In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral ${\prod }^{\to \to }(1+dr)$ with generator $dr=\lambda (d{A}^{†}\otimes dA-dA\otimes d{A}^{†})$. The method of construction is to approximate the product integral by a discrete double product ${\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\Gamma \left(R_{m,n}^{(j,k)}\right)=\Gamma \left({\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\left(R_{m,n}^{(j,k)}\right)\right)$ of second quantised rotations $R_{m,n}^{(j,k)}$ in different planes using the embedding of ${ℂ}^m\oplus ℂⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of ${ℂ}^m$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator functions of...

We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra $U_q\left(s{l}_2\left(ℂ\right)\right)$. We discuss two cases, according to whether the parameter $q$ is a root of unity. We show that the universal enveloping algebra of $s{l}_2\left(ℂ\right)$ embeds in a non-principal ultraproduct of $U_q\left(s{l}_2\left(ℂ\right)\right)$, where $q$ varies over the primitive roots of unity.

In [4] it is proved that a measure on a finite coarse-grained space extends, as a signed measure, over the entire power algebra. In [7] this result is reproved and further improved. Both the articles [4] and [7] use the proof techniques of linear spaces (i.e. they use multiplication by real scalars). In this note we show that all the results cited above can be relatively easily obtained by the Horn-Tarski extension technique in a purely combinatorial manner. We also characterize the pure measures...

The notion of a partially ordered partial abelian monoid is introduced and extensions of partially ordered abelian monoids by partially ordered abelian groups are studied. Conditions for the extensions to exist are found. The cases when both the above mentioned structures have the Riesz decomposition property, or are lattice ordered, are treated. Some applications to effect algebras and MV-algebras are shown.

A positive semi-definite block matrix (a state if it is normalized) is said to be separable if it is the sum of simple tensors of positive semi-definite matrices. A state is said to be entangled if it is not separable. It is very difficult to detect the border between separable and entangled states. The PPT (positive partial transpose) criterion tells us that the partial transpose of a separable state is again positive semi-definite, as was observed by M. D. Choi in 1982 from...

We give a new construction of semifinite factor representations of the diffeomorphism group of euclidean space. These representations are in canonical correspondence with the finite factor representations of the inductive limit unitary group. Hence, many of these representations are given in terms of quasi-free representations of the canonical commutation and anti-commutation relations. To establish this correspondence requires a generalization of complete positivity as developed in operator algebras....

We study a certain class of von Neumann algebras generated by selfadjoint elements ${\omega }_i=a_i+a{⁺}_i$, where $a_i,a{⁺}_i$ satisfy the general commutation relations: $a_{i}a{⁺}_j={\sum }_{r,s}{t}_j^i{r}_{s}a{⁺}_r{a}_s+{\delta }_{ij}Id$. We assume that the operator T for which the constants $t_j^i{r}_s$ are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the number of generators...