I've got difficulties concerning the above mentioned things.
I'm studying the type decomposition of (unital) von Neumann algebras. There often the result of a theorem is that one can decompose some von Neumann algebra $M$ into a direct sum of von Neumann algebras $(M_i)_i$, i.e. $$M=\oplus_{i\in I}M_i$$
In the proof then what one does it typically show that there are central pairwise orthogonal projections $\{p_i\}_i$ s.t. $\sum_ip_i=1$ and $p_iM=M_i$. See for example the chapter on type decomposition in the book of Kehe Zhu!
While the special intricacies of those proof often don't bother me very much, I didn't quite understand why the existence of such projections imply the desired result.
More concisely:
- In what sense is $\sum_ip_i=1$ valid? ( My guess SOT-convergence? )
- Why does the existence of such projections translate to a direct sum? I guess I'm looking for the exact isomorphism.
I've been thinking about this for a while now, and at least for the case that there are only 2 (thus probably by induction finitely many) projections $q,(1-q)$ the isomorphisms could be :
$$\Phi \colon qM\oplus(1-q)M\to M $$ $$\Psi \colon M\to qM\oplus(1-q)M $$
given by: $\Phi((a,b))=a+b$ and $\Psi(x)=(qx,(1-q)x)$
Is this correct? And if so is the isomorphism in the infinite case also
$$\oplus_{i\in I}p_iM \to M, \quad (a_i)_{i\in I}\mapsto \sum_ia_i\quad(1)$$ and $$M \to \oplus_{i\in I}p_iM, \quad a\mapsto (p_ia)_{i\in I}\quad (2)$$?
While I think that I understand that the second map is really mapping into the direct sum if one assumes that the projections add to one in the SOT-sense I think I don't see the wood for the trees concerning the well-definedness of the first.
Any help would be very much appreciated. Thanks!
Yes, it is exactly as you say. The sum pairwise orthogonal family of projections will always converge sot; this can be done directly (by thinking of orthonormal bases) or by using the result that an increasing, bounded, net of selfadjoints is sot-convergent.
And the isomorphisms are again as you say. The sot convergence of $\sum a_j$, where $a_j=p_ja_jp_j$ and the $\{p_j\}$ are pairwise orthogonal, follows as above.