scholarly journals The Crossed Product of Finite Hopf C*-Algebra and C*-Algebra

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1023
Author(s):  
Xiaomin Wei ◽  
Lining Jiang ◽  
Dianlu Tian
Keyword(s):  
Haar Measure ◽  
Linear Functional ◽  
Finite Dimension ◽  
Vital Role ◽  
Faithful Representation ◽  
Crossed Product ◽  
Complete Positivity ◽  
C Algebra ◽  
Positive Linear Functional

Let H be a finite Hopf C*-algebra and A a C*-algebra of finite dimension. In this paper, we focus on the crossed product A⋊H arising from the action of H on A, which is a ∗-algebra. In terms of the faithful positive Haar measure on a finite Hopf C*-algebra, one can construct a linear functional on the ∗-algebra A⋊H, which is further a faithful positive linear functional. Here, the complete positivity of a positive linear functional plays a vital role in the argument. At last, we conclude that the crossed product A⋊H is a C*-algebra of finite dimension according to a faithful ∗- representation.

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 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.

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.

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.

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