We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + αx+ βy + γ for q ∈ K∗ and α, β, γ ∈ K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y · x in te...

We introduce the notion of “local system of ZT -twisted vertex operators” on a Z2-graded vector space M , generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of ZT -twisted vertex operators on M has a vertex superalgebra structure with an automorphism σ of order T with M as a σ-twisted module. Then we prove that for a vertex (operator) superal...

The main purpose of this paper is to prove the following result. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators on X, let A(X) ⊆ L(X) be a standard operator algebra, and let T : A(X) → L(X) be an additive mapping satisfying the relation T (A2n+1) = 2n+1 ∑ i=1 (−1)i+1Ai−1T (A)A2n+1−i, for all A ∈ A(X) and some fixed integer n ≥ 1. In this case T ...

After some earlier work on single operators, von Neumann turned to families of operators in [1]. He initiated the study of rings of operators which are commonly called von Neumann algebras today. The papers which constitute the series “Rings of operators” opened a new field in mathematics and influenced research for half a century (or even longer). In the standard theory of modern operator alge...

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...

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

In this paper, we consider Rota–Baxter operators on involutive asso-ciative algebras. We define cohomology for involutivealgebras that governs the formal deformation of operator. This cohomologycan be seen as Hochschild a certain associativealgebra with coefficients in suitable bimodule. also relate thiscohomology dendriform Finally, show standard Fard–Guo construction functor from category alg...

In this paper, we introduce a notion of Lie operator algebras which as a generalization of ordinary Lie algebras is an analogy of operator groups. We discuss some elementary properties of Lie operator algebras. Moreover, we also prove a decomposition theorem for Lie operator algebras.

In this paper, we introduce a notion of generalized states from an EQ-algebra E1 to another EQ-algebra E2, which is a generalization of internal states (or state operators) on an EQ-algebra E. Also we give a type of special generalized state from an EQ-algebra E1 to E1, called generalized internal states (or GI-state). Then we give some examples and basic properties of generalized (internal) st...