# Talk:Von Neumann algebra

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How 'bout giving some concrete examples? Michael Hardy 02:07, 31 May 2004 (UTC)

## Separability bias

What do you mean bias? The previous version you reverted explicitly said much of the theory is still applicable in general (bicommutant theorem, Kaplansky density theorem etc.) The important fact to note is that hyperfinite von Neumann algebras (essentially classifiiable by COnnes' classification of hyperfinite factors) are all separably realizable.CSTAR 18:30, 3 Jan 2005 (UTC)

## High non-commutativity

I would like to delete the comment that the space of bounded linear transforms on a ${\displaystyle B(L^{2}(X,\mu ))}$ is highly non-commutative since it doesn't seem to mean much except stating that such an algebra is non-commutative. This is not true if we make everything we can trivial.HStel 13:42, 22 April 2006 (UTC)

Go ahead, delete.--CSTAR 14:40, 22 April 2006 (UTC)

O boy o boy, I did it! Herman Stel 22:51, 22 April 2006 (UTC)

Someone please include a definition of "locally compact measure space", does it mean a locally compact Hausdorff space with a measure that is a positive linear functional on continuous maps with compact support??? Kuratowski's Ghost 13:39, 16 July 2006 (UTC)

I couldn't find the term "locally compact measure space" anywhere in the article. Are you suggesting we include a definition of it in any case? --CSTAR 19:38, 17 July 2006 (UTC)

## The pathological measure space case

I'm not sure a footnote is the best way to do it, but in the interest of mathematical correctness, it is important to point out that ${\displaystyle L^{\infty }(X)}$ is not always a von Neumann algebra. Sure, the counterexamples might be pathological and uninteresting, but that doesn't mean incorrect statements are okay.

For the non-believers:

Let X be an uncountable set and μ a measure on the countable-cocountable algebra -- the counting measure will do fine. Choose ${\displaystyle A\subset X}$ uncountable with uncountable complement, and consider a net of operators in ${\displaystyle L^{\infty }(X)}$ that converge in the weak or strong topology towards the projection onto A (an element of ${\displaystyle B(L^{2}(X))}$: but that projection isn't measurable.

(Alternatively, note that the lattice of projections in ${\displaystyle L^{\infty }(X)}$ isn't order complete, but the lattice of projections in a von Neumann algebra is).

IIRC, the algebra of locally measurable essentially bounded functions on a locally finite measure space is a von Neumann algebra; I'm not sure whether there is any measure space not equivalent to a locally finite one, but if there is such a beast, I see no way of constructing a von Neumann algebra out of ${\displaystyle L^{\infty }(X)}$ short of considering its closure (but, of course, the resulting von Neumann algebra would be commutative, and thus isomorphic to ${\displaystyle L^{\infty }(X)}$ for some nice (i.e. locally finite) measure space).

RandomP 01:11, 13 May 2006 (UTC)

## Fixing the examples section

The examples section was rather error-ridden: here are the changes, and reasons for them:

• The essentially bounded functions on a measure space form a commutative (type I1) von Neumann algebra acting on the L2 functions.

Fails in the case of certain pathological measure spaces, see above.

• If we have any unitary representation of a group G on a Hilbert space H then the bounded operators commuting with G form a von Neumann algebra G′, whose projections correspond exactly to the closed subspaces of H invariant under G. The double commutator G′′ of G is also a von Neumann algebra.

I fail to see why this is discussed before von Neumann group algebras are.

• The crossed product of a von Neumann algebra by a discrete (or more generally locally compact) group is a von Neumann algebra.

The crossed product article currently mentions only von Neumann algebras, though if memory serves a similar construction for C*-algebras exists. I am unsure whether the two coincide for von Neumann algebras considered as C*-algebras.

• The von Neumann group algebra of a locally compact group G is the von Neumann algebra generated by the left translations of G acting on the L2 functions on G.

What exactly is the point of defining a von Neumann algebra as being "generated" by a certain set of operators? We haven't even explained how this is always possible, nevermind that it's hardly a good way to define the group von Neumann algebra.

Also, what's the point of considering locally compact rather than discrete groups? It makes the example much more complicated. (I believe the von Neumann group algebra of a locally compact group is isomorphic to that of the same group considered as discrete group).

• This comment is incorrect. The von Neumann algebras of the real line with its usual topology, and the real line with its discrete topology, are not isomorphic, since the former has separable predual while the latter does not. Or consider the rotation group in 3 dimensions, again with its usual topology and then with the discrete topology; the former gives rise to an injective von Neumann algebra, the latter does not. It is nonetheless true that the majority of work nowadays is done in the case where the group is discrete, but IMHO worth knowing that a more general definition can be given. NowhereDense (talk) 11:13, 24 November 2008 (UTC)

"L2 functions" is also a slightly inaccurate term when locally compact groups are considered: it might not be totally obvious that the Hilbert space is independent of the choice of Haar measure, besides the whole identification-of-equivalent-functions issue.

• Since a left-invariant Haar measure is unique up to multiplication by a scalar factor, L2 is unambiguous provided one states explicitly whether one is using a left or right invariant Haar measure. Not sure what the problem is supposed to be here. NowhereDense (talk) 11:13, 24 November 2008 (UTC)
• The tensor product of two von Neumann algebras is a von Neumann algebra. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The tensor product of two von Neumann algebras of types X and Y (I II or III) has type equal to the maximum of X and Y.

The tensor product of two von Neumann algebras is most certainly not a von Neumann algebra (except in some very simple cases). It can be completed to be a von Neumann algebra, and the interesting statement is that the result of this completion is independent of certain choices made to define it.

• The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead one usually chooses a state on each of the von Neumann algebras, uses this to define a state on the algebraic tensor product, which can be used to product a Hilbert space and a (reasonably small) von Neumann algebra. The factors of Powers and Araki-Woods were found like this. The term ITPFI stands for "infinite tensor product of finite type I factors". The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type I2 factors can have any type depending on the choice of states.

Von Neumann algebras are virtually always non-separable, in the norm topology. This most definitely no longer belongs in the examples section.

• Kriegers factors are constructed from ergodic actions.

What are constructed from what? This isn't even a stub.

If the removed examples can be salvaged, feel free to do so.

RandomP 01:44, 13 May 2006 (UTC)

The term "measure space" should indeed have read "σ-finite measure space". However many of the other comments above are caused by a misunderstanding of standard von Neumann algebra terminology. As this obviously confuses some readers I have added a section explaining some of the more confusing terms.
Please do not delete paragraphs that you find too terse or difficult. Instead you can add a "too technical" tag and put a request on the talk page for someone to provide more explanation.

R.e.b. 21:40, 16 May 2006 (UTC)

Sorry, I should have made it clearer that I was deliberately approaching this subject from the point of view of a student who does not know the usual terminology about von Neumann algebras -- the article von Neumann algebra should certainly not presuppose that.
The tensor product case should certainly go out. The definition isn't trivial, and it should certainly not be dispensed with as a mere question of terminology!
The name of the section is Examples, R.e.b. Better not to have an example at all than having one that makes the article not self-contained. I will certainly remove such examples if there's no obvious way to fix them.
RandomP 22:03, 16 May 2006 (UTC)
Oops, sorry. Just saw you didn't revert to the old wording of the tensor product example.
• I'd like a real section about the tensor products of von Neumann algebras; it is more than terminology.
• the infinite tensor product example needs hashing out. It's overly colloquial, and it's not terribly clear unless you know what's meant to begin with. I'm still not absolutely sure it belongs there, but no reason to throw it out as long as it's at the end and not actually wrong.
RandomP 22:12, 16 May 2006 (UTC)

## Type I-C*-algebras

Why is their definition in the von Neumann-article? I'm moving them to the C*-algebras. KennyDC 01:47, 16 September 2006 (UTC)

## Abstract von Neumann algebras

At some point we should say that von Neumann algebras can be characterized abstractly as a C*-algebra with a predual. We need to decide where to put this in.--CSTAR 17:05, 9 October 2006 (UTC)

Oops its already there!--CSTAR 17:12, 9 October 2006 (UTC)

## what's the dealio?

a comment that says, essentially:

an orthogonal sum of unitarily equivalent projections need not be unitarily equivalent, i.e. suppose A ~ B and C ~ D, where ~ means unitary equivalence, it's not necessarily true that A + C ~ B + D.

was removed with the edit summary claiming it's incorrect. why is it incorrect? Mct mht 05:18, 26 January 2007 (UTC)