• exists_predual : ∃ (X : Type ?) [_inst_1_1 : normed_add_comm_group X] [_inst_2 : normed_space ℂ X] [_inst_3 : complete_space X], nonempty (normed_space.dual ℂ X ≃ₗᵢ⋆[ℂ] M)

Sakai's definition of a von Neumann algebra as a C^* algebra with a Banach space predual.

So that we can unambiguously talk about these "abstract" von Neumann algebras in parallel with the "concrete" ones (weakly closed *-subalgebras of B(H)), we name this definition wstar_algebra.

Note that for now we only assert the mere existence of predual, rather than picking one. This may later prove problematic, and need to be revisited. Picking one may cause problems with definitional unification of different instances. One the other hand, not picking one means that the weak-* topology (which depends on a choice of predual) must be defined using the choice, and we may be unhappy with the resulting opaqueness of the definition.

• carrier : set (H →L[ℂ] H)
• mul_mem' : ∀ {a b : H →L[ℂ] H}, a ∈ self.carrier → b ∈ self.carrier → a * b ∈ self.carrier
• one_mem' : 1 ∈ self.carrier
• add_mem' : ∀ {a b : H →L[ℂ] H}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
• zero_mem' : 0 ∈ self.carrier
• algebra_map_mem' : ∀ (r : ℂ), ⇑(algebra_map ℂ (H →L[ℂ] H)) r ∈ self.carrier
• star_mem' : ∀ {a : H →L[ℂ] H}, a ∈ self.carrier → has_star.star a ∈ self.carrier
• centralizer_centralizer' : self.carrier.centralizer.centralizer = self.carrier

The double commutant definition of a von Neumann algebra, as a *-closed subalgebra of bounded operators on a Hilbert space, which is equal to its double commutant.

Note that this definition is parameterised by the Hilbert space on which the algebra faithfully acts, as is standard in the literature. See wstar_algebra for the abstract notion (a C^*-algebra with Banach space predual).

Note this is a bundled structure, parameterised by the Hilbert space H, rather than a typeclass on the type of elements. Thus we can't say that the bounded operators H →L[ℂ] H form a von_neumann_algebra (although we will later construct the instance wstar_algebra (H →L[ℂ] H)), and instead will use ⊤ : von_neumann_algebra H.