Question

Prove that in a complemented modular lattice either of chain condition applies the others

Answer 1

The paper [Murray and von Neumann, 1936] shows that there exist non-distributive, modular lattices different from the Hilbert lattice of a finite dimensional Hilbert space. Proving the existence of such a structure is part of what is known as the “classification theory of von Neumann algebras”, which has since 1936 become a classical chapter in the theory of operator algebras.16 The relevant – and surprising – result of this classification theory is that there exists a modular lattice of non-finite (linear) dimensional projections on an infinite dimensional Hilbert space, and that on this lattice there exists a (unique up to normalization) dimension function d that takes on every value in the interval [0, 1]. The von Neumann algebra generated by these projections is called the “type II1 factor von Neumann algebra” N.17 Furthermore, it can be shown that the unique dimension function on the lattice of projections of a type II1 factor comes from a (unique up to constant) trace τ defined on the factor itself – just like in the finite dimensional case, where too the dimension function is the restriction of the trace functional Tr to the lattice of projections.

Answer 2

Step-by-step explanation:

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