Let $R$ be a finite von Neumann algebra and $S$ be a von Neumann subalgebra of $R$. Does every normal tracial state on $S$ extend to a normal tracial state on $R$?



Extension of trace on von Neumann subalgebra
Let $C$ denote the complexes, and embed $C \times C$ in $M\_2 C$ ($2 \times 2$ matrices) as diagonal matrices. Then $M\_2 C$ has unique trace, but $C \times C$ has two extremal ones. 


