Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false?
Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq d} p_n=0$ for some $d\in \mathbb{N}$.
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1$\begingroup$ No. There exist lots of examples (e.g., all finite type II W* factors) with (lots of) descending sequences of nonzero projections $\{p_n \}$ with $p_k \leq p_{k+1}$ and the traces go to zero (so the infimum is zero. The condition you require is a type of noetherian condition, which in the AW* case, would force it to have all simple quotients by ideals to be finite dimensional, a very strong condition. $\endgroup$– David HandelmanCommented Aug 8, 2018 at 13:15
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$\begingroup$ @DavidHandelman, Thanks a lot, very helpful comment. $\endgroup$– ABBCommented Aug 8, 2018 at 13:22
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