All Questions
13 questions with no upvoted or accepted answers
20
votes
0
answers
827
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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
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
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
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
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}^\...
- 2,385
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
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
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
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
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
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