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Von Neumann algebra
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===Type I factors=== A factor is said to be of '''type I''' if there is a minimal projection ''E β 0'', i.e. a projection ''E'' such that there is no other projection ''F'' with 0 < ''F'' < ''E''. Any factor of type I is isomorphic to the von Neumann algebra of ''all'' bounded operators on some Hilbert space; since there is one Hilbert space for every [[cardinal number]], isomorphism classes of factors of type I correspond exactly to the cardinal numbers. Since many authors consider von Neumann algebras only on separable Hilbert spaces, it is customary to call the bounded operators on a Hilbert space of finite dimension ''n'' a factor of type I<sub>''n''</sub>, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I<sub>β</sub>.
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