The chiral model for self-dual gravity given by Husain in the context of the chiral equations approach is discussed. A Lie algebra corresponding to a finite dimensional subgroup of the group of symplectic diffeomorphisms is found, and then use for expanding the Lie algebra valued connections associated with the chiral model. The self-dual metric can be explicitly given in terms of harmonic maps...

The holonomy-flux *-algebra was recently proposed as an algebra of basic kinematical observables for loop quantum gravity. We show the conventional GNS construction breaks down when the the holonomyflux *-algebra is allowed to be a Jordan algebra of observables. To remedy this, we give a Jordan GNS construction for the holonomy-flux *-algebra that is based on trace. This is accomplished by assu...

We give a new and clever proof of the Russo–Dye theorem for JB∗-algebras, which depends on certain recent tools due to the present author. The proof given here is quite different from the known proof by J. D. M. Wright and M. A. Youngson. The approach adapted here is motivated by the corresponding C∗-algebra results due to L. T. Gardner, R. V. Kadison and G. K. Pedersen. Accordingly, it yields ...

A theorem is presented on existence and uniqueness up to the topological *-isomorphism of universal locally C*-algebra for an arbitrary locally JB-algebra. The abstract Banach associative symmetrical *-algebras over C, so called C*algebras, were introduces first by Gelfand and Naimark in []. In the present time the theory of C*-algebras become a vast portion of Functional Analysis having connec...

We prove that every surjective isometry from the unit sphere of space K(H), all compact operators on an arbitrary complex Hilbert H, onto real Banach Y can be extended to a linear K(H) Y. This is probably first example infinite dimensional non-commutative C*-algebra containing no unitaries and satisfying Mazur-Ulam property. also C*-algebras weakly JB*-triples satisfy

We prove that given a real JB*-triple E, and a real Hilbert space H , then the set of those bounded linear operators T from E toH , such that there exists a norm one functionalφ ∈ E∗ and corresponding pre-Hilbertian semi-norm ‖.‖φ on E such that ‖T (x)‖ ≤ 4 √ 2‖T‖ ‖x‖φ for all x ∈ E, is norm dense in the set of all bounded linear operators from E toH . As a tool for the above result, we show th...