# 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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### Defintion of a real algebraic space in Atiyah's K-theory and reality

In Atiyah's paper "K-theory and reality", p. 370, there is an example of a "real" algebraic space. Given the complex projective space $X=P(C^n)$, one considers the standard line-...
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### Spin$^c$-structure necessary for K-orientation

In Atiyah, Bott, and Shapiro's Clifford Modules, Theorem 12.3, they prove that a Spin$(k)$-structure (resp. Spin$^c(2k)$)-structure gives a KO(K)-orientation on the associated vector bundle of rank $k$...
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### Showing $K_0(F)=\mathbb{Z}$

In Morel's "$\mathbb{A}^1$-algebraic topology over a field", Milnor-Witt K-theory is defined to be the graded ring $K_*^{MW}(F)$ generated by the symbols, $[u]$ of degree $1$ and $\eta$ of ...
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### How to construct an $AF$-algebra $N$ with $K_0(N)=\mathbb Q$?

On the paper i'm reading, it says letting $N=\lim M_2\otimes ...\otimes M_{n!}$ will do. I wonder why it doesn't simply let $N=\lim M_2\otimes ...\otimes M_n$. Take arbitary $p\in N\otimes M_n$, there ...
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### What is $K_1(B(H))$? (operator K-theory)

Let $H$ be a (separable) infinite-dimensional Hilbert space. Can someone give me a reference or a proof why the identity $$K_1(B(H)) = 0$$ is true? The more elementary the proof, the happier I am. ...
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### identity matrices stably equivalent?

I'm reading the K-theory chapter in Murphy's C*-algebras and Operator Theory book. He defines stable equivalence to be $p \approx q$ if there exists positive integer $n$ such that  \begin{pmatrix}...
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### Is there a paper that explains Kasparov's KK theory in English?

I need to learn Kasparov's KK theory but the original paper is written in Russian. G.G. KASPAROV, The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571-...
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### K-Theory, how to start [duplicate]

This semester I'm studing courses about representation theory, ring theory, algebraic topology and homological algebra, and this huge mix between algebra and topology made my think about other ways to ...
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### Prove that $v=vv^*v$ for a partial isometry $v$.

Let $A$ be a $C^*$-algebra and let $v$ be a partial isometry so that $v^*v$ is a projection. Show that $v=vv^*v$. The hint in the book recommended setting $z=(1-vv^*)v$ and then computing $z^*z$. I ...
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What is the typical example $\pi \colon X \to B$ of a surjective morphism between topological (or $C^{\infty}$) manifolds, i.e. locally euclidean space via continuous bijections, where path lifting ...
If $A$ is a commutative unital ring and $E$ is a finite rank projective $A$-module there is a surjective $A$-linear map $\phi: A^n \rightarrow E$, with kernel $F:=ker(\phi)$ and $F\oplus E \cong A^n$ ...