About unitary group of a von Neumann algebra

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Is unitary group of a von Neumann algebra is locally compact in some topology? Can we make sense of integration with respect to Haar measure?




operator-algebras von-neumann-algebras




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    Is unitary group of a von Neumann algebra is locally compact in some topology? Can we make sense of integration with respect to Haar measure?




    operator-algebras von-neumann-algebras




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      Is unitary group of a von Neumann algebra is locally compact in some topology? Can we make sense of integration with respect to Haar measure?




      operator-algebras von-neumann-algebras




      share|cite|improve this question











      Is unitary group of a von Neumann algebra is locally compact in some topology? Can we make sense of integration with respect to Haar measure?





      operator-algebras von-neumann-algebras





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      mathlover

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          The unitary group is compact for finite von Neumann algebras of type I.
          It is also locally compact for the discrete topology.



          Other than these two cases, the set of unitary elements
          can be equipped with the ultraweak topology, which yields a compact topolpogical
          space but for which the operation of multiplication
          is not continuous, or it can be equipped with the norm or ultrastrong topologies,
          which are not locally compact, but do give a topological group.
          In either case, the Haar theorem is not applicable.






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            up vote
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            accepted










            The unitary group is compact for finite von Neumann algebras of type I.
            It is also locally compact for the discrete topology.



            Other than these two cases, the set of unitary elements
            can be equipped with the ultraweak topology, which yields a compact topolpogical
            space but for which the operation of multiplication
            is not continuous, or it can be equipped with the norm or ultrastrong topologies,
            which are not locally compact, but do give a topological group.
            In either case, the Haar theorem is not applicable.






            share|cite|improve this answer

























              up vote
              3
              down vote



              accepted










              The unitary group is compact for finite von Neumann algebras of type I.
              It is also locally compact for the discrete topology.



              Other than these two cases, the set of unitary elements
              can be equipped with the ultraweak topology, which yields a compact topolpogical
              space but for which the operation of multiplication
              is not continuous, or it can be equipped with the norm or ultrastrong topologies,
              which are not locally compact, but do give a topological group.
              In either case, the Haar theorem is not applicable.






              share|cite|improve this answer























                up vote
                3
                down vote



                accepted







                up vote
                3
                down vote



                accepted






                The unitary group is compact for finite von Neumann algebras of type I.
                It is also locally compact for the discrete topology.



                Other than these two cases, the set of unitary elements
                can be equipped with the ultraweak topology, which yields a compact topolpogical
                space but for which the operation of multiplication
                is not continuous, or it can be equipped with the norm or ultrastrong topologies,
                which are not locally compact, but do give a topological group.
                In either case, the Haar theorem is not applicable.






                share|cite|improve this answer













                The unitary group is compact for finite von Neumann algebras of type I.
                It is also locally compact for the discrete topology.



                Other than these two cases, the set of unitary elements
                can be equipped with the ultraweak topology, which yields a compact topolpogical
                space but for which the operation of multiplication
                is not continuous, or it can be equipped with the norm or ultrastrong topologies,
                which are not locally compact, but do give a topological group.
                In either case, the Haar theorem is not applicable.







                share|cite|improve this answer













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                share|cite|improve this answer











                answered 2 days ago









                Dmitri Pavlov

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