MONOTONE OPERATOR FUNCTIONS ON C*-ALGEBRAS

2005 ◽  
Vol 16 (02) ◽  
pp. 181-196 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
SERGEI SILVESTROV ◽  
JUN TOMIYAMA
Keyword(s):  
Classical Case ◽  
Linear Operators ◽  
Monotone Functions ◽  
Bounded Linear ◽  
C Algebra ◽  
C Algebras ◽  
Operator Monotone ◽  
Operator Functions ◽  
Operator Monotone Functions

The article is devoted to investigation of classes of functions monotone as functions on general C*-algebras that are not necessarily the C*-algebra of all bounded linear operators on a Hilbert space as in classical case of matrix and operator monotone functions. We show that for general C*-algebras the classes of monotone functions coincide with the standard classes of matrix and operator monotone functions. For every class we give exact characterization of C*-algebras with this class of monotone functions, providing at the same time a monotonicity characterization of subhomogeneous C*-algebras. We use this result to generalize characterizations of commutativity of a C*-algebra based on monotonicity conditions for a single function to characterizations of subhomogeneity. As a C*-algebraic counterpart of standard matrix and operator monotone scaling, we investigate, by means of projective C*-algebras and relation lifting, the existence of C*-subalgebras of a given monotonicity class.

scholarly journals AN INTERNAL CHARACTERIZATION OF COMPLETE POSITIVITY FOR ELEMENTARY OPERATORS

2002 ◽  
Vol 45 (2) ◽  
pp. 285-300
Author(s):  
Richard M. Timoney
Keyword(s):  
Linear Operators ◽  
Type I ◽  
Complete Positivity ◽  
Elementary Operator ◽  
Bounded Linear ◽  
C Algebra ◽  
C Algebras ◽  
Metric Operator ◽  
Elementary Operators

AbstractComplete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

scholarly journals A Note on Real Operator Monotone Functions

2020 ◽  
Author(s):  
Marcell Gaál ◽  
Miklós Pálfia
Keyword(s):  
Hilbert Space ◽  
Functional Calculus ◽  
Monotone Functions ◽  
Open Convex ◽  
Completely Positive ◽  
Bounded Linear ◽  
Operator Monotone ◽  
Hilbert Space Operators ◽  
Operator Monotone Functions

Abstract In this paper, we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or Hermitian parts) of bounded linear Hilbert space operators. We completely characterize such functions on open convex free domains in terms of ordinary operator monotone free functions on self-adjoint domains. Further assuming the more stringent free holomorphicity, we prove that all such functions are affine linear with completely positive nonconstant part. This problem has been proposed by David Blecher at the biannual OTOA conference held in Bangalore in December 2016.

ORTHOGONALITY PRESERVERS REVISITED

2009 ◽  
Vol 02 (03) ◽  
pp. 387-405 ◽  
Author(s):  
María Burgos ◽  
Francisco J. Fernández-Polo ◽  
Jorge J. Garcés ◽  
Antonio M. Peralta
Keyword(s):  
Linear Operator ◽  
Natural Product ◽  
Bounded Linear Operator ◽  
Complete Characterization ◽  
Bounded Linear ◽  
C Algebra ◽  
Jordan Homomorphism ◽  
C Algebras ◽  
Orthogonality Preservers

We obtain a complete characterization of all orthogonality preserving operators from a JB *-algebra to a JB *-triple. If T : J → E is a bounded linear operator from a JB *-algebra (respectively, a C *-algebra) to a JB *-triple and h denotes the element T**(1), then T is orthogonality preserving, if and only if, T preserves zero-triple-products, if and only if, there exists a Jordan *-homomorphism [Formula: see text] such that S(x) and h operator commute and T(x) = h•r(h) S(x), for every x ∈ J, where r(h) is the range tripotent of h, [Formula: see text] is the Peirce-2 subspace associated to r(h) and •r(h) is the natural product making [Formula: see text] a JB *-algebra. This characterization culminates the description of all orthogonality preserving operators between C *-algebras and JB *-algebras and generalizes all the previously known results in this line of study.

scholarly journals 2-LOCAL ISOMETRIES OF SOME OPERATOR ALGEBRAS

2002 ◽  
Vol 45 (2) ◽  
pp. 349-352 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Operator Algebras ◽  
Linear Operators ◽  
Primary 47B49 ◽  
Similar Statement ◽  
Bounded Linear ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Local Isometry

AbstractAs a consequence of the main result of the paper we obtain that every 2-local isometry of the $C^*$-algebra $B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ is an isometry. We have a similar statement concerning the isometries of any extension of the algebra of all compact operators by a separable commutative $C^*$-algebra. Therefore, on those $C^*$-algebras the isometries are completely determined by their local actions on the two-point subsets of the underlying algebras.AMS 2000 Mathematics subject classification: Primary 47B49

scholarly journals Kwong matrices and operator monotone functions on $(0,1)$

2014 ◽  
Vol 5 (1) ◽  
pp. 121-127 ◽  
Author(s):  
Juri Morishita ◽  
Takashi Sano ◽  
Shintaro Tachibana
Keyword(s):  
Monotone Functions ◽  
Operator Monotone ◽  
Operator Monotone Functions
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