T, $$
where $Delta: N to N otimes N$ is the natural comultiplication $Delta(lambda_g) = lambda_g otimes lambda_g$. Can something similar be said of the image of
(iii)? Is there a notion of "equivariant" bimodule?



Question 2: Given a normal cp map $phi: mathcal L G to mathcal L G$ we can associate to it a positive definite function $varphi(g) = tau(phi(lambda_g) lambda_g^ast)$. Is there an analogous
construction that gives an equivariant bimodule from a general one?




  • [AP]: http://www.math.ucla.edu/~popa/Books/IIun-v13.pdf



representation-theory operator-algebras c-star-algebras von-neumann-algebras bimodules




share|cite|improve this question























    up vote
    2
    down vote

    favorite













    TL;DR: Bimodules over a von Neumann algebra are commonly understood as
    a generalization of group representation. Indeed, when the von Neumann algebra $N$ is $mathcalL G$, the unitary representations embed into the
    bimodules.



    Question: Can we describe the image of this embedding? Which are the
    bimodules over $mathcalL G$ that come from a representation?




    Let $G$ be a discrete group. The four objects in the diagram bellow are:



    • $mathrmRep_cyc(G)$: its cyclic unitary representations, i.e. unitary representations $pi: G to U(H_pi)$ for which there is a vector $xi_pi in H_pi$ such that $pi(g) xi : g in G$ spans a dense subset of $H_pi$.

    • $mathcalP(G)$: The cone of positive definite functions on $G$.

    • $ _N H _N$: The space of all $N$-$N$-bimodules, i.e: Hilbert spaces with normal commuting left and right actions of $N$. In this case $N = mathcalL G$, the left regular von Neumann algebra of $G$. We ask the bimodules to be cyclic, i.e.: there has to be a $xi in H$ such that $x cdot xi cdot y$ is dense and that $xi$ is left $N$-bounded.

    • $mathcalCP(N to N)$: The cone of completely positive and normal maps.

    All those objects are closely related. For instance, there are bijective correspondences between the objects in each of the columns of the diagram bellow. The objects in the right column are interpreted as "generalizations" of those on the right column from the context of groups to that of von Neumann algebras. The arrows from the left column to the right one are injections.





    In particular the maps above are given by:




    • (i) Given a cyclic representation $pi$ there is a positive definite function $varphi$ given by $$varphi(g) = langle xi, pi(g) xi rangle.$$


    • (ii) Given a positive definite function $varphi$ we can obtain a rep. by a GNS-type construction. Define a positive form in $ell^1(G)$ given by $langle varphi, zeta^ast ast zeta rangle$. Quotienting out the nullspace and taking completions gives a Hilbert space with a cyclic representation of $G$.


    • (iii) Every unitary representation $pi$ gives rise to a $mathcalL G$-bimodule $ell^2(G) otimes H_pi$ with left and right actions given by:
      $$
      lambda_g cdot ( delta_k otimes xi ) cdot lambda_h = delta_g k h otimes pi(g) xi.
      $$


    • (iv) If $varphi$ is positive definite, the Fourier multiplier given by extension of $lambda_g mapsto varphi(g) lambda_g$ is normal and cp.


    • (v) Given a bimodule $H$ with a left $N$-bounded cyclic vector $xi$ we can construct a normal cp map by $x mapsto L_xi^ast x L_xi$


    • (vi) Given a normal cp map $phi: N to N$ we can construct a bimodule $L^2(N otimes_varphi N)$ by a GNS-type construction. Take $N otimes_alg N$ and complete it with respect to
      $$
      biglangle x_1 otimes y_1, x_2 otimes y_2 bigrangle = tau_N big( x_1^ast , phi(y_1^* y_2) , x_2 big).
      $$
      The actions are given by $x cdot (a otimes b) cdot y = x a otimes b y$. The construction above requires that $N$ is finite since it uses the trace (no problem since $N = mathcal L G$).

    More on (iii), (v) and (vi) can be found in [AP: Chapter 13].




    Question 1: The image of (iv) can be described as the $Delta$-equivariant cp maps, i.e. maps $T$ such that:
    $$ (T otimes id) cdot Delta = Delta cdot T, $$
    where $Delta: N to N otimes N$ is the natural comultiplication $Delta(lambda_g) = lambda_g otimes lambda_g$. Can something similar be said of the image of
    (iii)? Is there a notion of "equivariant" bimodule?



    Question 2: Given a normal cp map $phi: mathcal L G to mathcal L G$ we can associate to it a positive definite function $varphi(g) = tau(phi(lambda_g) lambda_g^ast)$. Is there an analogous
    construction that gives an equivariant bimodule from a general one?




    • [AP]: http://www.math.ucla.edu/~popa/Books/IIun-v13.pdf



    representation-theory operator-algebras c-star-algebras von-neumann-algebras bimodules




    share|cite|improve this question





















      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite












      TL;DR: Bimodules over a von Neumann algebra are commonly understood as
      a generalization of group representation. Indeed, when the von Neumann algebra $N$ is $mathcalL G$, the unitary representations embed into the
      bimodules.



      Question: Can we describe the image of this embedding? Which are the
      bimodules over $mathcalL G$ that come from a representation?




      Let $G$ be a discrete group. The four objects in the diagram bellow are:



      • $mathrmRep_cyc(G)$: its cyclic unitary representations, i.e. unitary representations $pi: G to U(H_pi)$ for which there is a vector $xi_pi in H_pi$ such that $pi(g) xi : g in G$ spans a dense subset of $H_pi$.

      • $mathcalP(G)$: The cone of positive definite functions on $G$.

      • $ _N H _N$: The space of all $N$-$N$-bimodules, i.e: Hilbert spaces with normal commuting left and right actions of $N$. In this case $N = mathcalL G$, the left regular von Neumann algebra of $G$. We ask the bimodules to be cyclic, i.e.: there has to be a $xi in H$ such that $x cdot xi cdot y$ is dense and that $xi$ is left $N$-bounded.

      • $mathcalCP(N to N)$: The cone of completely positive and normal maps.

      All those objects are closely related. For instance, there are bijective correspondences between the objects in each of the columns of the diagram bellow. The objects in the right column are interpreted as "generalizations" of those on the right column from the context of groups to that of von Neumann algebras. The arrows from the left column to the right one are injections.





      In particular the maps above are given by:




      • (i) Given a cyclic representation $pi$ there is a positive definite function $varphi$ given by $$varphi(g) = langle xi, pi(g) xi rangle.$$


      • (ii) Given a positive definite function $varphi$ we can obtain a rep. by a GNS-type construction. Define a positive form in $ell^1(G)$ given by $langle varphi, zeta^ast ast zeta rangle$. Quotienting out the nullspace and taking completions gives a Hilbert space with a cyclic representation of $G$.


      • (iii) Every unitary representation $pi$ gives rise to a $mathcalL G$-bimodule $ell^2(G) otimes H_pi$ with left and right actions given by:
        $$
        lambda_g cdot ( delta_k otimes xi ) cdot lambda_h = delta_g k h otimes pi(g) xi.
        $$


      • (iv) If $varphi$ is positive definite, the Fourier multiplier given by extension of $lambda_g mapsto varphi(g) lambda_g$ is normal and cp.


      • (v) Given a bimodule $H$ with a left $N$-bounded cyclic vector $xi$ we can construct a normal cp map by $x mapsto L_xi^ast x L_xi$


      • (vi) Given a normal cp map $phi: N to N$ we can construct a bimodule $L^2(N otimes_varphi N)$ by a GNS-type construction. Take $N otimes_alg N$ and complete it with respect to
        $$
        biglangle x_1 otimes y_1, x_2 otimes y_2 bigrangle = tau_N big( x_1^ast , phi(y_1^* y_2) , x_2 big).
        $$
        The actions are given by $x cdot (a otimes b) cdot y = x a otimes b y$. The construction above requires that $N$ is finite since it uses the trace (no problem since $N = mathcal L G$).

      More on (iii), (v) and (vi) can be found in [AP: Chapter 13].




      Question 1: The image of (iv) can be described as the $Delta$-equivariant cp maps, i.e. maps $T$ such that:
      $$ (T otimes id) cdot Delta = Delta cdot T, $$
      where $Delta: N to N otimes N$ is the natural comultiplication $Delta(lambda_g) = lambda_g otimes lambda_g$. Can something similar be said of the image of
      (iii)? Is there a notion of "equivariant" bimodule?



      Question 2: Given a normal cp map $phi: mathcal L G to mathcal L G$ we can associate to it a positive definite function $varphi(g) = tau(phi(lambda_g) lambda_g^ast)$. Is there an analogous
      construction that gives an equivariant bimodule from a general one?




      • [AP]: http://www.math.ucla.edu/~popa/Books/IIun-v13.pdf



      representation-theory operator-algebras c-star-algebras von-neumann-algebras bimodules




      share|cite|improve this question












      TL;DR: Bimodules over a von Neumann algebra are commonly understood as
      a generalization of group representation. Indeed, when the von Neumann algebra $N$ is $mathcalL G$, the unitary representations embed into the
      bimodules.



      Question: Can we describe the image of this embedding? Which are the
      bimodules over $mathcalL G$ that come from a representation?




      Let $G$ be a discrete group. The four objects in the diagram bellow are:



      • $mathrmRep_cyc(G)$: its cyclic unitary representations, i.e. unitary representations $pi: G to U(H_pi)$ for which there is a vector $xi_pi in H_pi$ such that $pi(g) xi : g in G$ spans a dense subset of $H_pi$.

      • $mathcalP(G)$: The cone of positive definite functions on $G$.

      • $ _N H _N$: The space of all $N$-$N$-bimodules, i.e: Hilbert spaces with normal commuting left and right actions of $N$. In this case $N = mathcalL G$, the left regular von Neumann algebra of $G$. We ask the bimodules to be cyclic, i.e.: there has to be a $xi in H$ such that $x cdot xi cdot y$ is dense and that $xi$ is left $N$-bounded.

      • $mathcalCP(N to N)$: The cone of completely positive and normal maps.

      All those objects are closely related. For instance, there are bijective correspondences between the objects in each of the columns of the diagram bellow. The objects in the right column are interpreted as "generalizations" of those on the right column from the context of groups to that of von Neumann algebras. The arrows from the left column to the right one are injections.





      In particular the maps above are given by:




      • (i) Given a cyclic representation $pi$ there is a positive definite function $varphi$ given by $$varphi(g) = langle xi, pi(g) xi rangle.$$


      • (ii) Given a positive definite function $varphi$ we can obtain a rep. by a GNS-type construction. Define a positive form in $ell^1(G)$ given by $langle varphi, zeta^ast ast zeta rangle$. Quotienting out the nullspace and taking completions gives a Hilbert space with a cyclic representation of $G$.


      • (iii) Every unitary representation $pi$ gives rise to a $mathcalL G$-bimodule $ell^2(G) otimes H_pi$ with left and right actions given by:
        $$
        lambda_g cdot ( delta_k otimes xi ) cdot lambda_h = delta_g k h otimes pi(g) xi.
        $$


      • (iv) If $varphi$ is positive definite, the Fourier multiplier given by extension of $lambda_g mapsto varphi(g) lambda_g$ is normal and cp.


      • (v) Given a bimodule $H$ with a left $N$-bounded cyclic vector $xi$ we can construct a normal cp map by $x mapsto L_xi^ast x L_xi$


      • (vi) Given a normal cp map $phi: N to N$ we can construct a bimodule $L^2(N otimes_varphi N)$ by a GNS-type construction. Take $N otimes_alg N$ and complete it with respect to
        $$
        biglangle x_1 otimes y_1, x_2 otimes y_2 bigrangle = tau_N big( x_1^ast , phi(y_1^* y_2) , x_2 big).
        $$
        The actions are given by $x cdot (a otimes b) cdot y = x a otimes b y$. The construction above requires that $N$ is finite since it uses the trace (no problem since $N = mathcal L G$).

      More on (iii), (v) and (vi) can be found in [AP: Chapter 13].




      Question 1: The image of (iv) can be described as the $Delta$-equivariant cp maps, i.e. maps $T$ such that:
      $$ (T otimes id) cdot Delta = Delta cdot T, $$
      where $Delta: N to N otimes N$ is the natural comultiplication $Delta(lambda_g) = lambda_g otimes lambda_g$. Can something similar be said of the image of
      (iii)? Is there a notion of "equivariant" bimodule?



      Question 2: Given a normal cp map $phi: mathcal L G to mathcal L G$ we can associate to it a positive definite function $varphi(g) = tau(phi(lambda_g) lambda_g^ast)$. Is there an analogous
      construction that gives an equivariant bimodule from a general one?




      • [AP]: http://www.math.ucla.edu/~popa/Books/IIun-v13.pdf




      representation-theory operator-algebras c-star-algebras von-neumann-algebras bimodules





      share|cite|improve this question










      share|cite|improve this question




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