scholarly journals Positive Herz–Schur multipliers and approximation properties of crossed products

2017 ◽  
Vol 165 (3) ◽  
pp. 511-532 ◽  
Author(s):  
ANDREW MCKEE ◽  
ADAM SKALSKI ◽  
IVAN G. TODOROV ◽  
LYUDMILA TUROWSKA
Keyword(s):  
Discrete Group ◽  
Crossed Products ◽  
Approximation Properties ◽  
Completely Positive ◽  
Haagerup Property ◽  
Schur Multipliers

For aC*-algebraAand a setXwe give a Stinespring-type characterisation of the completely positive SchurA-multipliers on κ(ℓ2(X)) ⊗A. We then relate them to completely positive Herz–Schur multipliers onC*-algebraic crossed products of the formA⋊α,rG, withGa discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, forA⋊α,rG.

HAAGERUP PROPERTY FOR -CROSSED PRODUCTS

2016 ◽  
Vol 95 (1) ◽  
pp. 144-148 ◽  
Author(s):  
QING MENG
Keyword(s):  
Conditional Expectation ◽  
Discrete Group ◽  
Crossed Products ◽  
Haagerup Property ◽  
Residually Finite ◽  
Tracial State ◽  
Finite Dimensional ◽  
Countable Discrete Group ◽  
Strong Property

Let $\unicode[STIX]{x1D6E4}$ be a countable discrete group that acts on a unital $C^{\ast }$-algebra $A$ through an action $\unicode[STIX]{x1D6FC}$. If $A$ has a faithful $\unicode[STIX]{x1D6FC}$-invariant tracial state $\unicode[STIX]{x1D70F}$, then $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}\circ {\mathcal{E}}$ is a faithful tracial state of $A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}$ where ${\mathcal{E}}:A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}\rightarrow A$ is the canonical faithful conditional expectation. We show that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property if and only if both $(A,\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6E4}$ have the Haagerup property. As a consequence, suppose that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property where $\unicode[STIX]{x1D6E4}$ has property $T$ and $A$ has strong property $T$. Then $\unicode[STIX]{x1D6E4}$ is finite and $A$ is residually finite-dimensional.

scholarly journals Induced Coactions of Discrete Groups on C*-Algebras

1999 ◽  
Vol 51 (4) ◽  
pp. 745-770 ◽  
Author(s):  
Siegfried Echterhoff ◽  
John Quigg
Keyword(s):  
Quotient Group ◽  
Discrete Group ◽  
Discrete Groups ◽  
Crossed Products ◽  
C Algebras ◽  
Morita Equivalent ◽  
Close Relationship

AbstractUsing the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction δ: D → D ⊗C*(G/N) of a quotient group G/N of a discrete group G to a C*-coaction Ind δ: Ind D → D ⊗C*(G) of G. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products Ind D ×IndδG and D ×δG/N are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

scholarly journals HILBERT C*-BIMODULES OF FINITE INDEX AND APPROXIMATION PROPERTIES OF C*-ALGEBRAS

2017 ◽  
Vol 60 (2) ◽  
pp. 321-331
Author(s):  
MARZIEH FOROUGH ◽  
MASSOUD AMINI
Keyword(s):  
Finite Index ◽  
Morita Equivalence ◽  
Discrete Groups ◽  
Crossed Products ◽  
Approximation Properties ◽  
C Algebras ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
The Stability

AbstractLet A and B be arbitrary C*-algebras, we prove that the existence of a Hilbert A–B-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by A and B. For this, we first study the stability of the WEP, QWEP, and LLP under Morita equivalence of C*-algebras. We present examples of Hilbert A–B-bimodules, which are not of finite index, while such properties are shared between A and B. To this end, we study twisted crossed products by amenable discrete groups.

FREE-PRODUCT GROUPS, CUNTZ-KRIEGER ALGEBRAS, AND COVARIANT MAPS

1991 ◽  
Vol 02 (04) ◽  
pp. 457-476 ◽  
Author(s):  
JOHN SPIELBERG
Keyword(s):  
Free Product ◽  
Compact Convex ◽  
Negative Answer ◽  
Crossed Product ◽  
Crossed Products ◽  
Explicit Computation ◽  
Completely Positive ◽  
Cyclic Groups ◽  
Crossed Product Algebra ◽  
Product Algebra

A construction is given relating a finitely generated free-product of cyclic groups with a certain Cuntz-Krieger algebra, generalizing the relation between the Choi algebra and 02. It is shown that a certain boundary action of such a group yields a Cuntz-Krieger algebra by the crossed-product construction. Certain compact convex spaces of completely positive mappings associated to a crossed-product algebra are introduced. These are used to generalize a problem of J. Anderson regarding the representation theory of the Choi algebra. An explicit computation of these spaces for the crossed products under study yields a negative answer to this problem.

scholarly journals Dynamical complexity and K-theory of Lp operator crossed products

2019 ◽  
pp. 1-33
Author(s):  
Yeong Chyuan Chung
Keyword(s):  
Compact Hausdorff Space ◽  
Discrete Group ◽  
Hausdorff Space ◽  
Crossed Products ◽  
Dynamical Complexity ◽  
Countable Discrete Group ◽  
K Theory ◽  
Assembly Map

We apply quantitative (or controlled) [Formula: see text]-theory to prove that a certain [Formula: see text] assembly map is an isomorphism for [Formula: see text] when an action of a countable discrete group [Formula: see text] on a compact Hausdorff space [Formula: see text] has finite dynamical complexity. When [Formula: see text], this is a model for the Baum–Connes assembly map for [Formula: see text] with coefficients in [Formula: see text], and was shown to be an isomorphism by Guentner et al.

scholarly journals P-Tensor Product for Group C*-Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 627
Author(s):  
Yufang Li ◽  
Zhe Dong
Keyword(s):  
Tensor Product ◽  
Free Group ◽  
Discrete Group ◽  
Tensor Products ◽  
Haagerup Property ◽  
C Algebras ◽  
P Tensor

In this paper, we introduce new tensor products ⊗ p ( 1 ≤ p ≤ + ∞ ) on C ℓ p * ( Γ ) ⊗ C ℓ p * ( Γ ) and ⊗ c 0 on C c 0 * ( Γ ) ⊗ C c 0 * ( Γ ) for any discrete group Γ . We obtain that for 1 ≤ p < + ∞ C ℓ p * ( Γ ) ⊗ m a x C ℓ p * ( Γ ) = C ℓ p * ( Γ ) ⊗ p C ℓ p * ( Γ ) if and only if Γ is amenable; C c 0 * ( Γ ) ⊗ m a x C c 0 * ( Γ ) = C c 0 * ( Γ ) ⊗ c 0 C c 0 * ( Γ ) if and only if Γ has Haagerup property. In particular, for the free group with two generators F 2 we show that C ℓ p * ( F 2 ) ⊗ p C ℓ p * ( F 2 ) ≇ C ℓ q * ( F 2 ) ⊗ q C ℓ q * ( F 2 ) for 2 ≤ q < p ≤ + ∞ .

scholarly journals Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

2011 ◽  
Vol 63 (5) ◽  
pp. 1161-1187 ◽  
Author(s):  
Stefan Neuwirth ◽  
Éric Ricard
Keyword(s):  
Lebesgue Space ◽  
Discrete Group ◽  
Orlicz Spaces ◽  
Fourier Multipliers ◽  
Von Neumann ◽  
Riesz Projection ◽  
Schur Multipliers ◽  
Schauder Bases ◽  
The Relationship

Abstract We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue– Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten–von- Neumann–Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum , the norm of the Hilbert transformand the Riesz projection on Schatten–von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten–von-Neumann classes with exponent less than 1.

scholarly journals Fullness of crossed products of factors by discrete groups

2019 ◽  
Vol 150 (5) ◽  
pp. 2368-2378 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Discrete Group ◽  
Amenable Group ◽  
Complete Characterization ◽  
Crossed Product ◽  
Discrete Groups ◽  
Sufficient Condition ◽  
Crossed Products ◽  
Product Factor ◽  
Jones Criterion ◽  
Arbitrary Factor

AbstractLet M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full.

scholarly journals APPROXIMATION PROPERTIES FOR CROSSED PRODUCTS BY ACTIONS AND COACTIONS

2001 ◽  
Vol 12 (05) ◽  
pp. 595-608 ◽  
Author(s):  
MAY M. NILSEN ◽  
ROGER R. SMITH
Keyword(s):  
Approximation Property ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Crossed Products ◽  
Approximation Properties ◽  
Locally Compact ◽  
Amenable Groups ◽  
C Algebras ◽  
Approximation Constant

We investigate approximation properties for C*-algebras and their crossed products by actions and coactions by locally compact groups. We show that Haagerup's approximation constant is preserved for crossed products by arbitrary amenable groups, and we show why this is not always true in the non-amenable case. We also examine similar questions for other forms of the approximation property.

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