An almost radial gauge $A^\mathrm{ar}$ of the electromagnetic potential is constructed for which $x\cdot A^\mathrm{ar}(x)$ vanishes arbitrarily fast in timelike directions. This class introduced by Dirac with purpose forming gauge-invariant quantities quantum electrodynamics. In case construction smeared operators $A^\mathrm{ar}(K)$ enabled a natural extension free field algebra earlier (repres...

The main theorems below are Theorem 9 and Theorem 11. As far as I know, Theorem 9 represents a slight improvement over what currently appears in the literature, and gives a fairly easy proof of Theorem 11 which is due to R. Howe and C. Moore [4]. In the semisimple case, the Howe-Moore result follows from [6] or [7]. The proofs appearing here are relatively elementary and some readers will recog...

We continue the analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and, following Sahlmann’s ideas [1], define a holonomy-flux ∗-algebra whose elements correspond to the elementary variables. There exists a natural action of autom...

Information algebra is algebraic structure for local computation and inference. Given an initial universe set and a parameter set, we show that a soft set system over them is an information algebra. Moreover, in a soft set system, the family of all soft sets with a finite parameter subset can form a compact information algebra.

In this paper the concept of unbounded Fredholm operators on Hilbert C∗modules over an arbitrary C∗-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert C∗-modules over C∗-algebras of compact operators. In the framework of Hilbert C∗-modules over C∗-algebras of compact operators, the index of an unbounded Fredholm operator and the...

In this paper, we introduce a Sheffer stroke Hilbert algebra by giving definitions of and algebra. After it is shown that the axioms are independent, given some properties algebraic structure. Then stated relationship between defining unary operation on Also, presented deductive system ideal It defined an generated subset algebra, constructed new adding element to its ideal.

Let A be a linear bounded operator in a Hilbert space H, N(A) and R(A) its null-space and range, and A∗ its adjoint. The operator A is called Fredholm iff dim N(A) = dim N(A∗) := n < ∞ and R(A) and R(A∗) are closed subspaces of H. A simple and short proof is given of the following known result: A is Fredholm iff A = B + F , where B is an isomorphism and F is a finite-rank operator. The proof co...

One of unsolved problems in quantum measurement theory is to characterize coexistence of quantum effects. In this paper, applying positive operator matrix theory, we give a mathematical characterization of the witness set of coexistence of quantum effects and obtain a series of properties of coexistence. We also devote to characterizing bijective morphisms on quantum effects leaving the witness...

In this paper, we investigate the structure of the multiplier module of a Hilbert module over a locally C∗-algebra and the relationship between the set of all adjointable operators from a Hilbert A -module E to a Hilbert A module F and the set of all adjointable operators from the multiplier module M(E) of E to the multiplier module M(F ) of F.

In this article, we throw a bridge between two objects which are unrelated at rst sight. One is the in nite dimensional Heisenberg algebra (simply the Heisenberg algebra, later) which plays a fundamental role in the representation theory of the a ne Lie algebras. The other is the Hilbert schemes of points on a complex surface appearing in the algebraic geometry. As we will explain soon, the Hei...