I'm learning about von Neumann algebras and have gotten to like the following excerpt of wikipedia article concerning von Neumann algebras because it gives a somewhat blurry bird's eye perspective on the whole "equivalence of projections"-stuff I learned about in my lectures.

Operators E in a von Neumann algebra for which E = EE = E* are called projections; they are exactly the operators which give an orthogonal projection of H onto some closed subspace. A subspace of the Hilbert space H is said to belong to the von Neumann algebra M if it is the image of some projection in M. This establishes a 1:1 correspondence between projections of M and subspaces that belong to M. Informally these are the closed subspaces that can be described using elements of M, or that M "knows" about.It can be shown that the closure of the image of any operator in M and the kernel of any operator in M belongs to M. Also, the closure of the image under an operator of M of any subspace belonging to M also belongs to M. (These results are a consequence of the polar decomposition).

Comparison theory of projections[edit]

The basic theory of projections was worked out by Murray & von Neumann (1936). Two subspaces belonging to M are called (Murray–von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if M "knows" that the subspaces are isomorphic). This induces a natural equivalence relation on projections by defining E to be equivalent to F if the corresponding subspaces are equivalent, or in other words if there is a partial isometry of H that maps the image of E isometrically to the image of F and is an element of the von Neumann algebra. Another way of stating this is that E is equivalent to F if E=uu* and F=uu for some partial isometry u in M. The equivalence relation ~ thus defined is additive in the following sense: Suppose E1 ~ F1 and E2 ~ F2. If E1 ⊥ E2 and F1 ⊥ F2, then E1 + E2 ~ F1 + F2. Additivity would not generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we say E is equivalent to F if uEu = F for some unitary u. The subspaces belonging to M are partially ordered by inclusion, and this induces a partial order ≤ of projections. There is also a natural partial order on the set of equivalence classes of projections, induced by the partial order ≤ of projections. If M is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below. A projection (or subspace belonging to M) E is said to be a finite projection if there is no projection F < E (meaning F ≤ E and F ≠ E) that is equivalent to E. For example, all finite-dimensional projections (or subspaces) are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself. However it is possible for infinite dimensional subspaces to be finite. Orthogonal projections are noncommutative analogues of indicator functions in L∞(R). L∞(R) is the ||·||∞-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the spectral theorem for self-adjoint operators. The projections of a finite factor form a continuous geometry.

Since the conceptual role of central projections is somewhat blurry to me I wonder whether there's someone here who could try to give me a feeling of why they are important (and rather not only quote theorems/definitions in which they play a part) As already said, I'm interested in a conceptual answer. Even if analogies might be somewhat way off, I might be interested!

Thanks in advance!


It is easily understood in the finite-dimensional case. A finite-dimensional von Neumann algebra is of the form $$\tag{$*$} M=\bigoplus_{j=1}^NM_{k_j}(\mathbb C). $$ We may see $M$ as $N$-tuples of matrices. The central projections are precisely the ones that given the decomposition: $p\in M$ is central if and only if $$ p=\bigoplus_j \alpha_j\,I_{k_j}, $$ where $\alpha_j\in\{0,1\}$, $j=1,\ldots,N$.

If you started knowing in abstract that $M$ is a finite-dimensional von Neumann algebra, the way to obtain the decomposition ($*$) is to let $p_1,\ldots,p_N$ be the minimal central projections (it is easy to show that they are unique). Then show that $p_jM$ is a factor for each $j$, and prove that a finite-dimensional factor is $M_k(\mathbb C)$.

In infinite dimension there is a version of the above, where now the direct sum becomes a direct integral, and the matrix algebras become factors.

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  • $\begingroup$ How one can choose the minimal projections $p_{1},p_{2},...p_{N}$ in $M$? $\endgroup$ – rkmath Jul 20 '18 at 5:35
  • $\begingroup$ You don't get to choose them. The set of minimal central projections in a von Neumann algebra is uniquely defined. It may be empty, but that cannot be the case in finite dimension. $\endgroup$ – Martin Argerami Jul 20 '18 at 5:39
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    $\begingroup$ If $q_1$ has no minimal projections below, you can get a strict sequence $q_1\geq q_2\geq\cdots $. Define $r_n=q_n-q_{n+1} $ and now $\{r_n\} $ are pairwise orthogonal and $\operatorname {span}\{r_n:\ n\} $ is infinite-dimensional. $\endgroup$ – Martin Argerami Jul 20 '18 at 5:57

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