Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

Representations of locally convex *-algebras

Abstract: Conditions for a functional to be admissible on a locally convex *-algebra are defined. Let F be an admissible positive Hermitian functional on a commutative locally convex *-algebra; then it is shown that there exists a representation of A into a Hilbert space. Sufficient conditions for a functional F to be representable are also given. 1. By a locally convex algebra A we shall mean an algebra A, over the complex numbers C, which has associated with it a Hausdorff topology T7 such that multiplication is separately continuous. A will be called a locally convex *-algebra if A has a continuous involution. If x is an element of A such that x*=x then x will be called Hermitian. An element x of A is said to be bounded if for some nonzero complex number 2, the set {(2x)n:n e N} is bounded. The set of bounded elements of A will be denoted by AO. Let B1 denote the collection of all closed, convex, circled sets B that are also bounded and idempotent. If B E B1, then A(B) will denote the subalgebra of A generated by B, i.e., A(B)= {2x:2 iE C, x E B}, and the equation 11x111 = inf{2,> 0: x E 2B} defines a norm which makes A(B) a normed algebra. A will be called pseudo-complete if each A(B) is a Banach algebra. For each x E A, the radius of boundedness of x, P(x), is defined by P(x)=inf{2>O:{(x/2) : n E N} is bounded} with oo=inf0. (For properties of P see [1].) Let A be a locally convex *-algebra, and let F be a linear functional on A. If F(x*)=(F(x)) for all x in A, F will be called Hermitian. If F(x*x)>O for all x in A, then F will be called a positive functional. 2. Admissible functionals. Before defining admissible functionals consider the following: LEMMA 1. Let A be a pseudo-complete locally convex *-algebra and let xo be any element of A such that P(xo)< 1. Then there exists an element Presented to the Society, February 22, 1968 under the title Representation theory for locally convex *-algebras; received by the editors December 15, 1970. AMS (MOS) subject classifications (1970). Primary 46H15.

Real Orthogonal Projections as Quantum Logic

Abstract: In this paper, we study linear operators on real and complex Euclidean spaces which are real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We can compare some partial order properties of the orthogonal and of the R-orthogonal projections. We prove that the set of all R-orthogonal projections in finite-dimensional complex space is a quantum logic.

QUADRATIC RESIDUE CODES OVER p-ADIC INTEGERS AND THEIR PROJECTIONS TO INTEGERS MODULO p e

Abstract: We give idempotent generators for quadratic residue codes over $p$-adic integers and over the rings $\mathbb{Z}_{p^e}$.

Study of idempotents in cyclic group rings over F2

Abstract: The existence of an idempotent generator for group codes or group ring codes in FqG plays a very important role in determining the minimal distance of the respective code. Some necessary and sufficient conditions for a group ring element to be an idempotent in F2Cn are investigated in this paper. The main result in this paper is the affirmation of the existence of finitely many basis idempotents which gives a full identification of all idempotents in every binary cyclic group ring F2Cn. All the basis idempotents in F2Cn are able to be found by partitioning the largest idempotent’s support.