All Questions
14 questions
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
$$[X,\mathcal{F}]\...
- 673
6
votes
1
answer
200
views
Coarse index of Dirac operator on $\mathbb{R}$
Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index
$$\text{Ind}(...
- 1,903
6
votes
2
answers
487
views
Induction theorems for finite-dimensional complex representations of infinite groups
Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
- 976
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
2
answers
862
views
Hilbert $C^*$-modules and approximate units
Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...
- 93
4
votes
1
answer
272
views
Fredholmness of an operator-valued Toeplitz operator
Let $f$ be an invertible element of $C({\mathbb{T}}; C_b(r,1))$, that is, there exists a $f^{-1}\in C({\mathbb{T}}; C_b(r,1))$ such that for all $z\in {\mathbb{T}}$, $f(z)f^{-1}(z)=1$ in $C_b(r,1)$.
...
- 105
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
1
answer
290
views
Approximation of continuous projections on a manifold by smooth idempotents
Every continuous vector bundle on a closed smooth manifold $M$ has a smooth structure. On the other hand, every vector bundle $E$ is the image of a trivial bundle $M\times\mathbb{C}^n$ under some ...
- 1,903
3
votes
0
answers
295
views
Tensor product of compact operators on Banach modules
Let $A$ and $B$ be Banach algebras. Consider a right Banach $A$-module, $E$, and a right Banach $B$-module, $F$, as well as a Banach algebra morphism $\pi\colon A\to\mathcal L_B(F)$ into the bounded $...
- 320
3
votes
0
answers
129
views
Equivariant $K$-homology with $G$-compact support
Let $G$ be a discrete countable group and let $A$ be $\sigma$-unital $G$-$C^*$-Algebra. For a proper locally compact Hausdorff $G$-space $X$ the equivariant $K$-homology with $G$ compact support and ...
- 31
3
votes
0
answers
128
views
Stable homotopy equivalence
Let $\alpha:A \rightarrow B$ be a *-homomorphism of $C^*$-algebras. Then $\alpha$ ist a stable homotopy equivalence if there exists a $*$-homomorphism $\beta: B \otimes \mathcal{K} \rightarrow A \...
- 31
2
votes
1
answer
323
views
Ideal spanned by matrix units isomorphic to compact operators
Hello,
Assume we have $(n+1)$ isometries $S_1,...,S_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum_{i=1}^{n+1}S_iS_i^*=I, S_i^*S_j=0$ (i.e. $S_i$ are the generators of the ...
- 23
2
votes
0
answers
116
views
Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 993