scholarly journals JB*-Algebras of Topological Stable Rank 1

2007 ◽  
Vol 2007 ◽  
pp. 1-24 ◽  
Author(s):  
Akhlaq A. Siddiqui
Keyword(s):  
Unit Ball ◽  
Von Neumann Algebras ◽  
Invertible Element ◽  
Extreme Points ◽  
Stable Rank ◽  
Von Neumann ◽  
Proper Subclass ◽  
Topological Stable Rank ◽  
Complex Spin ◽  
Invertible Elements

In 1976, Kaplansky introduced the classJB*-algebras which includes allC*-algebras as a proper subclass. The notion of topological stable rank 1 forC*-algebras was originally introduced by M. A. Rieffel and was extensively studied by various authors. In this paper, we extend this notion to generalJB*-algebras. We show that the complex spin factors are of tsr 1 providing an example of specialJBW*-algebras for which the enveloping von Neumann algebras may not be of tsr 1. In the sequel, we prove that every invertible element of aJB*-algebra𝒥is positive in certain isotope of𝒥; if the algebra is finite-dimensional, then it is of tsr 1 and every element of𝒥is positive in some unitary isotope of𝒥. Further, it is established that extreme points of the unit ball sufficiently close to invertible elements in aJB*-algebra must be unitaries and that in anyJB*-algebras of tsr 1, all extreme points of the unit ball are unitaries. In the end, we prove the coincidence between theλ-function andλu-function on invertibles in aJB*-algebra.

scholarly journals On the Geometry of the Unit Ball of aJB*-Triple

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Haifa M. Tahlawi ◽  
Akhlaq A. Siddiqui ◽  
Fatmah B. Jamjoom
Keyword(s):  
Unit Ball ◽  
Structural Properties ◽  
Extreme Points ◽  
Von Neumann ◽  
Regular Elements ◽  
Von Neumann Regular ◽  
Invertible Elements

We explore aJB*-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting ofC*-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in aJB*-triple; this indicates their structural richness. We initiate a study of the unit ball of aJB*-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. SomeC*-algebra andJB*-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended toJB*-triples.

On Completely Positive Maps Defined by an Irreducible Correspondence

1990 ◽  
Vol 33 (4) ◽  
pp. 434-441 ◽  
Author(s):  
C. Anantharaman-Delaroche
Keyword(s):  
Von Neumann Algebras ◽  
Convex Set ◽  
Extreme Points ◽  
Completely Positive Maps ◽  
Complex Matrices ◽  
Completely Positive ◽  
Von Neumann ◽  
Positive Maps

AbstractCompletely positive maps defined by an irreducible correspondence between two von Neumann algebras M and N are introduced. We give results about their structure and characterize, among them, those which are extreme points in the convex set of all unital completely positive maps from M to N. As particular cases we obtain known results of M. D. Choi [4] on completely positive maps between complex matrices and of J. A. Mingo [8] on inner completely positive maps.

scholarly journals Linear Maps on Factors which Preserve the Extreme Points of the Unit Ball

1998 ◽  
Vol 41 (4) ◽  
pp. 434-441 ◽  
Author(s):  
Vania Mascioni ◽  
Lajos Molnár
Keyword(s):  
Unit Ball ◽  
Unitary Operator ◽  
Extreme Points ◽  
Linear Preserver ◽  
Von Neumann ◽  
Linear Maps

AbstractThe aim of this paper is to characterize those linear maps from a von Neumann factor A into itself which preserve the extreme points of the unit ball of A. For example, we show that if A is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital *-homomorphism or a unital *-antihomomorphism.

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.

scholarly journals Some extremal properties in the unit ball of von Neumann algebras

1969 ◽  
Vol 21 (2) ◽  
pp. 175-181 ◽  
Author(s):  
Hisashi Choda ◽  
Yōichi Kijima ◽  
Yoshiomi Nakagami
Keyword(s):  
Unit Ball ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Extremal Properties

scholarly journals Commutativity Preserving Mappings of Von Neumann Algebras

1993 ◽  
Vol 45 (4) ◽  
pp. 695-708 ◽  
Author(s):  
Matej Brešar ◽  
C. Robert Miers
Keyword(s):  
Von Neumann Algebras ◽  
Invertible Element ◽  
Von Neumann ◽  
Additive Map ◽  
Jordan Isomorphism

AbstractA map θ: M —> N where M and N are rings is said to preserve commutativity in both directions if the elements a,b ∊ M commute if and only if θ(a) and θ(b) commute. In this paper we show that if M and N are von Neumann algebras with no central summands of type I1 or I2 and θ is a bijective additive map which preserves commutativity in both directions then θ(x) = cφ(x) +f(x) where c is an invertible element in ZN, the center of N, φ M —> N is a Jordan isomorphism of M onto N, and f is an additive map of M into ZN.

scholarly journals EXTREME POINT METHODS IN THE STUDY OF ISOMETRIES ON CERTAIN NONCOMMUTATIVE SPACES

2021 ◽  
pp. 1-22
Author(s):  
PIERRE DE JAGER ◽  
JURIE CONRADIE
Keyword(s):  
Extreme Point ◽  
Von Neumann Algebras ◽  
Extreme Points ◽  
Lorentz Spaces ◽  
Von Neumann ◽  
Noncommutative Spaces ◽  
Finite Von Neumann Algebras

Abstract In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$ , as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$ . The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.

The Approximate Jordan-Hahn Decomposition

1989 ◽  
Vol 41 (6) ◽  
pp. 1124-1146 ◽  
Author(s):  
Gottfried T. Rüttimann
Keyword(s):  
Banach Space ◽  
Von Neumann Algebras ◽  
Convex Set ◽  
Measure Theory ◽  
Extreme Points ◽  
Foundations Of Quantum Mechanics ◽  
The Other ◽  
Orthomodular Poset ◽  
Geometrical Aspect ◽  
Von Neumann

Non-commutative measure theory embraces measure theory on cr-fields of subsets of a set, on projection lattices of von Neumann algebras or JBW-algebras and on hypergraphs alike [20], [27], [33], [37], [39], [40], [41]. Due to the unifying structure of an orthoalgebra concepts can easily be transferred from one branch to the other. Additional conceptual inpetus is obtained from the logico-probabilistic foundations of quantum mechanics (see [6], [19], [21]).In the late seventies the author studied the Jordan-Hahn decomposition of measures on orthomodular posets and certain graphs. These investigations revealed an interesting geometrical aspect of this decomposition in that the Jordan-Hahn property of the convex set of probability charges on a finite orthomodular poset can be characterized in terms of the extreme points of the unit ball of the Banach space dual of the base normed space of Jordan charges.

scholarly journals Linear maps on von Neumann algebras preserving zero products on tr-rank

2002 ◽  
Vol 65 (1) ◽  
pp. 79-91 ◽  
Author(s):  
Cui Jianlian ◽  
Hou Jinchuan
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Subspace ◽  
Invertible Element ◽  
Bounded Linear ◽  
Von Neumann ◽  
Linear Map ◽  
Linear Maps ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras

In this paper, we give some characterisations of homomorphisms on von Neumann algebras by linear preservers. We prove that a bounded linear surjective map from a von Neumann algebra onto another is zero-product preserving if and only if it is a homomorphism multiplied by an invertible element in the centre of the image algebra. By introducing the notion of tr-rank of the elements in finite von Neumann algebras, we show that a unital linear map from a linear subspace ℳ of a finite von Neumann algebra ℛ into ℛ can be extended to an algebraic homomorphism from the subalgebra generated by ℳ into ℛ; and a unital self-adjoint linear map from a finite von Neumann algebra onto itself is completely tr-rank preserving if and only if it is a spatial *-automorphism.

scholarly journals Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$

2011 ◽  
Vol 109 (1) ◽  
pp. 114 ◽  
Author(s):  
Raymond Mortini ◽  
Rudolf Rupp
Keyword(s):  
Holomorphic Functions ◽  
Continuous Functions ◽  
Stable Rank ◽  
The Real ◽  
Real Algebra ◽  
Topological Stable Rank ◽  
Invertible Elements ◽  
Complex Valued ◽  
Compact Planar

We determine the generalized $E$-stable ranks for the real algebra, $C(\boldsymbol{D})_{\mathrm{sym}}$, of all complex valued continuous functions on the closed unit disk, symmetric to the real axis, and its subalgebra $A(\boldsymbol{D})_{\mathsf R}$ of holomorphic functions. A characterization of those invertible functions in $C(E)$ is given that can be uniformly approximated on $E$ by invertibles in $A(\boldsymbol {D})_{\mathsf R}$. Finally, we compute the Bass and topological stable rank of $C(K)_{\mathrm{sym}}$ for real symmetric compact planar sets $K$.

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