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Using direct integral decomposition, also known as reduction theory, one can reduce the problem to the case of a factor. A conditional expectation in this case is a state. Every factor admits a state, …

The unitary group of any purely infinite von Neumann algebra is contractible (this is a generalization of Kuiper's theorem due to Brüning and Willgerodt, “Eine Verallgemeinerung eines Satzes von N. Ku …

Such t_0 is unique if its support is at most p, where p is the support of ϕ. Note that we can replace t_0 by pt_0p and the support of pt_0p is at most p. Without this additional condition t_0 is high …

Morphisms of von Neumann algebra have very nice properties. More precisely, the kernel of a morphism f: M→N of von Neumann algebras is a σ-weakly closed two-sided ideal. Such ideals are in bijective c …

Yes. The kernel of F is an ultraweakly closed *-ideal of M generated by some central projection z. M splits as a direct sum of zM and (1-z)M. As a 2x2 matrix F has only two nonzero entries, one that …

This follows from the reduction theory for von Neumann algebras (alias direct integral decomposition). Any von Neumann algebra is a direct integral of factors (i.e., von Neumann algebras with a trivia …

No. Take e=0 and 0 < f < 1 such that both f and 1−f are infinite, with (1−f)~1. Then e≾f because 0≾f for any projection f. Also 1−e≾1−f because 1≾1−f, which holds by definition of f.

A von Neumann algebra is a $C^*$-algebra $A$ that admits a predual, i.e., a Banach space $A_*$ such that there is an isomorphism $A\to(A_*)^*$. A morphism of von Neumann algebras is a morphism of $C^ …

How the norm on L^{P} space related to weight φ? The L^p-spaces and their norms are independent of the choice of the weight φ. See, for instance, the exposition by Yamagami in “Algebraic Aspects …

This is false. Consider, for example, the case of M being the von Neumann algebra of bounded complex-valued functions on an infinite countable set I. It acts on the Hilbert space of square-summable fu …

The category of von Neumann algebras W* admits a variety of monoidal structures of three distinct flavors. (1) W* is complete and therefore you have a monoidal structure given by the categorical prod …

The spaces $[0,1]$, $[0,1]^2$, and $S^1$ are all isomorphic as measurable spaces, including their sets of measure 0, as required by the Gelfand-type duality for measurable spaces. For instance, the is …

Incidentally, the question can be generalized to arbitrary von Neumann algebras instead of B(H) and is closely related to its analog for Lie algebras: What is the set of all linear combinations of com …

There is a notion of compact element for any W*-algebra, namely, the two-sided ideal of compact elements is the norm-closure of the two-sided ideal of finite-rank elements, the latter being defined as …

Corollary 4.1.(i) in Johnstone's book Stone Spaces states that the category of realcompact spaces is dual to the full subcategory of the category of commutative rings consisting of rings of the form C …