Second preduals of tridual von Neumann algebras are geometrically unique
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The main result of the note claims that, if a von Neumann algebra Ais the third dual to some Banach space and thus, as is known, is the bidual of somevon Neumann algebra A, then the natural homomorphic projection of A onto theweakly closed two-sided ideal in A isomorphic to A (the kernel of this homomorphismcoincides with that of the canonical projection of A onto A dual to the canonicalembedding of the predual of A in the predual of A) is dened uniquely by A. Thus,the von Neumann algebra A, the image of the above projection, is not only uniquelydened (up to isomorphism) by its enveloping von Neumann algebra A but also theimage of the \canonical normal embedding" of A in A (onto the image of the aboveprojection) is uniquely dened, i.e., this weakly closed two-sided ideal in A isomorphicto A is uniquely dened geometrically. This enables us to rene some known resultson predual spaces of von Neumann algebras.
Abstract
The main result of the note claims that, if a von Neumann algebra Ais the third dual to some Banach space and thus, as is known, is the bidual of somevon Neumann algebra A, then the natural homomorphic projection of A onto theweakly closed two-sided ideal in A isomorphic to A (the kernel of this homomorphismcoincides with that of the canonical projection of A onto A dual to the canonicalembedding of the predual of A in the predual of A) is dened uniquely by A. Thus,the von Neumann algebra A, the image of the above projection, is not only uniquelydened (up to isomorphism) by its enveloping von Neumann algebra A but also theimage of the \canonical normal embedding" of A in A (onto the image of the aboveprojection) is uniquely dened, i.e., this weakly closed two-sided ideal in A isomorphicto A is uniquely dened geometrically. This enables us to rene some known resultson predual spaces of von Neumann algebras.
참고문헌 (14)
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