Making use of its smooth structure only, out a connected oriented $4$-manifold von Neumann algebra is constructed. As special four dimensional phenomenon this approximated by algebraic (i.e., formal) curvature tensors the underlying and itself hyperfinite factor ${\rm II}_1$ type hence unique up to abstract isomorphisms algebras. Nevertheless over fixed admits representation on Hilbert space su...

We prove a generalization of the polarization identity linear algebra expressing inner product complex space in terms norm, where field scalars is extended to an associative equipped with involution, and viewed as averaging operation over compact multiplicative subgroup scalars. Using this we general form Jordan-von Neumann theorem on characterizing spaces among normed spaces, when are taken al...

Let M be a von Neumann algebra and ρ : M → M be a ∗-homomorphism. Then ρ is called a centrally extendable ∗-homomorphism (CEH) if there is a maximal abelian subalgebra (masa) M of the commutant M of M and a surjective ∗-homomorphism φ : M → M such that φ(Z) = ρ(Z) for all Z in the center of M. A ∗-ρderivation δ : M → M is called a centrally extendable ∗-ρ-derivation (CED) if there is a masa M o...