All Questions
Tagged with noncommutative-geometry oa.operator-algebras
143 questions
3
votes
1
answer
190
views
A possible spectral characterization of commutative $C^*$ algebras
Let $A$ be a $C^*$ algebra. Assume that
the spectrum $Sp(a_1a_2\ldots a_{n-1}a_n)$ is unchanged as a set after a permutation of $a_i$'s. (unless possible emerge or removing 0 from the spectrum)
Does ...
- 356
6
votes
0
answers
126
views
How obtain the right definition of smooth elements in a $C^*$-algebra?
In Alain Connes' $C^*$-algèbres et géométrie différentielle (an English translation is here,), for a $C^*$-algebra $A$, we consider a $C^*$-dynamic system $(A,G,\alpha)$, where $G$ is a Lie group and $...
- 9,009
0
votes
1
answer
144
views
Is a NC sphere a (one point) compactification of a NC plane?
Inspired by this question About noncommutative sphere and inspired by the fact that the classical sphere is the one point compactificatiin of $\mathbb{R}^2$ we ask the question below:
Is the non ...
- 356
4
votes
1
answer
275
views
What are the norms of the generators of the standard Podleś sphere?
Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations
\begin{equation*}
\begin{split}
&a=a^*,~ ...
- 9,009
13
votes
0
answers
573
views
Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
0
votes
0
answers
157
views
Dependence of functional integral on the function space
In physics, the following functional integral is considered
\begin{gather}
Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf ))
\end{gather}
It is usually said that the integration is performed ...
- 593
7
votes
2
answers
1k
views
Is there a notion of point in noncommutative geometry?
It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
- 136
2
votes
1
answer
381
views
Lattices and noncommutative algebras in noncommutative geometry
This a question that I've asked in mathematics stack exchange without having received any response :
I am interested in the relation between lattices and noncommutative algebras in the context of ...
- 136
3
votes
1
answer
155
views
Is a compact set of extreme points contained in a compact face?
I have run into the following question in convex analysis, which I haven't found answered in the literature:
Suppose that $K$ is a "nice-enough" non-compact convex subset of a Hausdorff ...
- 135
-3
votes
1
answer
325
views
Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras
A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor.
They form a category with usual structures.
Question. Is this category equivalent to the category of $C^*$ algebras?
...
- 356
3
votes
0
answers
178
views
What's the purpose of the operator $(\Delta^{-1}+\lambda)^{-1}$ in Tomita-Takesaki modular theory?
I was reading Tomita-Takesaki modular theory (from all the books, and articles), the goal is to relate a von Neumann algebra $\mathcal{A}$ with its commutant $\mathcal{A}'$ on a Hilbert space $\...
- 433
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
6
votes
1
answer
166
views
Every element of a $W^*$-algebra is a linear combination of exponential unitaries?
I am trying to understand a proof in this paper (specifically theorem 5.4). In it, a fact is used that every element of the $W^*$-algebra $A$ is a linear combination of exponential unitaries.
I've ...
- 63
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
1
answer
194
views
Another formula for the Schwinger term — problems with a calculation
$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...
- 9,330
2
votes
0
answers
88
views
Non-existence of idempotent via evaluation of higher order cocycle on a tuple of idempotents
The Kaplansky conjecture and Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra $...
- 356
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
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
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
10
votes
1
answer
811
views
What is the significance of the Jiang Su algebra in classification of C$^*$ -algebras?
Something I've been thinking about for a while that I'm not sure I understand is why $\mathcal{Z}$ stability, as opposed to say $\mathcal{O}_\infty$-stability or even $\mathcal{K}$-stability is so ...
- 183
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) $.
Could you tell me ...
- 595
5
votes
0
answers
270
views
Kernels of completely positive maps
In the excellent book “$C^*$-algebras and their automorphism groups” by Pedersen there are results on the left ideals $L_\phi$ associated to states $\phi$ on a $C^*$-algebra $A$ and more results in ...
- 1,143
3
votes
0
answers
185
views
Non commutative Teichmuller theory
Perhaps the first example in Teichmuller theory is the following proposition:
Proposition: Let $1<r<R$. Then two annular region $U_r=\{z\in \mathbb{C}\bigm|1<|z|<r\}$ and $U_R=\{z\in \...
- 356
12
votes
1
answer
2k
views
Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
- 6,853
2
votes
0
answers
101
views
Construction of non-commutative torus using ergodic action of $\mathbb{T}^{n}$
It is well known that non-commutative torus can be constructed using universal C* algebra, by n unitary elements and twisted relations. It can also be constructed using ergodic action of torus group $\...
- 523
3
votes
0
answers
55
views
Dense subalgebra that is closed under unbounded derivation on noncommutative torus
Let $A_{\theta}$ be the noncommutative torus, we can define:
$$
A^{\infty}_{\theta}\mathrel{:=}\left\{\sum_{n,m\in\mathbb{Z}}a_{n,m}U^{n}V^{m} \,\middle\vert\, a_{n,m}\in S(\mathbb{Z}^{2})\right\}
$$
...
- 523
4
votes
1
answer
126
views
Why Der($A_{\theta}$) is spanned by two elements only?
In the work of Connes and Marcolli, on page 20, it state that:
Just as in the classical case of a (commutative) manifold, what ensures that the derivations
considered are enough to span the whole ...
- 523
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
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
1
vote
2
answers
444
views
Fredholm $C^*$-algebras
Let $H$ be a Hilbert space. A vector subspace $W\subset B(H)$ is called a Fredholm subspace if there is an upper bound for the absolute value of Fredholm index of all Fredholm operators $T$ in $W$.
...
- 356
2
votes
1
answer
164
views
Hilbert module over $C_0(\Lambda)$ as space of continuous sections of HIlbert bundle
Let $\Lambda$ be a manifold and $p:H\to\Lambda$ a continuous Hilbert bundle with $H(\lambda):=p^{-1}(\lambda)$. Suppose $\Gamma_0^0(\Lambda)$ is the space of continuous sections vanishing at infinity ...
- 123
5
votes
1
answer
217
views
Noncommutative symmetric spaces
I am recently studying ergodic actions of Lie groups acting on Riemannian symmetric spaces. Since I am also interested in operator algebras, it makes me wonder if there are some very natural ...
- 1,879
3
votes
0
answers
214
views
Reference on noncommutative PDE
I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...
- 31
6
votes
0
answers
242
views
For what kind of $C^*$ algebra $A$ every normal element $y\in A$ has a normal lift for every given surjective $C^*$ morphism $\phi:B\to A$
Is there a terminology for the following property of $C^*$ algebra $A$:
For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ...
- 356
2
votes
1
answer
272
views
A question on quantum tori
Let $\mathbb T_\theta^2$ be quantum tori generated by two unitary operators $u,v$. can $u,v$ be finite dimensional?
- 1,879
5
votes
1
answer
519
views
Embedding of Cuntz algebras $O_2\subseteq O_3$?
The Cuntz algebra $O_n$ is the (universal) C*-algebra generated by n-isometries $s_1,...,s_n$ such that
$$\sum_{i=1}^n s_is_i^\ast =\mathbf{1}, \ \hbox{and}\ s_i^\ast s_j=\delta_{ij} \mathbf{1}\ (\...
- 155
9
votes
1
answer
237
views
A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm
Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide?
A somewhat similar question is discussed here.
- 356
4
votes
0
answers
241
views
Non-commutative analogue of a certain fact in differential geometry
In the literature, is there a non-commutative analogue of the fact that every Riemannian manifold whose isometry group has sharp dimension must be a constant curvature manifold?
- 356
3
votes
1
answer
170
views
Reduced compact quantum group and left and right multiplication
Let $(A,\Delta)$ be a compact quantum group in the sense of Woronowicz, and let $A_0$ be its dense Hopf subalgebra. We can construct from the Haar state $h:A \to \mathbb{C}$ an inner product
$$
\...
- 1,144
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
0
votes
0
answers
97
views
Smooth sections of finite dimensional bundle and covering space
Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with ...
4
votes
1
answer
341
views
On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra
Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$.
Obviously the singularities of this systems are just the idempotents of the ...
- 356
8
votes
1
answer
739
views
A question regarding Kadison-Kaplansky idempotent conjecture (A nearest group element $g$ to a nontrivial self adjoint unitary element u )
Edit: According to answer and comments by Prof. Valette we edite the question.
The Kadison Kaplansky conjecture says:
Kadison-Kaplansky conjecture: If $G$ is a torsion-free discrete group then $C^*_{\...
- 356
1
vote
0
answers
111
views
Commutative subalgebras of $B(H)$ whose all automorphisms are in the form of unitary conjugation
Let $H$ be a complex Hilbert space.
Is there a compact Hausdorff space $X$ such that $C(X)$ is embeded in $B(H)$ and for every homeomorphism $\alpha$ of $X$ there exist a unitary operator $u\in B(H)$ ...
- 356
4
votes
0
answers
114
views
Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra
Let $H$ be a Hilbert space and $X$ be a compact Hausdorff space with a homeomorphism $\alpha: X \to X$. Assume that $C(X)$ is a commutative sub algebra of $B(H)$, namely $C(X)$ is embedded in $B(H)$...
- 356
0
votes
1
answer
212
views
Quantum (group) version of ${\mathbb Z}^n$?
As we know there are quantum analogue of tori called quantum tori generated by noncommuting operators $(A_1,\dots,A_n)$ with $A
_iA_j=A_jA_ie^{2\pi i\alpha}$ where $\alpha$ is a irrational number as a ...
- 1,879
6
votes
2
answers
297
views
Finite-dimensional Hilbert $C^*$-modules
Does there exist a classification, or characterization, of finite-dimensional Hilbert $C^*$-modules? More generally, does there exist a characterization of countable direct sums of finite-dimensional ...
- 993
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
5
votes
1
answer
228
views
Zero divisors in compact quantum groups
Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the polynomial Hopf algebra Pol$(\...
- 969