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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).

K-theory is the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.K-theory involves the construction of families of K-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes.

439 questions
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K-theory computation for algebra of bounded continuous functions on $[0,\infty)$

I want to compute the K-theory of $C_{b}[0,\infty)$, the algebra of bounded, uniformly continuous functions on $[0,\infty)$, by considering the exact sequence $0\rightarrow C_{b,0}(\bigcup_{n\geq 0}(2n-1,2n))\rightarrow C_{b}[0,\infty)\rightarrow…
cyc
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$ C_0(\mathbb{R}^2)$, $C(\mathbb{D})$, $C(\mathbb{T})$ and the index map

Consider the short exact sequence $$0 \longrightarrow C_0(\mathbb{R}^2) \overset{\varphi}\longrightarrow C(\mathbb{D}) \overset{\psi}\longrightarrow C(\mathbb{T}) \longrightarrow 0$$ I need to show that 1) $K_1(C(\mathbb{D}))=0$ 2)…
Miep
  • 167
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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 its restriction to the component of the…
D. Huang
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Show that $[u]_1$ belongs to $Im(K_1(\varphi))$ if and only if ...

Let $\varphi: A \to B$ be a surjective $^*$-homomorphism between unital $C^*$-algebras A and B, and let $u$ be a unitairy in $\mathcal{U}_n(B)$. I want to show that $[u]_1$ belongs to $Im(K_1(\varphi))$ if and only if there exist a natural number $m…
Miep
  • 167
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Elements in $K_0(A)$

Let A be a $C^*$-algebra, unital or not. I want to show that each element in $K_0(A)$ is of the form $$[p]_0 - \bigg[ \begin{pmatrix} 1_n & 0_n \\ 0_n & 0_n \\ \end{pmatrix} \bigg]_0$$ for some projection $p \in M_{2n}(\tilde A)$ satisfying the…
Miep
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Show that $K_0(A)$ is a countable abelian group when $A$ is a separable $C^*$-algebra.

I want to show that $K_0(A)$ is a countable abelian group when $A$ is a separable $C^*$-algebra and I know that this is the case when $A$ is a unital separable $C^*$-algebra as this has been shown earlier where the proof I have seen is similar to…
Miep
  • 167
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Let $X$ be an element in $\mathbb{E}(A,B)$, and there is a homomorphism between B and C then we can define an element in $\mathbb{E}(A,C).$

I am self-studying KK-theory. I came up with the following lemma: Let A, B and C be graded $C^{*}$-algebras, and $\phi: B \rightarrow C $ be an even *-homomorphism, and $X=(E, \pi , T)\in \mathbb{E}(A,B)$, the class of all Kasparov A-B-module. Then…
Emilly
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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 textbook giving a proof or a counterexample？and how…
user540663
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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 proved that the map $\delta$ :…
MacFly
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Fejer's theorem for Bundles

Fej$\mathbf{\acute{e}}$r's Theorem For Bundles: Let $X$ be a compact space and $L\rightarrow X$ be a line bundle with metric, giving rise to a sphere bundle $\sigma :S\rightarrow X$. Let further $F\rightarrow X$ a bundle with metric.Then every…
Bikram Banerjee
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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 is the definition of "polygonal loops?"
math112358
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$C_0(X)$ isomorphic to $C_0(X_1) \oplus C_0(X_2)$ when $X=X_1 \cup X_2$ for X locally compact Hausdorff

If X is a locally compact Hausdorff space and $X=X_1 \cup X_2$ where $X_1, X_2$ are disjoint open and closed subsets X, I want to show that $C_0(X)$ isomorphic to $C_0(X_1) \oplus C_0(X_2)$ I have that $K_0(A \oplus B) \cong K_0(A) \oplus K_o(B)$…
Miep
  • 167
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If $(E, \varphi, F) \in D(A, B)$, then $(E, \varphi, F)$ is homotopic to the 0-module.

I am reading proposition 17.2.3 Blackadar: If $ \varepsilon =(E, \varphi, T) \in \mathbb{D}(A, B)$, then $(E, \varphi, T)$ is homotopic to the $0$-module. I have some questions about the proof. We must find an element $ \bar{\varepsilon} =…
Emilly
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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 $[E]-[F]$, where $E,F$ are vector bundles over $X$. I…
user67803
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video lectures on topological K-theory?

I am trying to follow Karoubi's book on Topological K-theory and it would be great motivation if I had any video lectures to watch. Do you have any in mind? I would also appreciate any problem sets from a course on topological K-theory to guide my…
noname