In [1] Slawny formulated the C^* algebra of the CAR and CCR as twisted group C^* algebras for certain abelian groups. In this note we extend his work, in the case of the CAR algebra, by applying some recent results from the projective representation theory of abelian groups. We show that the Fock representation is monomial and construct pure infinite tensor product states via a group theoretic method. We also sketch some other constructions of irreducible representations including non-product ones.

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Product mappings between CAR-algebras are introduced. This notion is used to state and prove that generalised free completely positive mappings between CAR-algebras arise as central limits of general ones.

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An infinite-dimensional Lie algebra acting on solutions to the anti-self-dual equation on a four-dimensional Euclidean space is derived by means of the Riemann-Hilbert problem. Three types of Bäcklund transformations are considered in the framework of the Riemann-Hilbert problem.

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We study canonical pairs of self-adjoint operators P and Q whose restrictions to a common invariant dense domain \mathscr{D} of a Hilbert space are essentially self-adjoint and satisfy the Heisenberg commutation relation PQφ−QPφ=−iφ for φ∈\mathscr{D}. Under some additional assumption, we obtain a classification of such pairs. Moreover, we develop some methods for constructing canonical pairs of this class.

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The L_p-space L_p(M, φ₀) for a von Neumann algebra M and its faithful normal semifinite weight φ₀ is constructed as a linear space of closed linear operators acting on the standard representation Hilbert space H_φ0.
Any L_p-element has the polar and the Jordan decompositions relative to a positive part L_p⁺(M, φ₀). Any positive element in L_p-space has an interpretation as the (1/p)^th power φ^1/p of a φ∈M_*⁺ with its L_p-norm given by φ(1)^1/p.
The product of an L_p-element and an L_q-element is explicitly defined as an L_r-element with r⁻¹=p⁻¹+q⁻¹ provided 1{≤}r and the Hölder inequality is proved. Also L_p(M, φ₀) are shown to be isomorphic for the same p and different φ₀.
There exists a vector subspace D_φ0^∞ of the Tomita algebra associated with φ₀ and a p-dependent injection T_p:D_φ0^∞→L_p(M, φ₀) with dense range. The sesquilinear form on L_p(M, φ₀)×L_p' (M, φ₀), p⁻¹+(p')⁻¹=1 is a natural extension of the inner product of H_φ0 through the mapping T_p×T_p'.
The present work is an extension of our joint work with Araki to weights, and our L_p-spaces are isomorphic to those defined by Haagerup, Hilsum, Kosaki, and Terp.

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Using the properties of standard subspaces of the 1-particle space for the free Fermi field we prove the twisted duality for the von Neumann algebras associated with real closed subspaces. This is done by application of the Tomita-Takesaki theory.

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When the scalar quadratic variation of a martingale on a Riemannian manifold is finite almost surely, then the martingale converges almost surely in the one-point compactification of the manifold. A partial converse due to Zheng Wei-an is also proved. No curvature conditions on the manifold are required.

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