All Questions
21 questions
6
votes
0
answers
158
views
Questions about the $K$-theory of the algebraic standard Podleś sphere
Given $\theta \in \mathbb{R}$ irrational, the $K$-theory of the smooth noncommutative $2$-torus $C^\infty_\theta(\mathbb{T}^2)$ is well understood in relation to that of the corresponding $\mathrm{C}^\...
- 2,375
2
votes
0
answers
202
views
The trigonometric $C^*$-algebra
The trigonometric $C^*$-algebra is the universal $C^*$-algebra generated by $\mathcal{G}=\{x,y,z\}$ subject to relations \begin{align}x^2=x=x^*, &\quad y^2=y=y^*\\ [x, z]=y, &\quad [y,z]=...
- 356
2
votes
0
answers
90
views
The group of quasi unitary elements of a (simple) Banach algebra
For a Banach algebra $A$ with invertible group $G(A)$ we define the following group:
$$QG(A)=\{u\in G(A)\mid \;\text{the mapping}\; a\mapsto u^{-1} a u \;\text{is an isometry}\}$$
What is an ...
- 356
8
votes
2
answers
208
views
Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras
There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an ...
- 293
2
votes
0
answers
147
views
About the algebraic structure of the $G$-equivariant $KK$-theory
Let $ G $ be a second countable locally compact group.
Let $ A $ and $ B $ be two $G$-$C^*$-algebras.
Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $.
Could you tell me ...
- 595
3
votes
0
answers
166
views
"Somewhat connected" spaces or algebras
Before we state our question, we give a motivational simple example:
Put $X$ for disjoint union of two circles. However $X$ is not a connected space but it has an open dense subset $U$ such that $U$ ...
- 356
9
votes
0
answers
364
views
Geometric motivation behind the Fredholm module definition
If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
- 993
8
votes
1
answer
570
views
Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras
In Higson and Roe's Analytic K-homology, for a unital $C*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor.
Roughly, the group $K_0(A)$ is given by the ...
- 969
7
votes
0
answers
174
views
How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?
My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...
- 233
7
votes
1
answer
219
views
$*$-algebras, completions, and $K$-theory
What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...
- 993
4
votes
0
answers
247
views
Dense subalgebra of continuous functions with same K -theory
Suppose $X$ is a compact metric space. Is there a good candidate for a dense subalgebra $A\subseteq C(X)$, such that the inclusion induces an isomorphism in $K$-theory?
For example, if $X$ was a ...
- 410
4
votes
0
answers
398
views
Bott-type projections in $C^*$-algebras
Let $A$ be a unital $C^*$-algebra and $a\in A$. If $aa^*+1$ is invertible in $A$ then the element
$$\beta(a)=(aa^*+I)^{-1}\left(\begin{array}{cc}aa^* & a \\a^* & I\end{array}\right)$$
is an ...
- 41
8
votes
1
answer
422
views
K theory for pre $C^*$-algebras
In noncommutative geometry when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the ...
- 9,330
7
votes
0
answers
435
views
K theory as the fundamental group
There are several ways in which one can define $K$-theory for $C^*$-algebras: for $K_0(A)$ group two aproaches: algebraic (using idempotents) and topological (using projections, i.e. self-adjoint ...
- 9,330
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
4
votes
1
answer
362
views
$K$-Theory of finite dimensional Banach algebras
Is there a finite dimensional Banach algebra $A$ for which $K_{0}(A)$ is a finite group?
I asked this question in MSE but I received no answer
https://math.stackexchange.com/questions/1624250/...
- 356
3
votes
1
answer
293
views
Is the square diagram of index and exponential maps in $K$-theory of $C^*$-algebras anti-commutative?
Assume we have a $3\times 3$ grid with rows and columns being short exact sequences of $C^*$-algebras.
This gives a grid of 6-term exact sequences: 3 "horizontal" sequences and 3 "vertical" sequences,...
- 1,284
0
votes
0
answers
410
views
A noncommutative vector bundle
We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative ...
- 356
8
votes
2
answers
836
views
Projective modules over noncommutative tori?
It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...
- 493
6
votes
1
answer
550
views
A generalized K- theory via generalized idempotents
Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...
- 356
20
votes
0
answers
827
views
Can we define spectral triples using the language of rigged Hilbert spaces?
The traditional mathematical approach to quantum mechanics,
as developed by von Neumann, is based on Hilbert spaces and unbounded self-adjoint operators.
Another approach, which more closely resembles ...
- 37.8k