Let $E$ be a projection in a von Neumann algebra $R$. According to Kadison and Ringrose, $E$ is $\textit{properly infinite}$ if $E$ is infinite and for each central projection $P\in R$ either $PE=0$ or $PE$ is infinite. But I've seen other definitions that require $PE$ to be infinite for each nonzero central projection $P\in R$. It doesn't seem to me that these two definitions are equivalent, and if I am right then I believe there is a proof of a lemma in Kadison and Ringrose which is incorrect as written.
Definition of properly infinite projection in a von Neumann algebra
von-neumann-algebras
Best Answer
I don't know where you've seen your second definition, but it is a weird choice. The spirit of "properly infinite" is that it has no finite part. Like, in $B(H)\oplus B(H)$, the projection $E_{11}\oplus I$ is not properly infinite because it has a finite part in the first block; this precludes the possibility of halving it. But saying that $0\oplus I$ is not properly infinite makes little sense, as it satisfies any property a properly-infinite-in-the-second-sense projection would have (other than having central carrier different than the identity).
As for the Halving Lemma in Kadison-Ringrose, the distinction is immaterial as it doesn't change the proof.