How did Kadison obtain it here?

Solution 1:

This is a weird statement. Where exactly in KRII is it?

Without additional hypotheses, a type I$_n$ von Neumann algebra cannot have an abelian central summand if $n>2$.

Suppose that $R$ is type I$_n$ and $R=A\oplus M$, with $A$ abelian. By hypothesis there exist abelian projections $E_1,\ldots,E_n\in R$ and partial isometries $W_{kj}$ such that $W_{kj}^*W_{kj}=E_j$, $W_{kj}W_{kj}^*=E_k$. Let $P_A$ be the central projection corresponding to $A$. By definition $P_AE_k\in A$ and $P_AW_{kj}\in A$. So, using that $P_A$ is central and that $A$ is abelian, $$ P_AE_k=P_AW_{kj}W_{kj}^*=(P_AW_{kj})(P_AW_{kj}^*)=(P_AW_{kj}^*)(P_AW_{kj})=P_AW_{kj}^*W_{kj}=P_AE_j. $$ Then $$ P_AE_k=P_AE_kP_AE_j=P_AE_kE_j=0 $$ if $k\ne j$. This gives us $$ P_A=\sum_kP_AE_k=0, $$ and so $A=0$.