Von Neumann algebras and bicommutants

A net of operators converges strongly when its action converges in norm on every fixed Hilbert vector. It converges weakly when every scalar matrix coefficient converges. Operator-norm convergence implies strong convergence, which implies weak convergence, but neither reverse implication holds in general. Nets are the correct language on arbitrary Hilbert spaces; sequences suffice for many separable bounded examples but cannot silently replace every topological statement.

On the Hilbert space of square-summable sequences, let P_n project onto the first n standard basis vectors. The norm of (I−P_n)ξ tends to zero for each fixed ξ because its squared tail is summable. Thus P_n converges strongly to I. Yet the operator norm of I−P_n is always one, since a later basis vector is left unchanged by the difference. This example explains how an operator-topology closure can add an identity even when a norm closure cannot.

Let A be a unital *-subalgebra of B(H), containing the same identity as B(H). Von Neumann's bicommutant theorem states that its strong closure and weak closure both equal A-double-prime. This is not a claim that norm closure also equals them. The unit and the adjoint-closed algebra hypotheses are essential to this formulation; a random set of operators or an arbitrary nonselfadjoint algebra is not covered by simply placing two primes on its name.

The inclusion of either operator closure in the bicommutant follows from the continuity of commuting relations under these tests. The reverse strong-density direction is the substantive theorem, proved using approximation on finite collections of vectors. It is stated here rather than reproved. In finite dimensions all these topologies agree, so a finite-dimensional unital *-subalgebra containing the ambient identity equals its bicommutant. Infinite-dimensional examples are needed to appreciate the difference from C*-norm closure.

A von Neumann algebra contains suprema of bounded increasing nets of self-adjoint elements, with the usual operator order. A positive functional is normal when it preserves these increasing positive suprema. Equivalently, it is continuous for the ultraweak topology coming from the algebra's predual. The word normal here describes a functional or map; it is unrelated to an element commuting with its adjoint. Not every state on an infinite-dimensional von Neumann algebra is normal.

In a concrete representation, vector states are normal. On B(H), normal states are precisely those given by positive trace-class density operators of trace one. A von Neumann algebra admits a faithful normal state exactly when it is σ-finite, also called countably decomposable: every family of mutually orthogonal nonzero projections is countable. Later modular theory therefore explicitly assumes such a state, or a cyclic separating vector in a chosen representation. One must not infer the existence of that vector merely from writing M as operators on some Hilbert space.