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Questions tagged [k-theory]

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 Cayley transformation does to a $*$-homomorphism

We let $A$ be a $C^*$ algebra. We consider a grading on $A=C_0(\Bbb R) $ by even and odd functions whilst a grading on $M:=M_2(M_\infty(A))$ by diagonal and off diagonal elements given by grading ...
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Computing $K$-theory elements in a $C^*$ algebra $A$

Let $A$ be a unital $C^*$ algebra. Let $p,q$ be projections in $M_n(A)$. Then $[p]-[q]$ defines an element in $K_0(A)$. Now consider the matrices, the projections, $$ \left[ \begin{pmatrix} 1-p &...
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$K$ theory of $C^*$ algebra is different to algebraic $K$ Theory?

Is the $K_0$ group for a $C^*$ algebras $A$ same as that for the $K_0$ group of ring $A$ from algebraic $K$ theory? We assume $A$ is unital (I am not sure if this matters), i.e. what is an example ...
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$KK$-groups definitions

In $K$ Theory of Operator Algebras, page 144 and a paper by Skandalis, page 35 the $KK$ groups are defined differently: Both are triples $(E,\phi, F)$ but Skandalis does not require the condition ...
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K Theory of $C^*$ algebras I, Higson's notes

Let $B$ be a $C^*$ algebra, and $A(B)$ denote the algebra of bounded continuous functions from $[1,\infty)$ to $B$. Let $I(B)$ be the ideal of functions which vanish at infinity. Let $Q(B)$ be the ...
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$K$ Theory of Operators II, Higson's notes

What I want to understand is how the map defined in proof of Proposition 3.17 is an inverse of the defined map. Proposition 3.17: For any (ungraded) $C^*$-algebra $A$, the map $$\Phi:K_0(A) ...
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$K$-Theory of operators I, Higson notes

I am having trouble understanding the following statement: 3.20 Proposition, pg44: Let $D$ be a symmetric, odd graded elliptic operator on a graded vector bundle $S$ over a compact manifold. The ...
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What is the support of a vector bundle complex?

I'm doing a class on K-Theory, and I'm confused about what the support of a complex of vector bundles is. Consider the following complex: $$0\to V_1\to V_2\to \dots\to V_n\to 0$$ Assume that all ...
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Special elements in the $C^*$ algebra $A \otimes \mathcal{K}$.

Context: Let $A$ be an ungraded (not necessarily unital) $C^*$ algebra. $\mathcal{K}$ space of compact bounded operators on an infinite separable graded Hilbert space $H=H_0 \oplus H_1$. Consider ...
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Understanding the map from $K_0(A)$ to homotopy class of maps,

In page 42, lectures on operator $K$-theory The writer defines a map from $$\Phi: K_0(A) \rightarrow [\mathcal S,A\otimes \mathcal K]$$ Notation: $\mathcal S:=C_0(\Bbb R)$ and $\mathcal K$ is a ...
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$K$ theory of vector bundle, $K(V)$, is a $K(X)$ module

This is on page 67, definition 5.6 when the author defines the Thom homomorphism: $V$ is a hermitian vector bundle over a compact space. $K(V)$ is a $K(X)$ module. How does $K(X)$ act on $K(V)$? ...
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Induced $K$-theory maps between $C^*$ algebras.

So here is a construction outlined in Higson's note on index theory, pg 46 Let $\mathcal{K}$ denote a $C^*$ algebra of graded compact operators on a graded hilbert space $H=H_0\oplus H_1$. Let $...
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$C^*$ algebra of compact operators as a direct limit of matrix algebras?

This was written in Page 92, of Higson's Analytic $K$-Homology book. Let $H$ be a hilbert space. The $C^*$ algebra $K(H)$ of compact operators is the direct limit of a sequence $$M_2(\Bbb C) \...
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Short exact sequence in $K_0$ of non unital rings.

Where may I find some reference for explaining the basics of $K_0$ for non unital rings? I was reading this pdf by Max Karoubi. He stated two results. Let $A$ be any $k$-algebra, where $k$ is a ...
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$K_1(\mathbb{Z})$ is covered by diagonal matrices with only $\pm 1$ entries

Let me give some definitions first: We define the inclusion $GL(n,\mathbb{Z})\to GL(n+1,\mathbb{Z})$ as $A\mapsto \begin{pmatrix}A&0\\0&1\end{pmatrix}$ We call $E(n,\mathbb{Z})$ the subgroup ...
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K- theory of stably projectionless C* algebras

could anyone give me an example of a stably projectionless C*-algebra with non-zero $K_0$ group?
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K theory of projectionless C*-algebras

Is it possible to have a projectionless C*- algebra with non trivial K-theory? If so what would be such an example? I can't come up with any. p.s. By projectionless I mean non-unital aswell.
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An isomorphism of vector bundles over a manifold, $K(X)$,

Let $E_1, E_0$ be vector bundles over a manifold $X$. Let us suppose that $$E_1 - E_0 =0 \in K(X)$$ (I believe we also suppose $X$ to be compact so $K$-theory makes sense here.) Proposition: If $\...
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The complex topological $K$-theory spectrum is not an $H\mathbb{Z}$-module.

Why is it true that the complex topological $K$-theory spectrum is not an $H\mathbb{Z}$-module? I mean, why the nontriviality of the first $k$-invariant implies the claim above?
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How to show that this short exact sequence does not split

Consider the Short Exact sequence $0\rightarrow C_0((0,1))\rightarrow C([0,1])\rightarrow \mathbb{C}\bigoplus\mathbb{C}\rightarrow 0$ where the map from $C([0,1])\rightarrow \mathbb{C}\bigoplus\...
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A doubt from Atiyah's K-Theory

This is a question regarding the following statement on pg 43 of Atiyah's K-Theory. Using our construction of $K$ it follows that, if $X$ is a space, every element of $K(X)$ is of the form $[E]-[F]...
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Operator K-theory and Topological K-theory

I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_\infty (...
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Grothendieck group of tensor products

For two unital rings $R,S$, one can form the ring $R\otimes_\mathbb{Z} S$ (this is the coproduct in the category of rings). I was wondering whether the following formula holds for the Grothendieck ...
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Two types of Grothendieck groups for rings

For a Noetherian ring $R$, there seem to be two versions of zeroth K-theory one can associate to it: $K_0(R)$ the Grothendieck group of the exact category of projective modules and $G_0(R)$ the ...
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Given a map $X \to \text{GL}_2(\mathbb{R})$ how do I determine a flat connection on this Riemann surface?

I need help determining the Euler class of this vector bundle $\phi:E\to X$. The base space is the torus $X = \mathbb{R}^2/\mathbb{Z}^2$ and the fiber over each point, $f^{-1}(x) \simeq \mathbb{R}^2$....
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Existence of Thom Class

In page 133, Theorem 8.5.5. (The Thom isomoprhism theorem) Let $\pi:V \rightarrow X$ be a complex vector bundle of rank $n$, over al locally comapct space $X$. Let $$ 0 \rightarrow \pi^* \...
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Defining Bott Class by relative $K$-theory

I am really confused with this construction of Bott Class in Page 127, Example 8.4.12 If $V$ is a complex vector space of dimension $n$, we form the complex $$ 0 \rightarrow \wedge^0 V \...
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Relative $K$-theory, definition

On pg125 on proving that that the concordance classes quotiented by the acyclic ones froms a group, there is a lemma: Lemma 8.4.5 Let $(E,F,f)$ and $(E,F,g)$ be two $K$-cycles on $(X,Y)$. Assume ...
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Why we call monomial matrices?

Currently, I am reading Milnor’s book on algebraic K-theory, where he defined a monomial matrix over commutative ring with 1 to be a matrix of the form PD, where P is a permutation matrix and D is a ...
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Problems on exercise 7.G in the book “K-Theory and C*-Algebras”

I have a lot problems on exercise 7.G in the book K-Theory and C*-Algebras by Wegge-Olsen. $\newcommand{\C}{\mathbb{C}}$ $X\subset \mathbb{C}$? As I know the character space of $C^*(u_1,u_2)$ is ...
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$K_0(R)$ is generated by $[R]$?

Let $R$ be a unital associative ring. Let $F$ be the free abelian group on the set of all isomorphism classes $[P]$ of f.g. projective $R$-modules $P$. Let $K_0(R)$ be the quotient of $F$ modulo the ...
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KO theory v.s. ko theory

It looks that there are different types of topological K-theories, with similar names but they are totally different outputs for the same input. The first theory is called the KO theory. There are ...
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Commuting Elements in Tensor Products of C*-Algebras

I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen. Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a ...
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Rings and categories with zero Grothendieck group

I am interested in examples of rings (or triangulated categories) that have zero Grothendieck group but are somehow still interesting. More example, for what rings $R$ is the category of finitely-...
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Clarifying how we can identify $SK_1(R)$ with set of path components

Let $R$ be a commutative Banach algebra. Let $E_n$ denote the group generated by $n\times n$ elementary matrices. $SK_1(R)$ denotes the kernel of the induced determinant map $K_1(R) \to R^{\times}$. ...
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Decomposition of $K_0$ with Whitehead group

Let $Wh_0(G) ={K_0(\mathbb Z[G])}/{\mathbb Z}$, where $\mathbb Z[G]$ is the group ring of $G$ over $\mathbb Z$. Weibel states that the augmentation map $f: \mathbb Z[G] \to \mathbb Z$ induces the ...
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When $\mathbb Z$ is a direct summand of $K_0(R)$

Suppose that there is a ring homomorphism $R \to F$ where $F$ is a field. I'm trying to verify that $\mathbb Z$ is a direct summand of $K_0(R)$. We have an induced ring homomorphism $(K_0(R), \...
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Is there an example of a non-zero projection in a C$^{*}$-algebra that is infinite but not properly infinite?

For clarification: Given a projection $p$ in a C$^{*}$-algebra $A$, we say $p$ is infinite if there is a projection $q\in A$ satisfying $q\lneq p \sim q$; we say $p$ is properly infinite if there are ...
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Is the geometric realization of a pointed category contractible?

Given a pointed category $X$, is $|N(X)|$ contractible ($N(X)$ is the nerve of $X$)? Or are there counter-examples?
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Definition of algebraic K-theory space

Let $(C,wC)$ be a Waldhausen category. The algebraic K-theory space is the loop space of the classifying space of the simplicial pointed category $wS_*C$, i.e. of the topological realization of the ...
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Degreewise cofibration in $S_nC$

Given a category $C$, we have $S_nC=Fun(Ar[n],C)$ and given $A,B\in Ob(S_n(C))$, (i.e. A,B are functors from $Ar[n]$ to $C$), then what does a morphism $f:A\to B$ in $S_nC$ mean by degreewise ...
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Understanding Adams spectral sequence and Pontryagin-Thom isomorphism intuitively

The question is about understanding Adams spectral sequence intuitively and some of the meanings of its relations. In Adams spectral sequence, $$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTG), \...
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Atiyah's K Theory, pg 4

On page 4, of Atiyah's $K$ theroy he stated Suppose $V,W$ are real f.d. v.s, $E = X \times V$ and $F= X \times W$ are corresponding vector bundles. Then any homomoprhism $\varphi:E \rightarrow F$ ...
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How to calculate the $K_0$ and $K_1$ groups for $A$

Let $A=\{f\in C([0,1],M_n)\mid f(0)$ is scalar matrix $\}$. Then find the $K_0(A)$ and $K_1(A)$. I am trying to use the SES $J \rightarrow A \rightarrow A/J$ where $J$ can be taken as some closed ...
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is K theory functor continous with respect to inverse limit?

l know that K functor is continous with respect to direct limit, how about inverse limit ?does inverse limit exist in general in the caegory of topological space(or C star algebra). is there a ...
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Was Atiyah's proof of the odd order (Feit-Thompson) theorem false?

I read last year that Atiyah thought he had found a proof of the odd order theorem of only 12 pages, using $K$-theory, and that people were trying to figure out if it was correct or not. But I never ...
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Generators of $K_0(C(\mathbb{C}P^1))$

$\newcommand{\C}{\mathbb{C}}\newcommand{\R}{\mathbb{R}}$ I know that $K_0(C(\mathbb{C}P^1))\simeq K^0(\mathbb{C}P^1))\simeq K^0(S^2)\simeq \mathbb{Z}^2$ (we're talking about complex K-theory). This ...
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Index map on the Hilbert A-module $A\otimes H$

I'm working on K-theory. Let H be an infinite dimensional separable Hilbert space and $A$ a $C^{\star}$-algebra. Let put $\mathcal{Q}(H):=\mathcal{B}(H)/\mathcal{K}(H)$ the Calkin algebra. I've ...
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Projections in the Cuntz algebra which have the same $K_0$ class

Assume that $e,f$ are two projections in the Cuntz algebra $\mathcal{O}_n$ , which have the same $K_0$ class. Are they necessarily Murray von Neumann equivalent ? The following post is the ...
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trying to understand the connecting homomorphism between K theory groups

the connecting homomorphism from $K_{1}(A/J)$ to $K_{0}(J)$ is defined by the composition $(j_{*})^{-1} k_{*}$ where $j$ is the inclusion of $J$ to the mapping cone $C_{\pi}$, which induce isomorphism ...