# Role of central projections in theory of von Neumann algebras

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

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!

## 1 Answer

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.

• How one can choose the minimal projections $p_{1},p_{2},...p_{N}$ in $M$? – rkmath Jul 20 '18 at 5:35
• 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. – Martin Argerami Jul 20 '18 at 5:39
• 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. – Martin Argerami Jul 20 '18 at 5:57