ROHLIN FLOWS ON THE CUNTZ ALGEBRA ${\mathcal O}_2$

2002 ◽  
Vol 13 (10) ◽  
pp. 1065-1094 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Crossed Product ◽  
Free Flow ◽  
Cuntz Algebra ◽  
C Algebra

We prove that a quasi-free flow on [Formula: see text] has the Rohlin property if and only if the associated crossed product is purely infinite and simple and that the quasi-free flows on [Formula: see text] with the Rohlin property are cocycle conjugate with each other. The latter result also holds for any Cuntz algebra [Formula: see text] with m < ∞. We also give some remarks on Rohlin flows on a unital separable nuclear purely infinite simple C*-algebra.

scholarly journals Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

10.53733/90 ◽  
2021 ◽  
Vol 52 ◽  
pp. 109-143
Author(s):  
Astrid An Huef ◽  
Marcelo Laca ◽  
Iain Raeburn
Keyword(s):  
Regular Representation ◽  
Crossed Product ◽  
Ideal Structure ◽  
Toeplitz Algebra ◽  
Natural Numbers ◽  
Kms States ◽  
C Algebra ◽  
Affine Semigroup ◽  
Natural Dynamics ◽  
The Right

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

scholarly journals Discrete product systems with twisted units

1995 ◽  
Vol 52 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Marcelo Laca
Keyword(s):  
Dynamical System ◽  
Crossed Product ◽  
Type I ◽  
Crossed Products ◽  
Product System ◽  
C Algebra ◽  
Covariant Representations ◽  
Product Systems ◽  
Recent Theorem ◽  
I Factor

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

scholarly journals Operator algebras with hyperarithmetic theory

2020 ◽  
Author(s):  
Isaac Goldbring ◽  
Bradd Hart
Keyword(s):  
Word Problem ◽  
Operator Algebras ◽  
Cuntz Algebra ◽  
Finitely Generated ◽  
C Algebra ◽  
Finitely Generated Group ◽  
C Algebras ◽  
Existentially Closed ◽  
Solvable Word Problem ◽  
Solvable Word

Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).

scholarly journals Semigroups of local homeomorphisms and interaction groups

2007 ◽  
Vol 27 (6) ◽  
pp. 1737-1771 ◽  
Author(s):  
R. EXEL ◽  
J. RENAULT
Keyword(s):  
Compact Space ◽  
Crossed Product ◽  
C Algebra ◽  
Crossed Product Algebra ◽  
Product Algebra

AbstractGiven a semigroup of surjective local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. We also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymorphism.

A crossed product of the canonical anticommutation relations algebra in the Cuntz algebra

2014 ◽  
Vol 58 (8) ◽  
pp. 71-73 ◽  
Author(s):  
M. A. Aukhadiev ◽  
A. S. Nikitin ◽  
A. S. Sitdikov
Keyword(s):  
Crossed Product ◽  
Cuntz Algebra ◽  
Canonical Anticommutation Relations

scholarly journals The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers

2012 ◽  
Vol 154 (1) ◽  
pp. 119-126 ◽  
Author(s):  
SIEGFRIED ECHTERHOFF ◽  
MARCELO LACA
Keyword(s):  
Recent Result ◽  
Orbit Space ◽  
Crossed Product ◽  
Primitive Ideal ◽  
Prime Ideals ◽  
Ideal Space ◽  
C Algebra ◽  
Ring Of Integers ◽  
Affine Semigroup ◽  
Power Set

AbstractThe purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

Non-Commutative Deformations of C(T2) and K-Theory

1997 ◽  
Vol 08 (05) ◽  
pp. 555-571
Author(s):  
Cristina Cerri
Keyword(s):  
Positive Element ◽  
Crossed Product ◽  
C Algebra ◽  
Basic Properties ◽  
C Algebras ◽  
The Family ◽  
K Theory

For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

scholarly journals SPECTRAL INVARIANCE OF DENSE SUBALGEBRAS OF OPERATOR ALGEBRAS

1993 ◽  
Vol 04 (02) ◽  
pp. 289-317 ◽  
Author(s):  
LARRY B. SCHWEITZER
Keyword(s):  
Polynomial Growth ◽  
Operator Algebras ◽  
Crossed Product ◽  
Discrete Groups ◽  
Spectral Invariance ◽  
C Algebra ◽  
C Algebras ◽  
Schwartz Functions ◽  
Exact Sequences ◽  
Spectral Invariant

We define the notion of strong spectral invariance for a dense Fréchet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G ⋊ A is spectral invariant in the C*-crossed product G ⋊ B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one of these. Our theorem gives the spectral invariance of G ⋊ A if A is the set of C∞-vectors for the action of G on B, or if B = C0 (M), and A is a set of G-differentiable Schwartz functions [Formula: see text] on M. This gives many examples of spectral invariant dense subalgebras for the C*-algebras associated with dynamical systems. We also obtain relevant results about exact sequences, subalgebras, tensoring by smooth compact operators, and strong spectral invariance in L1 (G, B).

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.

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