K-theory is the study of invariants of large matrices, in a suitable sense. It has many variations: (algebraic-k-theory), (topological-k-theory), or in the study of (operator-algebras).

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What is the motivation of studying $P[A]$ in operator K-theory?

I am reading the last chapter of Murphy's $C^*$-algebras and operator theory. He defines $$P[A]=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {$p$ is a projection} \}$$ and construct the Grothendieck group ...
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$K_0$ group of Direct Sum of C*-Algebras

We know that for C*-algebras, $$ K_0(A\oplus B) \stackrel{(*)}{=} K_0(A)\times K_0(B)$$where the $\oplus$ on the L.H.S. is a direct sum of C*-algebras (derived from the notion of direct sum of ...
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K-theory of infinite direct product of copies of the algebra of compact operators

How does one compute the $K$-theory of an infinite (countable) direct product of copies of the algebra of compact operators on Hilbert space? Will it be an infinite direct sum of copies of ...
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Injectivity of index map for $K_1(S^1)$

This example/problem is from Valette's notes on the Baum-Connes conjecture (p. 45). The exercise is to prove that the (trivially equivariant) $K$-homology group $K_1(S^1)$ is $\mathbb{Z}$. For this, ...
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How does having a cycle in a quiver change the simple objects in the category of representations?

In theorem 1.12 on page 5 of http://www.math.utah.edu/dc/tilting.pdf, which states: Given a bounded acyclic quiver $(Q,R)$, the K-theory of it's representations is given by $\mathbb{Z}^{Q_0}$ why is ...
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Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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28 views

A question about ring has 1 stable range

I got this problem from my professor.It states that if $D$ is a division ring then ring of matrices $M_{n}(D)$ has 1 stable range. A ring called has $1$ stable range if we get $Ra+Rb=1$ for some $a,b ...
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Chern character in odd K-theory

I'm familiar with the definition of Chern character for a vector bundle. This leads to the definition of Chern character for $K$ theory (even theory) with values in even cohomology (the definition ...
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79 views

classifying space of a category

In his K-book, chapter IV, Weibel states the following as a “a straight forward application of Van Kampen's Theorem”: Lemma 3.4 Suppose that $T$ is a maximal tree in a small connected category ...
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Real topological K-theory of cyclic group

Letting $C_n$ be the cyclic group on $n$ elements we know through the use of the Atiyah-Segal completion theorem that $$ K^*(BG) = \pi_*(KU)[[t]]/((t+1)^n -1) $$ where $\pi_*(KU)=\mathbb{Z}[u^{\pm ...
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Finding a ring $\Lambda$ such that $\det:K_1(\Lambda) \rightarrow \Lambda^\times$ is not injective

Given a ring $\Lambda$, let $\text{GL}(\Lambda)$ be the direct limit of the general linear groups $\text{GL}_n(\Lambda)$, and similarly let $E(\Lambda)$ be the direct limit of the subgroup of the ...
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47 views

Multiplicative structure on algebraic K-theory

Let $R$ be a commutative ring. Using Quillen's $+$-construction, it is relatively easy to see that the algebraic K-theory of $R$, $K_*(R)$, admits a graded commutative product $$K_i(R)\otimes K_j(R) ...
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Can $xy$ and $yx$ lie in different connected components of the group of invertible elements of an algebra?

What is an example of a Banach or $C^{*}$ algebra $A$ which has two invertible elements $x, y$ such that $xy$ can not be connected to $yx$ in $G(A)$, the space of invertible elements of $A$. A ...
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$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : the semi group $V(A)$ of equivalent projections (under Murray Von Neumann equivalence) in $M_∞(A)$ is ...
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Idempotents which are not Murray-von Neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Murray-von Neumann equivalent to $e^{*}$?
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Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
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Determining a homomorphism between projective module groups

In Milnor's Introduction to Algebraic K-Theory, he writes on page 7: Suppose that $\Lambda$ can be mapped homomorphically into a field or skew-field $F$... Then we obtain a homomorphism $j_*: K_0 ...
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135 views

Proof of Rim's Theorem (Milnor)

I'm currently learning Algebraic K Theory from Milnor's "An Introduction to Algebraic K Theory" and am having trouble understanding his proof of Rim's theorem using a "Mayer-Vietoris" exact sequence. ...
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31 views

Homotopy fixed points of connective K-theory

Let $ku$ be the $p$-completion of the connective complex K-theory spectrum. The group $\Bbb{Z}_p^\times\cong \Delta \times \Bbb{Z}_p$ acts on $ku$, where $\Delta$ is the cyclic group of order $p-1$. ...
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What is the purpose of K-Theory?

I have recognized that there is a theory called K-Theory in mathematics is used also for applications in mathematical physics. There is existing algebraic K-Theory and topological K-Theory. Are ...
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26 views

Functoriality in $K$-theory for $C^*$-algebras or Banach algebras

I'm trying to clear up some confusion I'm having over how one establishes functoriality in $K$-theory for $C^*$-algebras or Banach algebras. Let me stick to $K_0$. Given a *-homomorphism (or bounded ...
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Proof that $K^*(BG)=K^*(BT)^W$.

I was wondering if anyone had a refrence for the fact that $K^*(BG)=K^*(BT)^W$ for $G$ a compact connected lie group, $T$ a maximal torus, and $W$ the associated Lie group. I was able to derive this ...
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The computation of $K_2(\mathbb Q)$

I have a small question on Tate's proof of the structure of the group $K_2\mathbb Q$, as found in e.g., Milnor's book "Introduction to Algebraic $K$-theory". The proof goes by showing that the map ...
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81 views

How to compute the K-group of this affine scheme?

By Bott periodicity, we know that $$K_{0}^{top}(S^2)=\mathbb{Z} \oplus \mathbb{Z}$$ But $S^2$ is defined by the equation $x^2+y^2+z^2=1$. If we write this abstractly as ...
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Definition of $K_1(A)$ for a Banach algebra $A$

When defining $K_1(A)$ for a Banach algebra $A$, one may consider $\bigcup_{n\in\mathbb{N}}\{x\in GL_n(A^+):x\equiv I_n\mod M_n(A)\}$ and take the quotient by the component containing the identity, or ...
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89 views

concept of the classification of $C^\ast$-algebras, introduction/overview

I don't have a specific mathematical problem at the moment but nevertheless I hope, my question is suitable for math.stackexchange. I'm interested in $C^\ast$-algebras and I would like to begin with ...
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1answer
75 views

Some questions about Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras

I have some questions about Joachim Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras, which is found in this paper. For this post, let us adopt the ...
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51 views

Analysing Exact Sequence

I have the following exact sequence $\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$. From here I want to conclude that ...
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A problem on isomorphic sections of locally trivial vector bundles

Let $E=(E,\pi,X)$ be a locally trivial vector bundle over a compact Hausdorff space $X$. Let $\Gamma(E)$ be the set of all sections in E. I am trying to prove that $E$ is ismorphic to the trivial ...
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K-theory,proper class,set,isomorphism types

Define $K$ as the free abelian group with generators $[A],[A'],[A'']$,the equivalence classes of isomorphism types, modulo $[A]=[A']+[A'']$ where $0\to A'\to A \to A''\to0$ is a s.e.s. of modules in ...
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117 views

K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the following book: Rordam, Larsen and Laustsen. I am having a problem with the the Exercise 3.4, which is: Let $X$ be any compact Housdorff space. In the ...
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What is the definition of polygonal loop?

How to prove "every loop can be uniformly approximated arbitrarily closed by a quotient of polynomial loops?" In Blackadar's book, the proof is completed with the help of "polygonal loops," but what ...
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119 views

Morita equivalence and KK-theory

Let $A,B,C$ be $C^\ast$-algebras. Suppose $B$ and $C$ to be strongly morita equivalent. Then $KK(A,B)\cong KK(A,C)$. Could someone provide a reference or proof of this fact? I guess the ...
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71 views

What is the Grothendieck group of finitely generated $R[G]$-modules?

Let $R$ be a ring with unity, $G$ a finite group and $R[G]$ the group ring. What is the definition of the Grothendieck group of finitely generated $R[G]$-modules? How is this connected to the ...
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Why is the K-theory of $X$ product of reduced $K$-theory and $\mathbb{Z}$

The reduced $K$-theory of $\tilde{K}(X)$ of the based space $X$ is the kernel of $d:K(X)\to \mathbb{Z}$, where $d$ is induced by $d:Vect(X)\to\mathbb{Z}$ that sends a vector bundle to the dimension of ...
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Computing the Grothendieck group of affine space.

For a Noetherian scheme $X$ the Grothendieck group $K(X)$ is defined as the free abelian group on coherent sheaves quotiented by the equivalence relation $\mathscr{F}=\mathscr{F}'+\mathscr{F}''$ for ...
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What is the quotient $\mathbb{T}^3/\mathbb{Z}_2$?

What is the quotient space $\mathbb{T}^3/\mathbb{Z}_2$, When $\mathbb{Z}_2$ acting on 3-dimensional torus $\mathbb{T}^3 := \mathbb{R}^3/\mathbb{Z}^3$ by sending $x$ to $-x$? Does anyone know how to ...
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1answer
67 views

Quillen's K-theory and ring homomorphisms

I am a beginner in algebraic K-theory and I want to make sure that I understand the following correctly: Let $f:A \to A'$ be an isomorphism of commutative rings. Denote by $\mathcal{P}(A)$ (resp. ...
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60 views

What is the 'Hom-description'?

I am trying to learn about the 'Hom-description' of the class group $Cl(A)$ of an $R_K$-order $A$ in $K[G]$ where $K$ is a number field with ring of integers $R_K$ and $G$ is a finite group. I've ...
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Exercises in Topological K-Theory (Atiyah)

I'm currently working through Michael Atiyah's K-Theory. The main problem I'm finding with it is that it does not have any exercises. Does anyone have a good collection of exercises to go along with ...
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When does a smooth projective variety X have a free Grothendieck group

Let $X$ be a smooth projective variety (e.g. Grassmannians). Since $X$ is smooth, the groups $G_0(X):=K_0(CohX)$ and $K_0(X):=K_0(VectX)$, the Grothendieck groups of coherent sheaves of modules on $X$ ...
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A question on algebraic K-theory- show that $K_1(R)\cong R^\times$ if $R$ is a field.

If $R$ is a ring we write $K_1(R)$ for the abelian group $K_1({\rm category\; of\; finitely\; generated\; projective\;R-modules})$. Swan's 'Algebraic K-Theory' defines $K_1$ in terms of the usual ...
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Algebraic results using lower K-theory as a blackbox

There is an algebraic K-theory seminar at my school and we are struggling to find applications from areas other than topology. We'd like a nice statement like "If X then Y" whose proof makes ...
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Holomorphic $K$-theory

$K$-theory is traditionally defined for arbitrary compact Hausdorff spaces. If instead we require the base to be a complex manifold and work with only holomorphic vector bundles, in what ways would ...
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106 views

$K$-theory for non-compact manifolds

It seems that usually (by which I mean, in every source I've looked at) people define the group $K^0(X)$ for $X$ compact Hausdorff. Sometimes they later extend this definition to all locally compact ...
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K-theory product trivial on $K^1$ due to stability?

Let $K^0$ be the topological complex vector bundle $K$-theory functor. On the one hand, we have that the product on $K^0(\Sigma X)$ vanishes by a Mayer-Vietoris argument. On the other hand, we define ...
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Higher homotopy groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
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vector bundles on $X \times S^2$ built via clutching functions

Let $X$ be a compact Hausdorff space and $E \to X$ a (complex) vector bundle over $X$. We can build a vector bundle on $X \times S^2$ by clutching functions, i.e an automorphism $f:E \times S^1 \to E ...
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Why is $K_{0}(C(\mathbb{D}))\rightarrow K_{0}(C(\mathbb{T}))$ injective?

There is the restriction map $\pi:C(\mathbb{D})\rightarrow C(\mathbb{T})$ where $\mathbb{D}$ is the closed unit disk and $\mathbb{T}$ is the unit circle. Why is $\pi_*:K_0(C(\mathbb{D}))\rightarrow ...