scholarly journals CONVEX MULTIVARIABLE TRACE FUNCTIONS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 631-648 ◽  
Author(s):  
ELLIOTT H. LIEB ◽  
GERT K. PEDERSEN
Keyword(s):  
Convex Function ◽  
Short Proof ◽  
Trace Function ◽  
C Algebra ◽  
Operator Mean ◽  
Lower Semi ◽  
Continuous Trace ◽  
Densely Defined

For any densely defined, lower semi-continuous trace τ on a C*-algebra A with mutually commuting C*-subalgebras A1, A2, … An, and a convex function f of n variables, we give a short proof of the fact that the function (x1, x2, …, xn)→ τ (f (x1, x2, …, xn)) is convex on the space [Formula: see text]. If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called ℓ-convexity, show how it applies to traces, and give some examples. In particular we show that the Kadison–Fuglede determinant is concave and that the trace of an operator mean is always dominated by the corresponding mean of the trace values.

scholarly journals Jensen's Trace Inequality in Several Variables

2003 ◽  
Vol 14 (06) ◽  
pp. 667-681 ◽  
Author(s):  
Frank Hansen ◽  
Gert K. Pedersen
Keyword(s):  
Symmetric Matrices ◽  
Trace Inequality ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Several Variables ◽  
Lower Semi ◽  
Continuous Trace ◽  
Abelian Fields ◽  
Densely Defined

For a convex, real function f, we present a simple proof of the formula [Formula: see text] valid for each tuple (x1,…, xm) of symmetric matrices in [Formula: see text] and every unital column (a1,…, am) of matrices, i.e. [Formula: see text]. This is the standard Jensen trace ine-quality. If f ≥ 0 it holds also for the unbounded trace on [Formula: see text], where [Formula: see text] is an infinite-dimensional Hilbert space. We then investigate the more general case where τ is a densely defined, lower semi-continuous trace on a C*-algebra [Formula: see text] and f is a convex, continuous function of n variables, and show that we have the inequality [Formula: see text] for every family of abeliann-tuples [Formula: see text], i.e. tuples of self-adjoint elements in [Formula: see text] such that [xik, xjk] = 0 for all i, j and k, where 1 ≤ k ≤ m, and every unital m-column (a1,…, am) in [Formula: see text], provided that the elements [Formula: see text] also form an abelian n-tuple. We even establish this result for weak* measurable, self-adjoint, abelian fields (xit)t∈T, 1 ≤ i ≤ n, i.e. [xit, xjt] = 0 for all i, j and t, and weak* measurable, unital column field (at)t∈T in [Formula: see text] paired with any trace or trace-like functional φ, i.e. one that contains the n-tuple (presumed abelian) with elements [Formula: see text] in its centralizer. This takes the form of the inequality [Formula: see text] We also study functions of n variables that are monotone increasing in each variable, and show in two important cases that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are abelian n-tuples with xi ≤ yi for each i and φ is a trace or a trace-like functional.

CONTINUOUS–TRACE C*-ALGEBRAS NOT ISOMORPHIC TO THEIR OPPOSITE ALGEBRAS

2001 ◽  
Vol 12 (03) ◽  
pp. 263-275 ◽  
Author(s):  
N. CHRISTOPHER PHILLIPS
Keyword(s):  
Homotopy Equivalent ◽  
C Algebra ◽  
C Algebras ◽  
Continuous Trace

We give examples of locally trivial continuous-trace C *-algebra not isomorphic to their opposite algebras. Our examples include a unital C *-algebra which is both stably isomorphic to and homotopy equivalent to its opposite algebra, a unital C *-algebra which is homotopy equivalent to but not stably isomorphic to its opposite algebra, and a unital C *-algebra which is not even stably homotopy equivalent to its opposite algebra.

A CONSTRUCTION FOR WEIGHTS ON C*-ALGEBRAS: DUAL WEIGHTS FOR C*-CROSSED PRODUCTS

1999 ◽  
Vol 10 (01) ◽  
pp. 129-157 ◽  
Author(s):  
J. QUAEGEBEUR ◽  
J. VERDING
Keyword(s):  
Dynamical System ◽  
Haar Measure ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebras ◽  
System A ◽  
Lower Semi ◽  
Continuous Weight ◽  
Densely Defined

A method for constructing densely defined lower semi-continuous weights on C*-algebras is presented. The method can be used to construct a "dual weight" on the C*-crossed product A×αG of a C*-dynamical system (A,G,α), starting from a relatively α-invariant densely defined lower semi-continuous weight on A. As an application we show that the Haar measure on the quantum E(2) group is a C*-dual weight.

scholarly journals The Pedersen Rigidity Problem

2019 ◽  
Vol 53 (supl) ◽  
pp. 237-244
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Morita Equivalence ◽  
Crossed Product ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
C Algebra ◽  
Dual Action ◽  
Continuous Trace

If is an action of a locally compact abelian group G on a C*-algebra A, Takesaki-Takai duality recovers (A, α) up to Morita equivalence from the dual action of Ĝ on the crossed product A × α G. Given a bit more information, Landstad duality recovers (A, α) up to isomorphism. In between these, by modifying a theorem of Pedersen, (A, α) is recovered up to outer conjugacy from the dual action and the position of A in M(A ×α G). Our search (still unsuccessful, somehow irritating) for examples showing the necessity of this latter condition has led us to formulate the "Pedersen Rigidity problem". We present numerous situations where the condition is redundant, including G discrete or A stable or commutative. The most interesting of these "no-go theorems" is for locally unitary actions on continuous-trace algebras.

scholarly journals A note on convex functions

1999 ◽  
Vol 22 (3) ◽  
pp. 525-534 ◽  
Author(s):  
Yu-Ru Syau
Keyword(s):  
Continuous Function ◽  
Euclidean Space ◽  
Convex Function ◽  
Convex Functions ◽  
Dimensional Euclidean Space ◽  
Strictly Convex ◽  
Lower Semi

In this paper, we give two weak conditions for a lower semi-continuous function on then-dimensional Euclidean spaceRnto be a convex function. We also present some results for convex functions, strictly convex functions, and quasi-convex functions.

scholarly journals Superstability of adjointable mappings on Hilbert c∗-modules

2009 ◽  
Vol 3 (1) ◽  
pp. 39-45 ◽  
Author(s):  
M. Frank ◽  
P. Găvruţa ◽  
M.S. Moslehian
Keyword(s):  
Banach Spaces ◽  
Linear Mapping ◽  
Stability Theorem ◽  
Classical Sense ◽  
C Algebra ◽  
The Stability ◽  
Densely Defined ◽  
Rassias Stability

We define the notion of ?-perturbation of a densely defined adjointable mapping and prove that any such mapping f between Hilbert A-modules over a fixed C*-algebra A with densely defined corresponding mapping g is A-linear and adjointable in the classical sense with adjoint g. If both f and g are every- where defined then they are bounded. Our work concerns with the concept of Hyers-Ulam-Rassias stability originated from the Th. M. Rassias' stability theorem that appeared in his paper [On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300]. We also indicate complementary results in the case where the Hilbert C?-modules admit non-adjointable C*-linear mappings.

scholarly journals THE CROSSED PRODUCT BY A POINTWISE UNITARY ACTION ON A C*-ALGEBRA WITH CONTINUOUS TRACE

2001 ◽  
Vol 44 (1) ◽  
pp. 215-218
Author(s):  
Klaus Deicke
Keyword(s):  
Compact Group ◽  
Elementary Proof ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Locally Compact ◽  
Primary 46L55 ◽  
C Algebra ◽  
Group A ◽  
Continuous Trace ◽  
Unitary Action

AbstractLet $G$ be a locally compact group, $A$ a continuous trace $C^*$-algebra, and $\alpha$ a pointwise unitary action of $G$ on $A$. It is a result of Olesen and Raeburn that if $A$ is separable and $G$ is second countable, then the crossed product $A\times_\alpha G$ has continuous trace. We present a new and much more elementary proof of this fact. Moreover, we do not even need the separability assumptions made on $A$ and $G$.AMS 2000 Mathematics subject classification: Primary 46L55

EXTERIOR EQUIVALENCE FOR POINTWISE UNITARY COACTIONS

2001 ◽  
Vol 12 (01) ◽  
pp. 63-79
Author(s):  
KLAUS DEICKE
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Equivalence Classes ◽  
Locally Compact ◽  
C Algebra ◽  
Isomorphism Classes ◽  
Continuous Trace

Let G be a second countable locally compact group and A a separable continuous trace C*-algebra. To each pointwise unitary coaction δ of G on A one can associate a proper G-bundle [Formula: see text], π × μ → π. We show that two pointwise unitary coactions δ and ∊ of G on A are exterior equivalent if and only if the proper G-bundles [Formula: see text] and [Formula: see text] are isomorphic. Thus, if A is stable, there exists a bijection between the isomorphism classes of proper G-bundles over [Formula: see text] and the exterior equivalence classes of pointwise unitary coactions of G on A. Moreover, when G is abelian we recover a theorem of Olesen and Raeburn.

scholarly journals Duality theorems for convex programming without constraint qualification

1984 ◽  
Vol 36 (2) ◽  
pp. 253-266 ◽  
Author(s):  
P. Kanniappan
Keyword(s):  
Convex Function ◽  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Convex Programming Problem ◽  
Duality Theorems ◽  
Finite Dimensional ◽  
Lower Semi ◽  
Continuous Convex Function

AbstractInvoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.

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