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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 4th 2011
   • (edited Nov 4th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to the [references section](http://ncatlab.org/nlab/show/von+Neumann+algebra#References) at [[von Neumann algebra]] a pointer to a recent/upcoming (?) [course by Jacob Lurie on von Neumann algebras](http://www.math.harvard.edu/~lurie/261y.html). I am being told that this is held in preparation of a treatment of the [Douglas-Henriques work on 2dCFT and TMF](http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionDouglasHenriques).

   I have added to the references section at von Neumann algebra a pointer to a recent/upcoming (?) course by Jacob Lurie on von Neumann algebras.

   I am being told that this is held in preparation of a treatment of the Douglas-Henriques work on 2dCFT and TMF.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 4th 2011
   • (edited Nov 4th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexAn old query, removed from [[von Neumann algebra]] > [[Tim van Beek]]: I'm confused by the remarks, to my knowledge, the situation is this: $W^*$-algebra is the abstract concept, von Neumann algebra is the concrete concept, meaning that the definition of a von Neumann algebra needs a Hilbert space $\mathcal{H}$, so that it can be defined as a e.g. weakly closed subalgebra of $\mathcal{L}(\mathcal{H})$, the algebra of all bounded linear operators on $\mathcal{H}$. Without the Hilbert space you can't say what the weak topology should be. > According to > * Schaefer, Helmut H.; Wolff, M.P.: Topological vector spaces. 2nd ed., Springer 1999 ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0983.46002&format=complete)) > the situation is then this: > Definition: > A $W^*$-algebra is a $C^*$-algebra whose underlying normed space is a dual Banach space. > Theorem: > Every $W^*$-algebra is $W^*$-isomorphic to a von Neumann algebra ("on a suitable Hilbert space" is added in corollary 3 in paragraph 7.1, which is redundant however) and vice versa. > Any objections to change the remarks accordingly?

   An old query, removed from von Neumann algebra

   Tim van Beek: I’m confused by the remarks, to my knowledge, the situation is this: $W^*$-algebra is the abstract concept, von Neumann algebra is the concrete concept, meaning that the definition of a von Neumann algebra needs a Hilbert space $\mathcal{H}$, so that it can be defined as a e.g. weakly closed subalgebra of $\mathcal{L}(\mathcal{H})$, the algebra of all bounded linear operators on $\mathcal{H}$. Without the Hilbert space you can’t say what the weak topology should be.

   According to

   • Schaefer, Helmut H.; Wolff, M.P.: Topological vector spaces. 2nd ed., Springer 1999 (ZMATH entry)

   the situation is then this:

   Definition: A $W^*$-algebra is a $C^*$-algebra whose underlying normed space is a dual Banach space.

   Theorem: Every $W^*$-algebra is $W^*$-isomorphic to a von Neumann algebra (“on a suitable Hilbert space” is added in corollary 3 in paragraph 7.1, which is redundant however) and vice versa.

   Any objections to change the remarks accordingly?