# Tagged Questions

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 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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### 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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### 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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### $K_0$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...
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 ...