The Experts below are selected from a list of 315 Experts worldwide ranked by ideXlab platform
Bernd Silbermann - One of the best experts on this subject based on the ideXlab platform.
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OnC*-Algebras Generated by Idempotents
Journal of Functional Analysis, 1996Co-Authors: Naum Krupnik, Steffen Roch, Bernd SilbermannAbstract:AbstractThe topic of the present paper is concrete Banach andC*-algebras which are generated by a finite number of Idempotents. Our first result is that, for each finitely generated Banach algebra A, there is a numbern0so that the algebra An×nof alln×nmatrices with entries in A is generated by three Idempotents whenevern⩾n0, and that An×nis generated by two Idempotents if and only ifn=2 and if A is singly generated. As an application we find that the algebraCn×n(K) of all continuous Cn×n-matrix-valued functions on a compactK⊂C with connected complement but without interior points, is generated by 2 or 3 Idempotents in casen=2 orn>2, respectively. This result is used to construct examples ofC*-algebras which are generated by 2 Idempotents but not 2 projections. For these algebras, the standard 2×2 matrix symbol fails to be symmetric. We finally show that eachC*-algebra satisfying a polynomial identity (in particular, eachC*-algebra generated by two Idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one
Naum Krupnik - One of the best experts on this subject based on the ideXlab platform.
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OnC*-Algebras Generated by Idempotents
Journal of Functional Analysis, 1996Co-Authors: Naum Krupnik, Steffen Roch, Bernd SilbermannAbstract:AbstractThe topic of the present paper is concrete Banach andC*-algebras which are generated by a finite number of Idempotents. Our first result is that, for each finitely generated Banach algebra A, there is a numbern0so that the algebra An×nof alln×nmatrices with entries in A is generated by three Idempotents whenevern⩾n0, and that An×nis generated by two Idempotents if and only ifn=2 and if A is singly generated. As an application we find that the algebraCn×n(K) of all continuous Cn×n-matrix-valued functions on a compactK⊂C with connected complement but without interior points, is generated by 2 or 3 Idempotents in casen=2 orn>2, respectively. This result is used to construct examples ofC*-algebras which are generated by 2 Idempotents but not 2 projections. For these algebras, the standard 2×2 matrix symbol fails to be symmetric. We finally show that eachC*-algebra satisfying a polynomial identity (in particular, eachC*-algebra generated by two Idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one
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 & Multilinear Algebra, 2014Co-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 m...
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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.
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.
Steffen Roch - One of the best experts on this subject based on the ideXlab platform.
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OnC*-Algebras Generated by Idempotents
Journal of Functional Analysis, 1996Co-Authors: Naum Krupnik, Steffen Roch, Bernd SilbermannAbstract:AbstractThe topic of the present paper is concrete Banach andC*-algebras which are generated by a finite number of Idempotents. Our first result is that, for each finitely generated Banach algebra A, there is a numbern0so that the algebra An×nof alln×nmatrices with entries in A is generated by three Idempotents whenevern⩾n0, and that An×nis generated by two Idempotents if and only ifn=2 and if A is singly generated. As an application we find that the algebraCn×n(K) of all continuous Cn×n-matrix-valued functions on a compactK⊂C with connected complement but without interior points, is generated by 2 or 3 Idempotents in casen=2 orn>2, respectively. This result is used to construct examples ofC*-algebras which are generated by 2 Idempotents but not 2 projections. For these algebras, the standard 2×2 matrix symbol fails to be symmetric. We finally show that eachC*-algebra satisfying a polynomial identity (in particular, eachC*-algebra generated by two Idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one
