Isometry groups of twisted reduced group C∗-algebras

2017 ◽  
Vol 28 (14) ◽  
pp. 1750101 ◽  
Author(s):  
Botao Long ◽  
Wei Wu
Keyword(s):  
Topological Group ◽  
Discrete Group ◽  
Isometry Group ◽  
Unitary Operator ◽  
Spectral Triple ◽  
Compact Lie Group ◽  
Norm Topology ◽  
Topological Characterization ◽  
C Algebras ◽  
Reduced Group

An isometry of a unital [Formula: see text]-algebra with respect to a spectral triple is a [Formula: see text]-automorphism of the [Formula: see text]-algebra given by the conjugation by a unitary operator which commutes with the Dirac operator. We give a semidirect product topological characterization on the isometry group of a twisted reduced group [Formula: see text]-algebra of a discrete group with respect to the standard spectral triple induced by a length function on the group. We prove that this isometry group is compact in the point-norm topology, and in particular, for a finitely generated discrete group, this isometry group is a compact Lie group in the point-norm topology. We also extend this result to a unital [Formula: see text]-algebra with a filtration, and prove that its isometry group is a compact topological group in the point-norm topology.

scholarly journals Discrete groups and simple C*-algebras

1991 ◽  
Vol 109 (3) ◽  
pp. 521-537 ◽  
Author(s):  
Erik Bédos
Keyword(s):  
Discrete Group ◽  
Free Groups ◽  
Free Subgroup ◽  
Discrete Groups ◽  
Full Group ◽  
C Algebras ◽  
Reduced Group

Let G denote a discrete group and let us say that G is C*-simple if the reduced group C*-algebra associated with G is simple. We notice immediately that there is no interest in considering here the full group C*-algebra associated with G, because it is simple if and only if C is trivial. Since Powers in 1975 [26] proved that all nonabelian free groups are C*-simple, the class of C*-simp1e groups has been considerably enlarged (see [1, 2, 6, 7, 12, 13, 14, 16, 24] as a sample!), and two important subclasses are so-called weak Powers groups ([6, 13]; see Section 4 for definition and examples) and the groups of Akemann-Lee type [1, 2], which are groups possessing a normal non-abelian free subgroup with trivial centralizer.

scholarly journals Boundary Action On Simple Reduced Group C*-Algebras

2021 ◽  
Vol 03 (01) ◽  
pp. 1-15
Author(s):  
V.A. Onuche ◽  
◽  
T.J Alabiand ◽  
O.F. Ajayi ◽  
◽  
...  
Keyword(s):  
Simple Group ◽  
Free Group ◽  
Discrete Group ◽  
Crossed Products ◽  
Ideal Structure ◽  
Stability Properties ◽  
C Algebras ◽  
Reduced Group ◽  
The Stability

A connection between boundary actions, ideal structure of reduced crossed products and C*-simple group is imminent.We investigate the stability properties for discrete group pioneered by powers and show that the non-abelian free group on two generators is C*-simple.Kalantar and Kennedy [32, Theorem 6.2] is now extended. Some examples are given using characterization of C*-simplicity obtained by Kalantar, Kennedy, Breuillard, and Ozawa [10, Theorem 3.1]

scholarly journals Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras

2019 ◽  
Author(s):  
Zhizhang Xie ◽  
Guoliang Yu
Keyword(s):  
Conjugacy Class ◽  
Discrete Group ◽  
Polynomial Growth ◽  
Algebraic Numbers ◽  
Eta Invariant ◽  
Trace Map ◽  
Novikov Conjecture ◽  
C Algebras ◽  
Eta Invariants ◽  
K Theory

Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $K_0$-group of its group $C^\ast$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum–Connes conjecture holds for a group, then Lott’s delocalized eta invariants take values in algebraic numbers. We also generalize Lott’s delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

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 Boundaries of reduced C * C^{*} -algebras of discrete groups

2017 ◽  
Vol 2017 (727) ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthew Kennedy
Keyword(s):  
Open Problem ◽  
Discrete Group ◽  
Discrete Groups ◽  
Algebraic Construction ◽  
C Algebras

AbstractFor a discrete groupThis operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove thatThe algebraIt is a longstanding open problem to determine which groups are

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 ≤ + ∞ .

On theorems of Thornton

1972 ◽  
Vol 6 (2) ◽  
pp. 211-212 ◽  
Author(s):  
R. Lalithambal
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Discrete Group ◽  
Torsion Group ◽  
Additional Hypothesis

The topology of a topological group in which the intersection of open sets is open is uniquely determined by a normal subgroup, and the group is uniquely an extension of an indiscrete group by a discrete group. This was proved by M.C. Thornton under the additional hypothesis that the group is a torsion group. The proofs here given make the more general facts almost trivial.

Property T and Amenable Transformation Group C*-algebras

2015 ◽  
Vol 58 (1) ◽  
pp. 110-114 ◽  
Author(s):  
F. Kamalov
Keyword(s):  
Compact Hausdorff Space ◽  
Discrete Group ◽  
Hausdorff Space ◽  
Transformation Group ◽  
Amenable Group ◽  
Transformation Groups ◽  
C Algebra ◽  
C Algebras ◽  
Property T ◽  
Tracial States

AbstractIt is well known that a discrete group that is both amenable and has Kazhdan’s Property T must be finite. In this note we generalize this statement to the case of transformation groups. We show that if G is a discrete amenable group acting on a compact Hausdorff space X, then the transformation group C*-algebra C*(X; G) has Property T if and only if both X and G are finite. Our approach does not rely on the use of tracial states on C*(X; G).

Sign in / Sign up

Export Citation Format

Share Document