The direct sum is a way to combine two or more vector spaces (or modules) into a new, larger vector space. In the context of von Neumann algebras, it allows for the decomposition of representations and helps in understanding the structure of the algebra by breaking it down into simpler components, which can be independently analyzed. This concept plays a crucial role in establishing the standard form of von Neumann algebras, where each part can be treated separately while still being part of a unified whole.
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