# Does there exists a von Neumann algebra which is neither of type I$_n$, II$_1$,II$_\infty$, nor III?

In the "Type decomposition" theorem 6.5.2 in Kadison-Ringrose, vol II, they proved that every von Neumann algebra can be decomposed as a direct sum of type $$I_n$$, type $$II_1$$, type $$II_\infty$$ and type $$III$$ parts (some of these summands may be zero). I understood this. My question is:

Does there exists a von Neumann algebra which is neither of type $$I_n, II_1,II_\infty, \text{ nor } III$$?

For example, consider $$\mathscr{R}:=L^\infty(X, \mu)$$ acting on $$\mathcal{H}:=L^2(X, \mu)$$. Then since $$\mathscr{R}$$ is abelian so every non-zero projection in it is abelian projection. Hence $$\mathscr{R}$$ cannot be of type $$II$$ nor $$III$$. Can I say from there that $$L^\infty(X, \mu)$$ is a type $$I$$ von Neumann algebra (without even finding a projeciton with central carrier $$I$$. In deed here $$\chi_X$$ is an abelian projection with central carrier $$I=\chi_X$$ proving $$L^\infty$$ or any abelian algebra is of type $$I$$.)?

• I am not sure if I understand your question. Every von Neumann algebra is a direct sum of algebras of type I, II and III. So of course if it is a sum of (non-zero) type I and type II algebras for example, it is neither of type I, II or III. Is that what you mean? Dec 16, 2021 at 7:54
• @MaoWao Yes that is my point....So such algebras (i.e. direct sum of non zero types) are neither of the each type?? Why??? Is this some thing very trivial; follows from the definition ?? Dec 16, 2021 at 9:25

A von Neumann algebra can have all the types as central summands. For instance take $$M$$ and $$N$$ of your favourite (different types), and form $$M\oplus N$$.
If you look at the proof of Theorem 6.5.2 in KRII, the central projection $$P_d$$ is constructed in such a way that $$(I-P_d)R$$ has no abelian projections. That makes it unique.
In a similar way, $$P_{c_1}$$ is constructed as the largest finite central projection in $$(I-P_d)R$$. And so on.
The point is that you don't get to choose. A von Neumann algebra $$R$$ is a direct sum of von Neumann algebras of fixed types.
That a direct sum of algebras of distinct types is of neither type is trivial to check. For instance if $$R=M\oplus N$$ with $$M$$ type I and $$N$$ type II$$_1$$, then there are not abelian projections under $$P_N$$, so $$R$$ is not type I. And there are abelian projections under $$P_M$$, so $$R$$ is not type II.