All Questions
14 questions
18
votes
0
answers
881
views
What is operator tmf?
One of the many wonderful things about K-theory, relative to other generalized cohomology theories, is that it can be defined for not-necessarily-commutative C*-algebras. The resulting construction, ...
- 118k
16
votes
1
answer
526
views
Equivariant Fredholm operators classify equivariant K-theory
Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm.
If $X$ is compact,
Atiyah-Jänich proved that
$$[X,\mathcal{F}]\...
- 673
10
votes
0
answers
325
views
H-space structure on the Calkin algebra
By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
- 7,637
8
votes
0
answers
493
views
Two pictures of K-theory and Bott periodicity
Let me recall the definition of the Bott periodicity isomorphism in the context of $C^*$-algebras. We take a (class of) projection $p \in M_n(A^+)$ and map it to the class of $M_n(A)$ valued loop $f_p$...
- 9,330
7
votes
2
answers
525
views
Integrality of the canonical trace and topology
Let $G$ be a discrete group and consider the reduced group C* algebra $C_r^\ast(G)$, viewed as an algebra of bounded operators on $\ell^2(G)$ by the regular representation. The canonical trace on $...
- 29.2k
6
votes
0
answers
181
views
Blocksum induces a unital H-space structure on the space of Fredholm operators
Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...
- 386
5
votes
1
answer
1k
views
Fredholm operators in $K$-theory?
Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ...
- 145
5
votes
1
answer
269
views
Equivalence of two pictures of odd $K$-theory
One can show that two functors $K^0$ and $K_0(C(-))$ from the category of compact topological spaces to the category of abelian groups are naturally equivalent. The first one is topological $K$-theory ...
- 9,330
5
votes
0
answers
232
views
The Segal Machine constructing spectra and topological $K$-Theory
I am currently looking into the Segal-Machine for constructing spectra. I am working with his original article . The first thing that confuses me is the spectrum that arises from a $\Gamma$-space is ...
- 51
4
votes
1
answer
507
views
A question on complex line bundle over $S^{2}$
Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$.
Assume that $\ell$ is a sub line bundle of ...
- 356
4
votes
0
answers
333
views
Baum Connes Conjecture [closed]
I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as it is formulated in topology to the conjecture as it is formulated in functional analysis. I ...
- 49
3
votes
0
answers
306
views
The Baum Connes Conjecture - the approach using spectra
In this article James Davis and Wolfgang Lück introduce a approach using spectra to formulate the Baum Connes Conjecture for a discrete group $G$. In order to do so, they construct a functor
$$\...
- 31
2
votes
0
answers
105
views
Multiplicativity of the analytic index (or of kernel bundle)
What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
- 1,173
2
votes
0
answers
254
views
isomorphism of Chern character in kk-theory
Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...
- 493