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André Leroy - One of the best experts on this subject based on the ideXlab platform.
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Decompositions into products of Idempotents
The Electronic Journal of Linear Algebra, 2015Co-Authors: Adel Alahmadi, S. K. Jain, André Leroy, A. SathayeAbstract:The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (â) that each right (left) singular element is a product of Idempotents, and (2) to consider the question: âwhen is a singular nonnegative square matrix a product of nonnegative Idempotent matrices?â The importance of the class of quasi- Euclidean rings in connection with the property (â) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154â170, 2014], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of Idempotents, generalizing the results of J. A Erdos [Glasgow Mathematical Journal, 8: 118â122, 1967] for matrices over fields and that of T. J. Laffey [Linear and Multilinear Algebra, 14:309â314, 1983] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of Idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative Idempotent matrices. We study as to when each singular matrix is a product of Idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two Idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative Idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of Idempotent matrices.
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Decomposition of Singular Matrices into Idempotents
Linear and Multilinear Algebra, 2013Co-Authors: Adel Alahmadi, S. K. Jain, André LeroyAbstract:In this paper we provide concrete constructions of Idempotents to represent typical singular matrices over a given ring as a product of Idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of Idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of Idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over B\'ezout domains as to when such a matrix is a product of Idempotent matrices.
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Decomposition of Singular Matrices into Idempotents
Linear and Multilinear Algebra, 2013Co-Authors: Adel Alahmadi, Surender Jain, André LeroyAbstract:In this paper we provide concrete constructions of Idempotents to represent typical singular matrices over a given ring as a product of Idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey ( Products of Idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of Idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over Bézout domains as to when such a matrix is a product of Idempotent matrices.
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Idempotents in ring extensions
Journal of Algebra, 2013Co-Authors: Pramod Kanwar, André Leroy, Jerzy MatczukAbstract:Abstract The aim of the paper is to study Idempotents of ring extensions R ⊆ S where S stands for one of the rings R [ x 1 , x 2 , … , x n ] , R [ x 1 ± 1 , x 2 ± 1 , … , x n ± 1 ] , R 〚 x 1 , x 2 , … , x n 〛 . We give criteria for an Idempotent of S to be conjugate to an Idempotent of R. Using our criteria we show, in particular, that Idempotents of the power series ring are conjugate to Idempotents of the base ring and we apply this to give a new proof of the result of P.M. Cohn (2003) [4, Theorem 7] that the ring of power series over a projective-free ring is also projective-free. We also get a short proof of the more general fact that if the quotient ring R / J of a ring R by its Jacobson radical J is projective-free then so is the ring R.
David R. Pitts - One of the best experts on this subject based on the ideXlab platform.
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Idempotents in nest algebras
Journal of Functional Analysis, 1991Co-Authors: David R. Larson, David R. PittsAbstract:Abstract The algebraic equivalence and similarity classes of Idempotents within a nest algebra Alg β are completely characterized. We obtain necessary and sufficient conditions for two Idempotents to be equivalent or similar. Our criterion yields examples illustrating pathology and also shows that to each equivalence class of Idempotents there corresponds a “dimension function” from β × β into N ∪ { ∞ }. We complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of Idempotents. Also, to each sequence of dimension functions, there corresponds a commuting sequence of Idempotents. A criterion is obtained for when an Idempotent is similar to a subIdempotent of another. The mapping which sends an equivalence class (or Idempotent) to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a Type I W ∗ -algebra to its center valued trace. We prove that almost commuting, similar Idempotents are homotopic; this contrasts with the situation in certain C ∗ -algebras. Using this, we show that similar, simultaneously diagonalizable Idempotents are homotopic, and in the continuous nest case, every diagonal Idempotent is homotopic to a core projection.
James East - One of the best experts on this subject based on the ideXlab platform.
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Idempotent generation in the endomorphism monoid of a uniform partition
Communications in Algebra, 2016Co-Authors: Igor Dolinka, James EastAbstract:Denote by 𝒯n and 𝒮n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ℰn = {1} ∪ (𝒯n∖𝒮n). Let 𝒯(X, 𝒫) denote the monoid of all transformations of the finite set X preserving a uniform partition 𝒫 of X into m subsets of size n, where m, n ≥ 2. We enumerate the Idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ⟨ E ⟩ generated by the Idempotents E = E(𝒯(X, 𝒫)). We show that S = S1 ∪ S2, where S1 is a direct product of m copies of ℰn, and S2 is a wreath product of 𝒯n with 𝒯m∖𝒮m. We calculate the rank and Idempotent rank of S, showing that these are equal, and we also classify and enumerate all the Idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary Idempotent generating sets of ℰn.
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Enumeration of Idempotents in diagram semigroups and algebras
Journal of Combinatorial Theory Series A, 2015Co-Authors: Igor Dolinka, James East, Athanasios Evangelou, Dg Fitzgerald, Nicholas Ham, James Hyde, Nicholas LoughlinAbstract:We give a characterisation of the Idempotents of the partition monoid, and use this to enumerate the Idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of Idempotents in each monoid as well as various R -, L - and D -classes. We also apply our results to determine the number of Idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
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Idempotent generation in the endomorphism monoid of a uniform partition
arXiv: Group Theory, 2014Co-Authors: Igor Dolinka, James EastAbstract:Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal P)$ denote the set of all transformations of the finite set $X$ preserving a uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where $m,n\geq2$. We enumerate the Idempotents of $\mathcal T(X,\mathcal P)$, and describe the subsemigroup $S=\langle E\rangle$ generated by the Idempotents $E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1$ is a direct product of $m$ copies of $\mathcal E_n$, and $S_2$ is a wreath product of $\mathcal T_n$ with $\mathcal T_m\setminus \mathcal S_m$. We calculate the rank and Idempotent rank of $S$, showing that these are equal, and we also classify and enumerate all the Idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary Idempotent generating sets of $\mathcal E_n$.
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the semigroup generated by the Idempotents of a partition monoid
Journal of Algebra, 2012Co-Authors: James East, Dg FitzgeraldAbstract:We study the Idempotent generated subsemigroup of the partition monoid. In the finite case this subsemigroup consists of the identity and all the singular partitions. In the infinite case, the subsemigroup is described in terms of certain parameters that measure how far a partition is from being a permutation. As one of several corollaries, we deduce Howieʼs description from 1966 of the semigroup generated by the Idempotents of a full transformation semigroup.
Julien Giol - One of the best experts on this subject based on the ideXlab platform.
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From a formula of Kovarik to the parametrization of Idempotents in Banach algebra.
Illinois Journal of Mathematics, 2007Co-Authors: Julien GiolAbstract:If p,q are Idempotents in a Banach algebra A and if p+q-1 is invertible, then the Kovarik formula provides an Idempotent k(p,q) such that pA=k(p,q)A and Aq=Ak(p,q). We study the existence of such an element in a more general situation. We first show that p+q-1 is invertible if and only if k(p,q) and k(q,p) both exist. Then we deduce a local parametrization of the set of Idempotents from this equivalence. Finally, we consider a polynomial parametrization first introduced by Holmes and we answer a question raised at the end of his paper.
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Segments of bounded Idempotents on a Hilbert space.
Journal of Functional Analysis, 2005Co-Authors: Julien GiolAbstract:Let H be a separable Hilbert space. We prove that any two homotopic Idempotents in the algebra may be connected by a piecewise affine Idempotent-valued path consisting of 4 segments at most. Moreover, we show that this constant is optimal provided H has infinite dimension. We also explain how this result is linked to the problem of finding common complements for two closed subspaces of H.
A. M. Bikchentaev - One of the best experts on this subject based on the ideXlab platform.
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trace and integrable operators affiliated with a semifinite von neumann algebra
Doklady Mathematics, 2016Co-Authors: A. M. BikchentaevAbstract:New properties of the space of integrable (with respect to the faithful normal semifinite trace) operators affiliated with a semifinite von Neumann algebra are found. A trace inequality for a pair of projections in the von Neumann algebra is obtained, which characterizes trace in the class of all positive normal functionals on this algebra. A new property of a measurable Idempotent are determined. A useful factorization of such an operator is obtained; it is used to prove the nonnegativity of the trace of an integrable Idempotent. It is shown that if the difference of two measurable Idempotents is a positive operator, then this difference is a projection. It is proved that a semihyponormal measurable Idempotent is a projection. It is also shown that a hyponormal measurable tripotent is the difference of two orthogonal projections.
