Finite sums of idempotents and logarithmic residues on connected domains

We show that every finite sum of idempotents in a Banach algebra can be represented as the logarithmic residue of some analytic Banach algebra valued function defined on any, given bounded Cauchy domain. Moreover, using this, we can construct a non-invertible analytic Banach algebra valued function which is defined on any given bounded Cauchy domain and whose logarithmic residue is equal to zero., Consequently, the classical theorem concerning logarithmic residues fails in the general situation for all domains, in particular for connected domains.     

The strong law of large numbers for k-means and best possible nets of Banach valued random variables

Summary Let B be a uniformly convex Banach space, X a B-valued random variable and k a given positive integer number. A random sample of X is substituted by the set of k elements which minimizes a criterion. We found conditions to assure that this set converges a.s., as the sample size increases, to the set of k-elements which minimizes the same criterion for X.     

Fundamental semigroups having a band of idempotents

The construction by Hall of a fundamental orthodox semigroup W _B from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup S _B that plays the role of W _B for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S)=B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice. The Hall semigroup W _B is a subsemigroup of S _B , as are the (weakly) idempotent connected semigroups V _B and U _B . We show how the structure of S _B can be used to extract information relating to arbitrary weakly B-abundant semigroups with (C). This work was carried out during a visit to Lisbon of the second author funded by the London Mathematical Society and while the first author was a member of project POCTI/0143/2003 of CAUL financed by FCT and FEDER.     

Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following.Let p ₁,...,p _k be idempotents in a Banach algebra B, and assume p ₁+...+p _k =0.Does it follow that p _j =0,j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four.     

On (0,1)-matrix semigroups

In this paper we count the number of elements and idempotents in certainD-classes of the semigroup B_n of all binary relations on a set of n elements, namely whose those row and column rank do not exceed 3. This is an announcement of the results; detailed proofs of all results are contained in [2]. The author presented the abstract of this paper in person at Tacoma, Washington, the 676 Meeting of the Amer. Math. Soc., June 20, 1970.     

Central idempotents of the bifree and left-free double Burnside ring

We determine the blocks, i.e., the primitive central idempotents, of the bifree double Burnside ring and the left-free double Burnside ring, as well as the primitive central idempotents of the algebras arising from scalar extension to ?.     

Regular semigroups with skew pairs of idempotents

Let S be a regular semigroup with set of idempotents E(S) . Given x,y ∈ S , we say that (x,y) is a skew pair if x y \notin E(S) whereas y x ∈ E(S) . Here we use this concept to characterise certain regular Rees matrix semigroups.     

The Riccati algebraic equation in a locallyC ⋆-algebra

The stabilizability and complete stabilizability of a pair of elements and the structure of the set of selfadjoint solutions of the algebraic Riccati equation is studied inC ^*-algebras and locallyC ^*-algebras which includes certain algebras of unbounded operators. Under some mild assumptions, selfadjoint solutions are in one-to-one correspondence with a particular set of idempotents.     

Logarithmic residues in Banach algebras

Letf be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivativef′f ^?1 around a Cauchy domainD vanishes. Does it follow thatf takes invertible values on all ofD? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.     

Characterizations and representations of the group inverse involving idempotents

This paper is to present some results on the group invertibility of products and differences of idempotents. In addition, some formulae for the group inverse of sums, differences and products of idempotents are established by using some given idempotents.     

Multilinear function series and transforms in free probability theory

The algebra Mul?B? of formal multilinear function series over an algebra B and its quotient SymMul?B? are introduced, as well as corresponding operations of formal composition. In the setting of Mul?B?, the unsymmetrized R- and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R- and T-transforms (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.     

Linear associative algebras with finitely many idempotents

Let A be a linear (i.e., finite-dimensional) associative algebra with unity defined over K, an algebraically closed field. Then A with respect to its multiplication is an algebraic monoid over k, denoted by A_M, and with respect to the the bracket forms a Lie algebra over K, denoted by A_L. The following theorem is established A_M is nilpotent as an algebraic monoid (equivalentlyA_L is so as a Lie algebra) if and only if the set of idempotents of A is finite if and only if all irreducible closed submonoids of codimension 1 are nilpotent.     

The Drazin inverses of sum and difference of idempotents

In this note, the Drazin inverses of sum and difference of idempotents on a Hilbert space are established under some conditions.     

Ordinal sums and idempotents of copulas

We prove that the ordinal sum of n-copulas is always an n-copula and show that every copula may be represented as an ordinal sum, once the set of its idempotents is known. In particular, it will be shown that every copula can be expressed as the ordinal sum of copulas having only trivial idempotents. As a by-product, we also characterize all associative copulas whose n-ary forms are n-copulas for all n.     

Quotient rings of algebras of functions and operators

Abstract. Various quotient rings of rings B of Banach algebra A-valued continuous functions on a completely regular Hausdorff Space X are constructed in terms of continuous functions defined on dense open subsets of X taking values in the maximal quotient ring of the Banach algebra A. This extends the results proved by N. J. Fine, L., Gillman and J. Lambek (1965) for the case of A, the field of real numbers. The pattern is similar and utilizes as well as generalises the results proved for algebras of multipliers of B by C. A. Akemann, G. K. Pedersen and J. Tomiyama (1973). The techniques combine those from algebra, analysis and topology. The details of the cases when A is the normed division algebra of real quaternions or the operator algebra B(H) of a Hilbert space H are given to illustrate our results. Received February 1, 1999; in final form June 18, 1999 / Published online May 8, 2000     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

Mappings preserving the idempotency of products of operators

We obtain a general form of a surjective (not assumed additive) mapping φ, preserving the nonzero idempotency of a certain product, being defined (a) on the algebra of all bounded linear operators B(X), where X is at least three-dimensional real or complex Banach space, (b) on the set of all rank-one idempotents in B(X) and (c) on the set of all idempotents in B(X). In any of the cases it turns out that φ is additive and either multiplicative or antimultiplicative.     

Exchange rings,units and idempotents

An associative ring R with identity is semiperfect if and only if every element of R is a sum of a unit and an idempotent, and R contains no infinite set of orthogonal idempotents. A ring which contains no infinite set of orthogonal idempotents is an exchange ring if and only if every element is a sum of a unit and an idempo-tent