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

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.

scholarly journals On the action of the unitary group on the projective plane over a local field

1997 ◽  
Vol 62 (3) ◽  
pp. 371-397 ◽  
Author(s):  
Harm Voskuil
Keyword(s):  
Projective Plane ◽  
Local Field ◽  
Unitary Group ◽  
Residue Field ◽  
Equivariant Map ◽  
Rank One ◽  
Characteristic 2 ◽  
Set Of Points ◽  
Stable Points ◽  
Split Torus

AbstractLet G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

scholarly journals On classification of typical representations for GL3(F)

2019 ◽  
Vol 31 (4) ◽  
pp. 917-941
Author(s):  
Santosh Nadimpalli
Keyword(s):  
Local Field ◽  
Residue Field ◽  
Principal Series

Abstract Let F be any non-Archimedean local field with residue field of cardinality {q_{F}} . In this article, we obtain a classification of typical representations for the Bernstein components associated to the inertial classes of the form {[\operatorname{GL}_{n}(F)\times F^{\times},\sigma\otimes\chi]} with {q_{F}>2} , and for the principal series components with {q_{F}>3} . With this we complete the classification of typical representations for {\operatorname{GL}_{3}(F)} , for {q_{F}>2} .

scholarly journals Local–global principle for reduced norms over function fields of -adic curves

2017 ◽  
Vol 154 (2) ◽  
pp. 410-458 ◽  
Author(s):  
R. Parimala ◽  
R. Preeti ◽  
V. Suresh
Keyword(s):  
Local Field ◽  
Function Field ◽  
Function Fields ◽  
Residue Field ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Hasse Principle ◽  
Central Simple ◽  
Global Principle ◽  
Reduced Norm

Let $K$ be a (non-archimedean) local field and let $F$ be the function field of a curve over $K$. Let $D$ be a central simple algebra over $F$ of period $n$ and $\unicode[STIX]{x1D706}\in F^{\ast }$. We show that if $n$ is coprime to the characteristic of the residue field of $K$ and $D\cdot (\unicode[STIX]{x1D706})=0$ in $H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$, then $\unicode[STIX]{x1D706}$ is a reduced norm from $D$. This leads to a Hasse principle for the group $\operatorname{SL}_{1}(D)$, namely, an element $\unicode[STIX]{x1D706}\in F^{\ast }$ is a reduced norm from $D$ if and only if it is a reduced norm locally at all discrete valuations of $F$.

scholarly journals Wild ramification and the characteristic cycle of an ℓ-adic sheaf

2009 ◽  
Vol 8 (4) ◽  
pp. 769-829 ◽  
Author(s):  
Takeshi Saito
Keyword(s):  
Local Field ◽  
Cotangent Bundle ◽  
Euler Number ◽  
Residue Field ◽  
Geometric Method ◽  
Characteristic Class ◽  
Wild Ramification ◽  
Characteristic Cycle

AbstractWe propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an ℓ-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number.

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

2015 ◽  
Vol 117 (2) ◽  
pp. 186 ◽  
Author(s):  
Magnus Dahler Norling
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
One Dimensional ◽  
C Algebras ◽  
Morita Equivalent ◽  
K Theory

We use a recent result by Cuntz, Echterhoff and Li about the $K$-theory of certain reduced $C^*$-crossed products to describe the $K$-theory of $C^*_r(S)$ when $S$ is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that $C^*_r(S)$ is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

scholarly journals Rankin–Selberg gamma factors of level zero representations of GLn

2019 ◽  
Vol 31 (2) ◽  
pp. 503-516 ◽  
Author(s):  
Rongqing Ye
Keyword(s):  
Local Field ◽  
Residue Field ◽  
Converse Theorem ◽  
Level Zero ◽  
Gamma Factor ◽  
Cuspidal Representations

AbstractFor a p-adic local field F of characteristic 0, with residue field {\mathfrak{f}}, we prove that the Rankin–Selberg gamma factor of a pair of level zero representations of linear general groups over F is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over {\mathfrak{f}}. Our results can be used to prove a variant of Jacquet’s conjecture on the local converse theorem.

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

1999 ◽  
Vol 19 (3) ◽  
pp. 611-625 ◽  
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.

scholarly journals TENSOR-PRODUCT COACTION FUNCTORS

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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