scholarly journals Toeplitz algebras and infinite simple $C^*$-algebras associated with reduced group $C^*$-algebras.

1997 ◽  
Vol 81 ◽  
pp. 86 ◽  
Author(s):  
Shuang Zhang
Keyword(s):  
C Algebras ◽  
Reduced Group

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 Traces on reduced group C* -algebras

2017 ◽  
Vol 49 (6) ◽  
pp. 988-990
Author(s):  
Matthew Kennedy ◽  
Sven Raum
Keyword(s):  
C Algebras ◽  
Reduced Group

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.

Embedding Theorems in Group C*-Algebras†

1983 ◽  
Vol 26 (2) ◽  
pp. 157-166 ◽  
Author(s):  
Tan-Yu Lee
Keyword(s):  
Open Subgroup ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Embedding Theorems ◽  
Group Representations ◽  
Locally Compact ◽  
C Algebras ◽  
Reduced Group ◽  
Natural Way

AbstractLet G be a locally compact group and H an open subgroup of G. The embeddings of group C*-algebras associated with H into the group C*-algebras associated with G are studied. Three conditions for the embeddings given in terms of C*-norms of the group algebras, group representations and positive definite functions are shown to be equivalent. As corollary, we prove that the full C*-algebra of H can be embedded into the full C*-algebra of G in a natural way as well as the case for the reduced group C*-algebras. We also show that the embeddings hold for their duals and double duals.

C*-ALGEBRAS GENERATED BY A PROJECTION AND THE REDUCED GROUP C*-ALGEBRAS ${C^*_r\Bbb Z*\Bbb Z_n}$ AND ${C^*_r\Bbb Z_m*\Bbb Z_n}$

2005 ◽  
Vol 16 (05) ◽  
pp. 533-554
Author(s):  
SHUANG ZHANG
Keyword(s):  
Closed Ideal ◽  
Quotient Algebra ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Ideal Formula ◽  
C Algebras ◽  
Reduced Word ◽  
Reduced Group

Let Γ=ℤm * ℤn or ℤ * ℤn, and let Γ(h) be the subtree consisting of all reduced words starting with any reduced word h ∈ Γ\{e}. We prove that the C*-algebra [Formula: see text] generated by [Formula: see text] and the projection Ph onto the subspace ℓ2(Γ(h)) has a unique nontrivial closed ideal ℐ, ℐ is *-isomorphic to [Formula: see text], and the quotient algebra [Formula: see text] is *-isomorphic to either [Formula: see text] or [Formula: see text] depending on the last letter of h. We also prove that [Formula: see text] is a purely infinite, simple C*-algebra if the last letter of h is a generator of ℤ, and that [Formula: see text] has a unique nontrivial closed ideal [Formula: see text] if the last letter of h is a generator of ℤn; furthermore, [Formula: see text] is *-isomorphic to [Formula: see text] and [Formula: see text] is again a purely infinite, simple C*-algebra. As consequences, all the C*-algebras above have real rank zero, and [Formula: see text] is nuclear for any h ≠ e.

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