Power commuting traces of bilinear maps on invertible elements

2020 ◽  
pp. 2150023
Author(s):  
Willian Franca ◽  
Nelson Louza
Keyword(s):  
Bilinear Maps ◽  
Simple Ring ◽  
Linear Maps ◽  
Invertible Elements ◽  
Unital Simple ◽  
Unit Formula

Let [Formula: see text] be a unital simple ring. Under a mild technical restriction on [Formula: see text], we characterize bilinear mappings [Formula: see text] satisfying [Formula: see text], and [Formula: see text] for all unit [Formula: see text] and [Formula: see text], where [Formula: see text]. As an application we describe bijective linear maps [Formula: see text] satisfying [Formula: see text] for all invertible [Formula: see text]. Precisely, we will show that [Formula: see text] is an isomorphism.

scholarly journals Linear maps on *-algebras acting on orthogonal elements like derivations or anti-derivations

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4543-4554 ◽  
Author(s):  
H. Ghahramani ◽  
Z. Pan
Keyword(s):  
Linear Map ◽  
Unital Algebra ◽  
C Algebras ◽  
Orthogonality Conditions ◽  
Linear Maps ◽  
Unital Simple

Let U be a unital *-algebra and ? : U ? U be a linear map behaving like a derivation or an anti-derivation at the following orthogonality conditions on elements of U: xy = 0, xy* = 0, xy = yx = 0 and xy* = y*x = 0. We characterize the map ? when U is a zero product determined algebra. Special characterizations are obtained when our results are applied to properly infinite W*-algebras and unital simple C*-algebras with a non-trivial idempotent.

scholarly journals Extremally rich C*-crossed products and the cancellation property

1998 ◽  
Vol 64 (3) ◽  
pp. 285-301 ◽  
Author(s):  
Ja A Jeong ◽  
Hiroyuki Osaka
Keyword(s):  
Finite Abelian Group ◽  
Extreme Points ◽  
Stable Rank ◽  
Crossed Product ◽  
Closed Unit Ball ◽  
Crossed Products ◽  
Cancellation Property ◽  
Rank One ◽  
Invertible Elements ◽  
Unital Simple

AbstractA unital C*-algebra A is called extremally rich if the set of quasi-invertible elements A-1 ex (A)A-1 (= A-1q) is dense in A, where ex(A) is the set of extreme points in the closed unit ball A1 of A. In [7, 8] Brown and Pedersen introduced this notion and showed that A is extremally rich if and only if conv(ex(A)) = A1. Any unital simple C*-algebra with extremal richness is either purely infinite or has stable rank one (sr(A) = 1). In this note we investigate the extremal richness of C*-crossed products of extremally rich C*-algebras by finite groups. It is shown that if A is purely infinite simple and unital then A xα, G is extremally rich for any finite group G. But this is not true in general when G is an infinite discrete group. If A is simple with sr(A) =, and has the SP-property, then it is shown that any crossed product A xαG by a finite abelian group G has cancellation. Moreover if this crossed product has real rank zero, it has stable rank one and hence is extremally rich.

Special clean elements in rings

2019 ◽  
Vol 19 (11) ◽  
pp. 2050208
Author(s):  
Dinesh Khurana ◽  
T. Y. Lam ◽  
Pace P. Nielsen ◽  
Janez Šter
Keyword(s):  
Fixed Point ◽  
Fixed Point Set ◽  
Spectral Decompositions ◽  
Decomposition Formula ◽  
Point Set ◽  
Invertible Elements ◽  
Unit Formula

A clean decomposition [Formula: see text] in a ring [Formula: see text] (with idempotent [Formula: see text] and unit [Formula: see text]) is said to be special if [Formula: see text]. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott–Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution.

scholarly journals Strict Positivity of Operators and Inflated Schur Products

2018 ◽  
Vol 10 (6) ◽  
pp. 30
Author(s):  
Ching-Yun Suen
Keyword(s):  
Main Diagonal ◽  
Completely Positive ◽  
Strict Positivity ◽  
Linear Maps ◽  
Positive Matrices ◽  
Positive Linear Maps ◽  
Invertible Elements ◽  
Equivalent Conditions

In this paper we provide a characterization of strictly positive matrices of operators and a factorization of their inverses. Consequently, we provide a test of strict positivity of matrices in . We give equivalent conditions for the inequality . We prove a theorem involving inflated Schur products [4, P. 153] of positive matrices of operators with invertible elements in the main diagonal which extends the results [3, P. 479, Theorem 7.5.3 (b), (c)]. We also discuss strictly completely positive linear maps in the paper.

On the Topologically Invertible Elements of a Topological Algebra

2007 ◽  
Vol 107 (1) ◽  
pp. 73-80
Author(s):  
Hugo Arizmendi-Peimbert ◽  
Angel Carrillo-Hoyo
Keyword(s):  
Topological Algebra ◽  
Invertible Elements

scholarly journals An Algebraic Brascamp–Lieb Inequality

2021 ◽  
Author(s):  
Jennifer Duncan
Keyword(s):  
General Class ◽  
Recent Progress ◽  
Algebraic Groups ◽  
Hölder’S Inequality ◽  
Duke Math ◽  
Rational Maps ◽  
Affine Invariant ◽  
Linear Maps ◽  
Nonlinear Maps ◽  
Funct Anal

AbstractThe Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

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