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Bernhard Burgstaller

Publications and source records attributed to Bernhard Burgstaller.

At least 19 recordsLinked to original sources
arXiv2026

The universal property of graded $KK^G$-theory

A universal category-theoretical characterization of groupoid equivariant $KK^G$-theory for ${\mathbb{Z}}_2$-graded $C^*$-algebras is established, by observing the ``$KK$-axiom'' that for each $[s,{\cal E} \oplus B, \mathbb{F}] \in KK^G(A,B)$, the `corner-embedding' $*$-homomorphism ${\bf j}: B \rightarrow {\sf cl} \big({\cal K}_B({\cal E} \oplus B) + s(A) + \mathbb{F} \cdot s(A) \big)$ is invertible in $KK^G$. This $KK$-axiom and homotopy-invariance characterize graded $KK^G$-theory universally and completely, thus directly extending the well-known characterization of $KK$-theory for ungraded $C^*$-algebras via stability, homotopy invariance and splitexactness by Higson.

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arXiv2025

Computing the $K$-homology $K$-theory product in splitexact algebraic $KK$-theory

Explicit formulas are indicated that compute the product $z \cdot w$ of a level-one element $z \in KK^G(A,{\bf C})$ and any element $w \in KK^G({\bf C},B)$ in splitexact algebraic $KK^G$-theory, or $KK^G$-theory for $C^*$-algebras, with very special $G$-actions. We also make such products accessible to linear-split half-exact $kk$-theory by verifying the existence of a functor from algebraic splitexact $KK$-theory to $kk$-theory.

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arXiv2025

Corner embeddings into algebras of compact operators in $K$-theory

We show that in $K$-theory-like categories many corner embeddings into a discrete algebra of compact operators are invertible, and consequently functors on splitexact algebraic $KK$-theory are faithful if and only if they are faithful on the subcategory generated by the homomorphisms.

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arXiv2024

On the $K$-theory in splitexact algebraic $KK$-theory

It is proven that in the universal splitexact equivariant algebraic $KK$-theory for algebras, the $K$-theory groups coincide with classical $K$-theory in the sense of Phillips. This partially answers a question raised by Kasparov.

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arXiv2023

Information content in formal languages

Motivated by creating physical theories, formal languages $S$ with variables are considered and a kind of distance between elements of the languages is defined by the formula $d(x,y)= \ell(x \nabla y) - \ell(x) \wedge \ell(y)$, where $\ell$ is a length function and $x \nabla y$ means the united theory of $x$ and $y$. Actually we mainly consider abstract abelian idempotent monoids $(S,\nabla)$ provided with length functions $\ell$. The set of length functions can be projected to another set of length functions such that the distance $d$ is actually a pseudometric and satisfies $d(x\nabla a,y\nabla b) \le d(x,y) + d(a,b)$. We also propose a "signed measure" on the set of Boolean expressions of elements in $S$, and a Banach-Mazur-like distance between abelian, idempotent monoids with length functions, or formal languages.

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arXiv2021

A kind of $KK$-theory for rings

A group equivariant $KK$-theory for rings will be defined and studied in analogy to Kasparov's $KK$-theory for $C^*$-algebras. It is a kind of linearization of the category of rings by allowing addition of homomorphisms, imposing also homotopy invariance, invertibility of matrix corner embeddings, and allowing morphisms which are the opposite split of split exact sequences. We demonstrate the potential of this theory by proving for example equivalence induced by Morita equivalence and a Green-Julg isomorphism in this framework.

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arXiv2019

Aspects of equivariant $KK$-theory in its generators and relations picture

We give a new proof of the universal property of $KK^G$-theory with respect to stability, homotopy invariance and split-exactness for $G$ a locally compact group, or a locally compact (not necessarily Hausdorff) groupoid, or a countable inverse semigroup which is relatively short and conceptual. Morphisms in the generators and relations picture of $KK^G$-theory are brought to a particular simple form.

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arXiv2019

Some remarks in $C^*$- and $K$-theory

This note consists of three unrelated remarks. First, we demonstrate how roughly speaking $*$-homomorphisms between matrix stable $C^*$-algebras are exactly the uniformly continuous $*$-preserving group homomorphisms between their genral linear groups. Second, using the Cuntz picture in $KK$-theory we bring morphisms in $KK$-theory represented by generators and relations to a particular simple form. Third, we show that for an inverse semigroup its associated groupoid is Hausdorff if and only if the inverse semigroup is $E$-continuous.

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arXiv2017

An elementary Green imprimitivity theorem for inverse semigroups

A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let $G$ be an inverse semigroup, $H$ a finite sub-inverse semigroup of $G$ and $A$ a $G$-algebra or a $H$-algebra. Then the crossed product $A \rtimes H$ is Morita equivalent to a certain crossed product $B \rtimes G$.

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arXiv2017

Attempts to define a Baum--Connes map via localization of categories for inverse semigroups

Meyer and Nest showed that the Baum--Connes map is equivalent to a map on $K$-theory of two different crossed products. This approach is strongly categorial in method since its bases is to regard Kasparov's theory $KK^G$ as a triangulated category. We have tried to translate this approach to the realm of inverse semigroup equivariant $C^*$-algebras but can prove the existence of a Baum--Connes map only under some unverified additional assumptions which we however strongly motivate. Some of our results may be of independent interest, for example Bott periodicity, the definition of induction functors, the definition of a completely novel compatible $L^2(G)$-space, a Cuntz picture of $KK^G$, and the verification that $KK^G$ is a triangulated category.

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arXiv2017

The universal property of inverse semigroup equivariant $KK$-theory

Higson proved that every homotopy invariant, stable and split exact functor from the category of $C^*$-algebras to an additive category factors through Kasparov's $KK$-theory. By adapting a group equivariant generalization of this result by Thomsen, we generalize Higson's result to the inverse semigroup equivariant setting.

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arXiv2016

A note on a certain Baum--Connes map for inverse semigroups

Let $G$ denote a countable inverse semigroup. We construct a kind of a Baum--Connes map $K(\tilde A \rtimes G) \rightarrow K(A \rtimes G)$ by a categorial approach via localization of triangulated categories, developed by R. Meyer and R. Nest for groups $G$. We allow the coefficient algebras $A$ to be in a special class of algebras called fibered $G$-algebras. This note continues and fixes our preprint "Attempts to define a Baum--Connes map via localization of categories for inverse semigroups".

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arXiv2016

The generators and relations picture of $KK$-theory

This is half an overview article since what we describe here is essentially known. We describe $KK$-theory by generators and relations in a formal sum of formal products of $*$-homomorphisms and some synthetical morphisms. What comes out is a category. The Kasparov product is then just the composition of morphisms. This description may be interesting to anyone who wants a quick and elementary definition of $KK$-theory. The description could also be used for other categories of algebras than $C^*$-algebras endowed with group actions, for example, $C^*$-algebras equipped with an action by a semigroup, a category et cetera.

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arXiv2015

Inverse semigroup equivariant $KK$-theory and $C^*$-extensions

In this note we extend the classical result by G. G. Kasparov that the Kasparov groups $KK_1(A,B)$ can be identified with the extension groups $\mbox{Ext}(A,B)$ to the inverse semigroup equivariant setting. More precisely, we show that $KK_G^1(A,B) \cong \mbox{Ext}_G(A \otimes {\cal K}_G,B \otimes {\cal K}_G)$ for every countable, $E$-continuous inverse semigroup $G$. For locally compact second countable groups $G$ this was proved by K. Thomsen, and technically this note presents an adaption of his proof.

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arXiv2014

A Green--Julg isomorphism for inverse semigroups

For every finite unital inverse semigroup $S$ and $S$-$C^*$-algebra $A$ we establish an isomorphism between $KK^S(\mathbb{C},A)$ and $K(A \rtimes S)$. This extends the classical Green--Julg isomorphism from finite groups to finite inverse semigroups.

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arXiv2013

On freely generated semigraph $C^*$-algebras

For special universal $C^*$-algebras associated to $k$-semigraphs we present the universal representations of these algebras, prove a Cuntz--Krieger uniqueness theorem, and compute the $K$-theory. These $C^*$-algebras seem to be the most universal Cuntz--Krieger like algebras naturally associated to $k$-semigraphs. For instance, the Toeplitz Cuntz algebra is a proper quotient of such an algebra.

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arXiv2012

Equivariant $KK$-theory of $r$-discrete groupoids and inverse semigroups

For an $r$-discrete Hausdorff groupoid ${\cal G}$ and an inverse semigroup $S$ of slices of ${\cal G}$ there is an isomorphism between ${\cal G}$-equivariant $KK$-theory and compatible $S$-equivariant $KK$-theory. We use it to define descent homomorphisms for $S$, and indicate a Baum--Connes map for inverse semigroups. Also findings by Khoshkam and Skandalis for crossed products by inverse semigroups are reflected in $KK$-theory.

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