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Joachim Cuntz

Publications and source records attributed to Joachim Cuntz.

At least 19 recordsLinked to original sources
arXiv2025

Split exactness, operator homotopy and stable uniqueness in KK

We develop important properties of the KK-functor on the basis of split exactness. In particular we discuss two slightly different short proofs for the existence of the Kasparov product and its associativity. We use the approach with quasihomomorphisms to obtain a short proof of the fact that operator homotopy implies homotopy . Using an idea of Gabe-Szabo we also deduce from this the 'stable uniqueness theorem' of Dadarlat-Eilers by a very short argument.

math.OA↗
arXiv2022

The image of Bott periodicity in cyclic homology

We analyze the relationship between Bott periodicity in topological $K$-theory and the natural periodicity of cyclic homology. This is a basis for understanding the multiplicativity, in odd dimensions, of a bivariant Chern-Connes character from $K$-theory to cyclic theory.

math.KT↗
arXiv2017

Weak completions, bornologies and rigid cohomology

Let $V$ be a complete discrete valuation ring with residue field $k$ of positive characteristic and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using completions of bornological $V$-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field $k$ that computes its rigid cohomology in the sense of Berthelot.

math.AG↗
arXiv2017

Nonarchimedean bornologies, cyclic homology and rigid cohomology

Let $V$ be a complete discrete valuation ring with residue field $k$ and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using spectral radius estimates for bounded subsets in complete bornological $V$-algebras. This leads us to a functorial chain complex for commutative $k$-algebras that computes Berthelot's rigid cohomology. This chain complex is related to the periodic cyclic homology of certain complete bornological $V$-algebras.

math.AG↗
arXiv2015

C*-algebras associated with algebraic actions

This is a survey of work in which the author was involved in recent years. We consider C*-algebras constructed from representations of one or several algebraic endomorphisms of a compact abelian group - or, dually, of a discrete abelian group. In our survey we do not try to describe the entire scope of the methods and results obtained in the original papers, but we concentrate on the important thread coming from the action of the multiplicative semigroup of a Dedekind ring on its additive group. Representations of such actions give rise to particularly intriguing problems and the study of the corresponding C*-algebras has motivated many of the new methods and general results obtained in this area.

math.OA↗
arXiv2015

Witt Vector Rings and the Relative de Rham Witt Complex

In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring $R$ and it is an important fact that the resulting objects are independent of all choices. The approach via presentations also sheds new light on our previous description of the ring of $p$-typical Witt vectors of a perfect $\mathbb{F}_p$-algebra as a completion of a semigroup algebra. We develop this description in different directions. For example, we show that the semigroup algebra can be replaced by any free presentation of $R$ equipped with a linear lift of the Frobenius automorphism. Using the result in the appendix by Umberto Zannier we also extend the description of the Witt vector ring as a completion to all $\bar{\mathbb{F}}_p$-algebras with injective Frobenius map.

math.RA↗
arXiv2013

An alternative to Witt vectors

The ring of Witt vectors associated to a ring R is a classical tool in algebra. We introduce a ring C(R) which is more easily constructed and which is isomorphic to the ring of Witt vectors W(R) for a perfect F_p-algebra R. It is obtained as the completion of the monoid ring ZR, for the multiplicative monoid R, with respect to the powers of the kernel of the natural map from ZR to R.

math.NT↗
arXiv2013

On the K-theory of the C*-algebra generated by the left regular representation of an Ore semigroup

We compute the K-theory of C*-algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions. Our result describes the K-theory of these semigroup C*-algebras in terms of the K-theory for the reduced group C*-algebras of certain groups which are typically easier to handle. Then we apply our result to specific semigroups from algebraic number theory.

math.OA↗
arXiv2012

C*-algebras of Toeplitz type associated with algebraic number fields

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\cT[R]$. It is an extension of the ring C*-algebra $\cA[R]$ studied previously by the first named author in collaboration with X.Li. In contrast to $\cA[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\rtimes R^\times$ on $\ell^2 (R\rtimes R^\times)$. The algebra $\cT[R]$ carries a natural one-parameter automorphism group $(σ_t)_{t\in\Rz}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The "partition functions" are partial Dedekind $ζ$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $ζ$-functions, which we then use to show uniqueness of the $β$-KMS state for each inverse temperature $β\in(1,2]$.

math.OA↗
arXiv2012

C*-algebras associated with endomorphisms and polymorphisms of compact abelian groups

A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the $K$-theory of these algebras and use it to compute the $K$-groups in a number of interesting examples.

math.OA↗
arXiv2012

On the K-theory of crossed products by automorphic semigroup actions

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies a certain Toeplitz condition and that the Baum-Connes conjecture holds for G. We prove a formula describing the K- theory of the reduced crossed product A \rtimesα,r P by any automorphic action of P. This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasi-lattice ordered semigroups. We also use the results to show that for certain semigroups P, including the ax + b-semigroup for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras of P coincide, although the structure of these algebras can be very different.

math.OA↗
arXiv2009

K-theory for ring C*-algebras attached to function fields

We compute the K-theory of ring C*-algebras for polynomial rings over finite fields. The key ingredient is a duality theorem which we had obtained in a previous paper. It allows us to show that the K-theory of these algebras has a ring structure and to determine explicit generators. Our main result also reveals striking similarities between the number field case and the function field case.

math.OA↗
arXiv2008

The Regular C*-algebra of an Integral Domain

To each integral domain R with finite quotients we associate a purely infinite simple C*-algebra in a very natural way. Its stabilization can be identified with the crossed product of the algebra of continuous functions on the "finite adele space" corresponding to R by the action of the ax+b-group over the quotient field Q(R). We study the relationship to generalized Bost-Connes systems and deduce for them a description as universal C*-algebras with the help of our construction.

math.OA↗