On certain finite-dimensional algebras generated by two idempotents

This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)^m=(QP)^m and (PQ)^m-1≠(QP)^m-1. The main result is the classification of all these algebras, implying that for each m?2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given.     

Algebras generated by two idempotents

It is shown that a noncommutative simple algebra generated over a field F by two idempotents is necessarily the ring of 2×2 matrices over a simple extension of F, and that every matrix ring over a field K can be generated over K by three idempotents.     

On a function algebra which is not generated by its idempotents

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 3–7, January, 1992.     

On the identities in algebras generated by linearly connected idempotents

We investigate the problem of the existence of polynomial identities (PI) in algebras generated by idempotents whose linear combination is equal to identity. In the case where the number of idempotents is greater than or equal to five, we prove that these algebras are not PI-algebras. In the case of four idempotents, in order that an algebra be a PI-algebra, it is necessary and sufficient that the sum of the coefficients of the linear combination be equal to two. In this case, these algebras are F₄-algebras.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 782–795, June, 2004.     

On Coxeter functors for some classes of algebras generated by idempotents

For an algebra generated by linearly connected idempotents, we describe the construction of Coxeter functors on the category of representations of the algebra in the category of Banach spaces.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 996–1001, July, 2004.     

Left selfdistributive rings generated by idempotents

Left selfdistributive rings (i.e., xyz = xyxz) which are semidirect sums of boolean rings and rings nilpotent of index at most 3 are studied.     

On group topologies and idempotents in weak almost periodic compactifications

Communicated by Karl H. Hofmann     

Symmetrization of the symbol in Banach algebras generated by idempotents

It is known [2] that a Banach algebra generated by 2N idempotents with relations (1), (2) is an algebra with matrix symbol of order 2N. This symbol is completely and explicitly defined via the spectrums of two (indicator) elements (3) and (5). But in the case when is a c^*-algebra, the symbol constructed in [2] is notsymmetric. Moreover in order to construct asymmetric symbol, one needs some additional information about the algebra , even for the case whenp _j=p _j ^* (j=1, ..., 2N). These additional conditions and the explicit form ofsymmetric symbol are described in this paper.     

Invertibility symbol for a Banach algebra generated by two idempotents and a shift

A Banach algebra generated by two idempotentsp, r, identitye and a shiftv which satisfy the conditionspv=vp andrv=v(e–r) is investigated. It is proved that all irreducible representations of algebraA are two- and four-dimensional. The explicit form of these representations is obtained. An Invertibility Symbol is constructed. Some examples are considered.     

Algebras generated by idempotents and the symbol calculus for singular integral operators

It is proved that in Banach algebras generated by two idempotents and, perhaps, by a certain flip operator the standard identity F₄ is fulfilled. The maximal ideal space of such algebras is determined and the corresponding symbol is given. By means of local techniques these results are applied to obtain a symbol calculus for singular integral operators with Carleman shift (changing the orientation) in weighted Banach spaces.     

On the semiring of cone preserving maps

If K is a proper cone in Rⁿ, then the cone of all linear operators that preserve K, denoted by π(K), forms a semiring under usual operator addition and multiplication. Recently J.G. Horne examined the ideals of this semiring. He proved that if K₁, K₂ are polyhedral cones such that π(K₁) and π(K₂) are isomorphic as semirings, then K₁ and K₂ are linearly isomorphic. The study of this semiring is continued in this paper. In Sec. 3 ideals of π(K) which are also faces are characterized. In Sec. 4 it is shown that π(K) has a unique minimal two-sided ideal, namely, the dual cone of π(K¹), where K¹ is the dual cone of K. Extending Horne's result, it is also proved that the cone K is characterized by this unique minimal two-sided ideal of π(K). The set of all faces of π(K) inherits a quotient semiring structure from π(K). Properties of this face-semiring are given in Sec. 5. In particular, it is proved that this face-semiring admits no nontrivial congruence relation iff the duality operator of π(K) is injective. In Sec. 6 the maximal one-sided and two-sided ideals of π(K) are identified. In Sec. 8 it is shown that π(K) never satisfies the ascending-chain condition on principal one-sided ideals. Some partial results on the question of topological closedness of principal one-sided ideals of π(K) are also given.     

On idempotents of a class of commutative rings

Abstract

In the present work, a procedure for determining idempotents of a commutative ring having a sequence of ideals with certain properties is presented. As an application of this procedure, idempotent elements of various commutative rings are determined. Several examples are included illustrating the main results.     

On the primitive idempotents of distance-regular graphs

Let Γ denote a distance-regular graph with diameter d3. Let E, F denote nontrivial primitive idempotents of Γ such that F corresponds to the second largest or the least eigenvalue. We investigate the situation that there exists a primitive idempotent H of Γ such that EF is a linear combination of F and H. Our main purpose is to obtain the inequalities involving the cosines of E, and to show that equality is closely related to Γ being Q-polynomial with respect to E. This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs.