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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