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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Definition of second topological $K$-group of a Banach algebra

The question is a about the definition of the second topological $K$-group of a Banach algebra $A$. I was reading a text of Alain Valette (Prop. 3.3.7) where he proves that $$ K_1(SA) \cong \pi_1(\...
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Unitary element in an AF $C^*$-algebra can be approximated by sequence of unitaries

Let $A$ be a unital AF $C^*$-algebra. Write $A=\overline{\bigcup_{k\in \mathbb{N}}A_k}$ where each $A_k$ is a unital (with the same unit of $A$) finite dimensional $C^*$ subalgebra. Suppose $u\in A$ ...
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prove that elements in $K_1(A)$ coincide

Let $A$ be a unital $C^*$-algebra $u\in A$ unitary and $s\in A$ isometry. I already proved that $sus^*+(1-ss^*)$ is an unitary. Why is $[u]_1=[sus^*+(1-ss^*)]_1\in K_1(A)$? Basic definitions: ...
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Compute the positive part of $K_0(A)$ where $A$ is a simple AF algebra

I'm trying to understand the following example from my lecture notes: Define $A_n=M_{F_n}(\Bbb{C})\oplus M_{F_{n+1}}(\Bbb{C})$ where $F_n$ defined by $F_1=1, \ F_2=2, \ F_{n+2}=F_{n}+F_{n+1}$, i.e., ...
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Bott projection as $K_1$ class

Consider the Bott projection (described in Exercise 5.I of Wegge-Olsen's book $K$-theory and $C^*$-algebras) given by $b(z)=\frac{1}{1+|z|^2}\begin{pmatrix} 1 & \bar{z} \\ z & |z|^2 \end{...
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30 views

Classifying Unitaries on the Circle with K-Theory

Let $S^n$ be the $n$-sphere. I'm trying to understand the "meaning" of the $\mathbb{Z}$ factors in $$ K_0(C(S^{2n+1}))\cong\mathbb{Z}$$ and $$ K_0(C(S^{2n}))\cong\mathbb{Z}\oplus\mathbb{Z}$$ So $S^n$...
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128 views

Direct limit of certain $C^*$ algebras is simple

Let $X$ be a compact Hausdorff space. Let $(x_n)$ be a sequence in $X$.Assume $X$ has no isolated points. Define $A_n = C(X, M_{2^n}(\mathbb{C}) )$ and define $\phi_{n+1,n} : A_n \to A_{n+1}$ by $$\...
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definition of Kasparov modules, what is a degree 1 operator?

I read the book "K-theory for operator algebras" and I have a question about the definition of Kasparov modules for graded $C^*$-algebras $A$ and $B$. Definition: Let $A$ and $B$ graded $C^*$-...
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128 views

Results from $K$-theory which could be used to motivate an undergraduate to study it

From the notes on $K$-theory from Allen Hatcher, one notices that $K$-theory was used to prove that the only division algebras over $\mathbb{R}$ are the real, complex, quaternion and octonion algebras ...
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Algebraic K-theory: induced maps

Let $f:A \to B$ be a homomorphism of rings. One way to define algebraic $K$ theory ($K_0$ group) is to consider the Grothendieck group of isomorphism classes of all finitely generated projective ...
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30 views

K-theory of $C_0(X)$

Suppose that $X$ is some contractible space. I want to determine the K-theory of $C_0(X)$, i.e. the continuous functions on $X$ which vanish at infinity. But I do not know where to begin.
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stability of $K_0$ ($C^*$-algebras), question about the tensor product $K(H)\otimes A$.

I have a small question about stability of $K_0$: If $A$ is a $C^*$-algebra and $H$ is a separable infinite dimensional Hilbert space then $$K_0(A)\cong K_0(K(H)\otimes A),$$ where $K(H)$ denotes the ...
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The reduced norm map $\operatorname{Nrd}: K_1(A)\to K^\times$

Let $K_1(A)$ be the Grothendieck $K_1$-group of the category of finitely generated projective $A$-modules where $A$ is a central simple $K$-algebra. I'd be grateful if someone could tell me if the ...
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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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22 views

$K_0$ 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 $\mathbb{Z}$...
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25 views

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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30 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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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 $\pi_*(KU)=\mathbb{Z}[u^{\pm 1}]$...
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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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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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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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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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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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1answer
85 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 $$X:=Spec\mathbb{R}[x,y,z]/(x^2+...
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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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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
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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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1answer
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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 $K_0(\mathcal{T}...
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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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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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131 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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76 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 ...