Can anyone reveal me what was von Neumann and Murray's original idea behind the definition of Murray-von Neumann equivalence of two projections in a von Neumann algebra?
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2 Answers
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The basic intuition is that the equivalence of projections reflects an algebraic way of characterizing when two projections in $B(H)$ have the same rank.
So one has an abstract notion of "rank", which works without the underlying substrate of the Hilbert space. Murray and von Neumann were very successful in using this idea to study II$_1$-factors.
The idea is so powerful that it resurfaces in K-theory.
answered Jul 19, 2018 at 15:21
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I can not tell what Murray and von Neumann had in mind, but a piece of intuition that i like is that if $\pi: \Gamma \to U(H)$ is a unitary representation a natural question is to classify its irreducible subrepresentations. But such subrepresentations are in bijective correspondence with the projections of $N = \pi[\Gamma]'$. Two such subrepresentations are unitary equivalent iff their associated projections are Murray-von Neumann equivalent.
answered Jul 19, 2018 at 15:15
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$\begingroup$ In this point of view why one focus on the case where $N$ is a factor? $\endgroup$– rkmathCommented Jul 25, 2018 at 6:48
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$\begingroup$ You don't need to focus on that case, this analogy works for all groups $\endgroup$ Commented Jul 25, 2018 at 9:13