All Questions
16 questions
3
votes
1
answer
181
views
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras, is there any good notion of "normal bundle of $B$ in $A$"?
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras (maybe more restricted kind of star algebra), is there any good notion of "normal bundle of $B$ in $A$"? By a "...
- 31
4
votes
1
answer
133
views
A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$
Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
- 356
1
vote
0
answers
106
views
A locally convex $C^*$ algebraic structure on the disk algebra
A locally convex $C^*$ algebra is a locally convex topological vector space $A$ whose topology is generated by a familly of complete $C^*$ semi norm and $A$ is a $*$ algebra. Moreover all algebra ...
- 356
7
votes
0
answers
158
views
$C^*$ algebras whose nontrivial projections form a non empty compact connected set
Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set?
Is there an example of this situation such that ...
- 356
7
votes
3
answers
676
views
Noncommutative torus as a von Neumann algebra
Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...
- 9,330
4
votes
2
answers
254
views
$K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $*$-homomorphism which is injective on a dense $*$-sub-algebra of $A$. Can such a map have non-trivial kernel,...
- 969
2
votes
1
answer
124
views
Twisted canonical commutation relations
I am dealing with universal C*-algebra generated by $x,y$ with the following relations: $xy = qyx$, $x^{*}y = qyx^{*}$, $y^{*}x = qxy^{*}$, $x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$, $y^{*}y = q^2yy^{*} - (...
- 161
6
votes
2
answers
690
views
Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?
Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces.
So it is ...
- 24.9k
5
votes
1
answer
249
views
$C^{*}$-correspondences viewed as generalized endomorphisms
I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers ...
- 367
7
votes
2
answers
742
views
$H^{*}$ algebras as a generalization of $C^{*}$ algebras
Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties:
$\forall \lambda \...
- 356
2
votes
1
answer
386
views
Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes [closed]
Question
I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
- 123
1
vote
1
answer
331
views
An unconventional definition of the $ C^{*} $-algebraic reduced crossed product
Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system, i.e., $ A $ is a $ C^{*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by ...
- 307
2
votes
0
answers
209
views
Number of connected components of a $C^{*}$ algebra
Inspired by the concept in the following post
What are these compact sets called?
We introduce the following concept:
Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent ...
- 356
17
votes
2
answers
2k
views
Realisation of the noncommutative torus as a universal $ C^{*} $-algebra
One of the most basic examples in noncommutative geometry is the so-called noncommutative torus, denoted here by $ \mathbb{T}_{\theta} $. As far as I know, there are several equivalent constructions ...
- 9,330
4
votes
1
answer
568
views
Crossed product of a non unital C*-algebra
Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align*}
&\alpha:C_0(X)\rightarrow C_0(X)\\\
&\alpha(f)=f\circ T
\end{align*}
is an automorphism. ...
- 43
8
votes
3
answers
2k
views
Definition of a von Neumann algebra
Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in ...
- 37.8k