On the $K$-Theory Groups of Certain Crossed Product $C*$-A1gebras

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.

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The $K$-Theory of Some Reduced Inverse Semigroup $C^*$-Algebras

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-22866 ◽

2015 ◽

Vol 117 (2) ◽

pp. 186 ◽

Cited By ~ 2

Author(s):

Magnus Dahler Norling

Keyword(s):

Inverse Semigroup ◽

The cohomology and $K$-theory of commuting homeomorphisms of the Cantor set

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385799130189 ◽

1999 ◽

Vol 19 (3) ◽

pp. 611-625 ◽

Cited By ~ 17

Author(s):

ALAN FORREST ◽

JOHN HUNTON

Keyword(s):

Higher Order ◽

Cantor Set ◽

Crossed Product ◽

Mapping Torus ◽

C Algebra ◽

Third Invariant ◽

K Theory ◽

Torsion Free ◽

Order Continuous

Given a $\mathbb{Z}^d$ homeomorphic action, $\alpha$, on the Cantor set, $X$, we consider the higher order continuous integer valued dynamical cohomology, $H^*(X,\alpha)$. We also consider the dynamical $K$-theory of the action, the $K$-theory of the crossed product $C^*$-algebra $C(X)\times_{\alpha}\mathbb{Z}^d$. We show that these two invariants are essentially equivalent. We also show that they only take torsion free values. Our work links the two invariants via a third invariant which is based on topological complex $K$-theory evaluated on an associated mapping torus.

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TENSOR-PRODUCT COACTION FUNCTORS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788720000063 ◽

2020 ◽

pp. 1-16

Author(s):

S. KALISZEWSKI ◽

MAGNUS B. LANDSTAD ◽

JOHN QUIGG

Keyword(s):

Tensor Product ◽

Recent Work ◽

Analogous Result ◽

Crossed Product ◽

Discrete Groups ◽

Crossed Products ◽

Fixed Action ◽

Product Action ◽

K Theory

Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$ .

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Compact operators and algebraic K-theory for groups which act properly and isometrically on Hilbert space

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2014-0154 ◽

2018 ◽

Vol 2018 (734) ◽

pp. 265-292

Author(s):

Guillermo Cortiñas ◽

Gisela Tartaglia

Keyword(s):

Hilbert Space ◽

Approximation Property ◽

Compact Operators ◽

Crossed Product ◽

K Theory

AbstractWe prove theK-theoretic Farrell–Jones conjecture for groups with the Haagerup approximation property and coefficient rings andC^{*}-algebras which are stable with respect to compact operators. We use this and Higson–Kasparov’s result that the Baum–Connes conjecture holds for such a groupG, to show that the algebraic and theC^{*}-crossed product ofGwith a stable separableG-C^{*}-algebra have the sameK-theory.

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On the K-theory of boundaryC*-algebras of Ã2groups

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is011005004jkt158 ◽

2011 ◽

Vol 9 (3) ◽

pp. 521-536

Author(s):

Oliver King ◽

Guyan Robertson

Keyword(s):

Projective Plane ◽

Local Field ◽

Residue Field ◽

Crossed Product ◽

Class 1 ◽

K Theory

AbstractLet Γ be an Ã2subgroup of PGL3(), whereis a local field with residue field of orderq. The module of coinvariantsC(,ℤ)Γis shown to be finite, whereis the projective plane over. If the group Γ is of Tits type and ifq≢ 1 (mod 3) then the exact value of the order of the class [1]K0in the K-theory of the (full) crossed productC*-algebraC(Ω) ⋊ Γ is determined, where Ω is the Furstenberg boundary of PGL3(). For groups of Tits type, this verifies a conjecture of G. Robertson and T. Steger.

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THE C1-Invariance of the Godbillon-Vey Map in Analytical K-Theory

Canadian Journal of Mathematics ◽

10.4153/cjm-1987-061-8 ◽

1987 ◽

Vol 39 (5) ◽

pp. 1210-1222

Author(s):

Toshikazu Natsume

Keyword(s):

Cohomology Class ◽

Discrete Group ◽

Chern Character ◽

Cyclic Cohomology ◽

Crossed Product ◽

Additive Map ◽

K Theory

An action α of a discrete group Γ on the circle S1 as orientation preserving C∞-diffeomorphisms gives rise to a foliation on the homotopy quotient S1Γ, and its Godbillon-Vey invariant is, by definition, a cohomology class of S1Γ([1]). This cohomology class naturally defines an additive map from the geometric K-group K0(S1, Γ) into C, through the Chern character from K0(S1, Γ) to H*(S1, Γ Q).Using cyclic cohomology, Connes constructed in [2] an additive map, GV(α), which we shall call the Godbillon-Vey map, from the K0-group of the reduced crossed product C*-algebra C(S1) ⋊ αΓ into C. He showed that GV(α) agrees with the geometric Godbillon-Vey invariant through the index map μ from K0(S1, Γ) to K0(C(S1) ⋊ αΓ).

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Enveloping Σ-C -algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC -algebras

Proceedings of the Indian Academy of Sciences - Section A ◽

10.1007/bf02829785 ◽

2006 ◽

Vol 116 (2) ◽

pp. 161-173 ◽

Cited By ~ 1

Author(s):

Subhash J. Bhatt

Keyword(s):

Crossed Product ◽

C Algebra ◽

Differential Structure ◽

Fréchet Algebra ◽

K Theory

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-ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS

Journal of the Australian Mathematical Society ◽

10.1017/s144678872000049x ◽

2021 ◽

pp. 1-34

Author(s):

KENGO MATSUMOTO

Keyword(s):

Compatibility Condition ◽

Special Property ◽

Crossed Product ◽

Topological Conjugacy ◽

Dimension Group ◽

Bratteli Diagrams ◽

K Theory ◽

Shift Automorphism ◽

Smale Spaces ◽

Edge Labeling

Abstract This paper is a continuation of the paper, Matsumoto [‘Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras’, J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $ , there exists a $\lambda $ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ , and define a $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product . It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $ -graph bisystems come from two-sided subshifts, these $C^*$ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$ -algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$ . The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts.

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