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2
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1answer
120 views
Examples of Eilenberg-type Swindles
I am compiling a list of 'swindles' in the style of the Eilenberg-Mazur swindle. I've already got some swindles in K-theory, the Mazur Swindle and the proof of the Cantor–Bernstein–Schroeder theorem. ...
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0answers
37 views
What properties of a C*-algebra are reflected in their $K_0$ groups?
Title says it all. I've been looking at K-groups of a few C* algebras. My very rough understanding is that these groups reflect (in some appropriate sense) the algebraic structure that we get when ...
2
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1answer
216 views
8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
Periodicity modulo 8 appears in the classification of real Clifford algebras $C\ell_{p,q}(\mathbb{R})$ (usualy refered to as the "Clifford Clock"), in real Bott periodicity and in the definition of a ...
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What is the grading of $x(x-1)R[x]$? Loopspace for Karoubi-Villamayor K-theory.
I am reading the chapter on Karoubi-Villamayor K-theory in Weibel's K-book. In particular he defines $\Omega R=(x^2-x)R[x]$ for a ring. This will eventually lead to a model for the loopspace
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2answers
65 views
Characteristic polynomial of the unique automorphism of the zero module
Is there any convention which makes sense of the characteristic polynomial of the unique automorphism of the zero module?
This might seem like an odd question but it matters to me. The background ...
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0answers
62 views
Products in $C^*$-algebra $K$-theory
Let $A_1$ and $A_2$ be unital $C^*$-algebras. If $p_1 \in M_{n_1}(A_1)$ and $p_2 \in M_{n_2}(A_2)$ are projections then $p_1 \otimes p_2 \in M_{n_1 n_2}(A_1 \otimes A_2)$ is also a projection, ...
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1answer
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Computing $K$ - theory groups of certain $C^{\ast}$- algebras
I am trying to show that the $K$-theory groups of the following $C^*$-algebra $A$ vanish:
Let $\mathcal{H}$ be a separable Hilbert space. Now consider the subalgebra of ...
4
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1answer
82 views
Operator K-theory and the unitary group
Suppose that $A$ is a C*-algebra whose unitary group is contractible (e.g. $B(H)$ or more generally the stable multiplier algebra of any C*-algebra). It is clear from the definition that $K_1(A) = ...
4
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1answer
169 views
Effect of permuting coordinates on K-theory
Let $A$ be a C*-algebra and let $f: \mathbb{R}^n \to \mathbb{R}^n$ be the linear map which permutes the coordinates via a permutation $\sigma$. There is an induced map $K_0(C_0(\mathbb{R}^n) \otimes ...
3
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1answer
135 views
Applications of Elliott's theorem concerning the classification of AF-algebras
An AF-algebra is a $C^* $-algebra which is the inductive limit of an inductive sequence of finite-dimensional $C^*$-algebras.
Elliott's theorem concerning the classification of AF-algebras says that ...
3
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1answer
191 views
What's the map $BU \times \mathbb{Z} \to \prod K(\mathbb{Z},n)$ representing the total Chern class?
Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total ...
