Reading the paper here, it mentions on the very first page that "The requirement of 'closed'-ness is imposed because we want to think of operator spaces as 'quantized (or non-commutative) Banach spaces,'" the 'closed'-ness here referring to their definition of an operator space as a closed subspace of a Hilbert space. My questions, then, are why does a closed subspace lend itself to a description as a quantization, and why does the word non-commutative show up here?
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1I think the word `non-commutative' appears because quantum operators are noncommutative. Closed-ness is just required for the operators of interest to be a Banach space. This is natural, so that you can add operators and take limits. – TLDR Jun 24 '15 at 18:29
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Related: http://physics.stackexchange.com/q/20822/2451 and links therein. – Qmechanic Jun 24 '15 at 18:35
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Echoing comment by @Couchyam: Closedness/completeness is a technical assumption (as opposed to a fundamental requirement or paradigm). It is typically imposed to simplify mathematical proofs. In particular, it should not be viewed as the culprit of quantization and non-commutativity. – Qmechanic Jun 24 '15 at 19:06