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Die Arbeit schließt eine Lücke im Preprint 'On the spinor norm and A_0(X, K_1) for quadrics' von Markus Rost. Die Ergebnisse von Rost wurden von Vladimir Voevodsky beim Beweis der Milnor-Vermutung benutzt.

Quasidiagonality is a matricial approximation property which asymptotically captures the multiplicative structure of C* -algebras. Quasidiagonal C* -algebras must be stably finite. It has been conjectured by Blackadar and Kirchberg that stably finiteness implies quasidiagonality for the class of separable nuclear C* -algebras. It has also been conjectured that separable exact quasidiagonal C* -algebras are AF embeddable. In this thesis, we study the behavior of these conjectures in the context of extensions 0 → I → E → B → 0. Specifically, we show that if I is exact and connective and B is separable, nuclear, and quasidiagonal (AF embeddable), then E is quasidiagonal (AF embeddable). Additionally, we show that if I is of the form C(X) ⊗ K for a compact metrizable space X and B is separable, nuclear, quasidiagonal (AF embeddable), and satisfies the UCT, then E is quasidiagonal (AF embeddable) if and only if E is stably finite.

This thesis deals with Panin and Walter's motivic spectrum BO. This spectrum is constructed using the real and quaternionic Grassmannians RG(r,n) and HGr(r,n) respectively, over schemes were 2 is invertible. We show that the construction of BO does not need the invertibility of 2. We also show that this spectrum is cellular over any base.

No description available.

This thesis is split up into two parts: the first concerns certain applications of the de la Harpe-Skandalis determinant to K-theory of appropriately regular C*-algebras. The second is concerned with (unital) inclusions of C*-algebras which satisfy a strong tensorial absorption condition. The first chapter following the preliminary section is joint work with Aaron Tikuisis [ST23], while the following chapters are independent. The penultimate chapter is [Sar23b] and the last chapter is essentially [Sar23a]. In the first chapter following the preliminaries, we examine the interplay between the algebraic K₁-group and the unitary algebraic K₁-group of a unital C*-algebra. We prove that for an abundance of unital C*-algebras, the algebraic K₁-group splits naturally as a direct sum of the unitary algebraic K₁-group and the space of continuous real-valued affine functions on the trace simplex. We further prove that if one considers Hausdorffized variants, then for any unital C*-algebra, there is a natural splitting of the Hausdorffized algebraic K₁-group in terms of the Hausdorffized unitary algebraic K₁-group and the space of continuous real-valued affine functions on the trace simplex. Moreover, this a splitting of topological groups. The following chapter studies how certain group homomorphisms between unitary groups of C*-algebras induce maps on the trace simplex. In particular, we show that a contractive group homomorphism between unital C*-algebras which sends the circle to the circle, induces a map between their trace simplices. Under mild regularity conditions these further induce maps between Elliott invariants. As a consequence we show that certain inclusions of C*-algebras are in a correspondence with certain inclusions of unitary groups. Finally we investigate what we call "D-stable inclusions" of C*-algebras, where D is strongly self-absorbing. We give a systematic study and prove that such inclusions between unital, separable, D-stable C*-algebras exist, are abundant, and are non-trivial.

We interpret the generalised gauge symmetry introduced in string theory and M-Theory as a special case of Grothendieck's stability equivalence relation in the definition of the 0th K-group and we calculate the Euler number of the elliptic de Rham complex twisted by a flat connection. Then using Polyakov's classical equivalence of flat bundles with non-linear sigma models we define a new topological invariant for foliations using techniques from noncommutative geometry, in particular the Connes' pairing between K-Theory and cyclic cohomology. This new invariant classifies foliations up to Morita equivalence.

Neste trabalho,nos apresentamos a K-teoria algébrica a qual é um ramo da álgebra que associa para cada anel comutativo comunidade R, uma sequencia de grupos abelianos ditos de n-ésimos K-grupos do anel R, denotada por Kn(R) . A meados da década de 1950,Alexander Grothendieck da a definição do K0(R) de um anel R. Em 1962, Hyman Bass e Stephen Schanuel apresenta a primeira definição adequada do K1(R) de um anel R. Em 1970, Daniel Quillen da uma definição geral dos K-grupos de um anel R a partir da +- construção do espaço classificante BGL(R). Nosso interesse é o estudo dos K-grupos sobre o anel de inteiros OF sobre um corpo numérico F. Usando alguns resultados de homologia dos grupos lineares, neste trabalho daremos a definição do mapa regulador de Borel. / In this paper,we present the algebraic K-theory,which is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R, denoted by Kn(R). The mid-1950s, Alexander Grothendieck gave a definition of the K0(R) of any ring R. In1962, Hyman Bass and Stephen Schanuel gave the first adequate definition of K1 of any ring R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +- construction of the classifying space BGL(R). Our interest is the study of the K-groups on the ring of integers OF over a number field F. Using some results of homology of linear groups, this work will give the definition of Borel\'s regulator map.

Nesta tese vamos apresentar dois resultados a respeito de K-teoria de álgebras C^{*} de classes de operadores pseudodiferenciais que são globalmente definidos em \\mathbb^. O primeiro resultado é a prova da regularidade da função \\eta para operadores clássicos com símbolos de Shubin. Vamos mostrar que a álgebra de operadores pseudodiferenciais em \\mathbb^ com símbolos de Shubin permite a construção de potências complexas e um tipo de traço de Kontsevich-Vishik numa forma muito similar àquela feita para variedades compactas, com definições até mais simples. Mostraremos, então, que podemos definir as funções \\zeta e \\eta também para esses símbolos. Finalmente mostraremos como o conhecimento de fatos simples sobre a sua K-teoria permitem a prova da regularidade da função \\eta. Para variedades compactas, esse resultado tem muitas implicações. Acreditamos assim que ele também possa ser interessante para os estudos de operadores globais em \\mathbb^. O segundo resultado é o cálculo da K-teoria de operadores limitados gerados por operadores de Boutet de Monvel SG de ordem (0,0) e tipo zero em \\mathbb_{+}^. Boutet de Monvel introduziu a álgebra que leva o seu nome para estudar o índice de operadores elípticos de fronteira em variedades compactas com bordo. Mais recentemente uma nova abordagem foi proposta por Melo, Nest, Schrohe e Schick para obter resultados sobre o índice de Fredholm usando a K-teoria de álgebras C^{*}, uma ferramenta que não era disponível ainda quando Boutet de Monvel desenvolveu sua álgebra. Nossa ideia foi, então, mostrar como calcular a K-teoria de álgebras de Boutet de Monvel com símbolos SG em \\mathbb_{+}^, em que os símbolos SG são uma classe de símbolos globalmente definidos em \\mathbb^. Acreditamos que isso possa ser útil também ao estudo de problemas elípticos de fronteira para operadores de Boutet de Monvel com símbolos SG em certas classes de variedades não compactas. / We are going to present two results concerning K-theory of C^{*} algebras of classes of pseudodifferential operators that are globally defined in \\mathbb^. The first result is the proof of the regularity of the \\eta function for classical operators with Shubin symbols. We are going to show that the algebra of classical pseudodifferential operators in \\mathbb^ with Shubin symbols allows the construction of complex powers and a kind of Kontsevich-Vishik trace in a very similar way as on compact manifolds, with even easier definitions. Then we show that we can define the \\zeta and \\eta functions also for these symbols. Finally we will show how the knowledge of simple facts about the K-theory of pseudodifferential operators with Shubin\'s symbols allows the proof of the regularity of the \\eta function at 0. For compact manifolds, this regularity is a result that has many implications. Therefore it may also be interesting for global operators in \\mathbb^. The second result is the evaluation of the K-theory of bounded operators generated by SG Boutet de Monvel operators of order (0,0) and type 0 in \\mathbb_^. Boutet de Monvel introduced his algebra to study the index of elliptic boundary value problems on compact manifolds. More recently a new approach was proposed by Melo, Nest, Schrohe and Schick to obtain results about the index of Fredholm operators using the K-theory of C^ algebras, a tool which was not well known when Boutet de Monvel published his work. The idea here is to show how one can evaluate the K-theory of the Boutet de Monvel operators with SG symbols in \\mathbb_^, where SG symbols is a class of symbols globally defined in \\mathbb^. We believe that this can be useful to the study of index of Fredholm problems also in the case of Boutet de Monvel operators with SG symbols in some classes of non-compact manifolds.

Neste trabalho,nos apresentamos a K-teoria algébrica a qual é um ramo da álgebra que associa para cada anel comutativo comunidade R, uma sequencia de grupos abelianos ditos de n-ésimos K-grupos do anel R, denotada por Kn(R) . A meados da década de 1950,Alexander Grothendieck da a definição do K0(R) de um anel R. Em 1962, Hyman Bass e Stephen Schanuel apresenta a primeira definição adequada do K1(R) de um anel R. Em 1970, Daniel Quillen da uma definição geral dos K-grupos de um anel R a partir da +- construção do espaço classificante BGL(R). Nosso interesse é o estudo dos K-grupos sobre o anel de inteiros OF sobre um corpo numérico F. Usando alguns resultados de homologia dos grupos lineares, neste trabalho daremos a definição do mapa regulador de Borel. / In this paper,we present the algebraic K-theory,which is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R, denoted by Kn(R). The mid-1950s, Alexander Grothendieck gave a definition of the K0(R) of any ring R. In1962, Hyman Bass and Stephen Schanuel gave the first adequate definition of K1 of any ring R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +- construction of the classifying space BGL(R). Our interest is the study of the K-groups on the ring of integers OF over a number field F. Using some results of homology of linear groups, this work will give the definition of Borel\'s regulator map.