Algebras of observables of nearly canonical physical theories. I (Q1071287)

The paper expands and generalizes earlier work of Shirokov. Jordan- algebras of functions are considered and a group of shifts is introduced. Results on shift invariance are obtained. Earlier results are extended to an infinite-dimensional setting. The proofs involve the following type of functional equations \[ \rho (x,y)\rho '(x+y,z)+\rho (y,z)\rho '(y+z,x)+\rho (z,x)\rho '(z+x,y)=0 \] where \(\rho \in C^ 2(\Omega \times \Omega)\), \(\Omega\) is a neighborhood of 0 in \({\mathbb{R}}^ n\), \(\rho '(x,y)=(\frac{\partial}{\partial x_ i}-\frac{\partial}{\partial y_ i})\rho (x,y),\) and \[ \rho (x,y)=ch\sum^{n}_{i,j=1}x_ iB_{ij}y_ jU(x)U(y)U^{-1}(x+y) \] with \((B_{ij})\) arbitrary anti- symmetric, and U(x)\(\neq 0\), \(x\in \Omega\), \(U\in C^ 2(\Omega)\), \(U(0)=1.\) A related functional equation is introduced for the function \(\pi\) (\(\cdot,\cdot)\) given by \[ \pi (x,y)=\exp \sum^{n}_{i,j=1}x_ iA_{ij}y_ jU(x)U(y)U^{-1}(x+y) \] where \((A_{ij})\) is now an arbitrary matrix. The functions \(\rho\) and \(\pi\) enter into formulas for integral operators of the form \[ (\rho AB)(x)=\int_{{\mathbb{R}}^{2n}}e^{i(k+p)x}\tilde A(k)\tilde B(p)(r(k,p)dkdp \] where \(\sim\) denotes Fourier transform.