Questions tagged [noncommutative-topology]

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A topological concept dual to compactness

We say that a subset A in a topological space X is anti-compact if every covering of A by closed sets has a finite subcover. Clearly if X is Hausdorff then all anti-compact subsets of X are finite. ...
p modabberi's user avatar
13 votes
1 answer
1k views

What is the commutative analogue of a C*-subalgebra?

Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff ...
Rasmus's user avatar
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10 votes
0 answers
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Noncommutative condensed sets

Ignoring set-theoretic problems, we can see condensed sets as sheaves of compact Hausdorff spaces. Using Gelfand Duality we obtain an equivalence of categories \begin{align*} \mathrm{CHaus}^{\mathrm{...
Luiz Felipe Garcia's user avatar
8 votes
1 answer
270 views

Noncommutative Fredholm operators

Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely ...
ernest's user avatar
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7 votes
1 answer
321 views

Separability of the C*-algebra in the definition of K-homology

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book ...
truebaran's user avatar
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7 votes
0 answers
236 views

Non Commutative Hyperspaces

Let $X$ be a compact metric space. Recall that the hyperspace $2^{X}$ is the set of all non empty compact subsets of $X$ with the Hausdorff metric. Assume that $\mathcal{C}$ is the category of all ...
Ali Taghavi's user avatar
6 votes
2 answers
658 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 ...
Alexander Chervov's user avatar
6 votes
1 answer
467 views

What are the sub $C^*$-algebras of $C(X,M_n)$?

Let $X$ be a locally compact Hausdorff topological space, denote by $M_n$ the $C^*$-algebra of complex $n\times n$ matrices, by $C_0(X,M_n)$ the $C^*$-algebra of continuous functions on $X$ with ...
Uwe Franz's user avatar
  • 2,191
6 votes
1 answer
858 views

Sources for exact triangles in triangulated categories.

The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (of C*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides ...
Rasmus's user avatar
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5 votes
1 answer
864 views

Proper continuous image of metrizable space

Motivated by the following post, "Gelfand duality" and the fact that "a Hausdorff continuous image of a compact metric space is metrizable", we ask: What is a counter example of two locally ...
Ali Taghavi's user avatar
5 votes
1 answer
338 views

NonCommutative Baire theorem

The classical Baire theorem says that the intersection of a sequence of open dense subsets of $X$, is dense, if the space is compact Hausdorff. In the language of $C^{*}$ algebras this is ...
Ali Taghavi's user avatar
5 votes
0 answers
206 views

range of trace on projections: beyond rotation algebras

In a rotation algebra, $A_\theta=C(S^1)\rtimes \mathbb{Z}$, there is a tracial state $\tau$ coming from the invariant measure $\mu$ on the circle. There is a projection $p\in M_n( A_\theta)$ (we can ...
vap's user avatar
  • 330
5 votes
0 answers
240 views

"Definitive" Noncommutative Space

Let $Y$ be a (locally compact) non-Hausdorff topological space. I want to know if there is a necessary and/or sufficient condition for $Y=X/G$, that is, $Y$ is the orbit space of a locally compact ...
Andrew's user avatar
  • 59
4 votes
1 answer
311 views

A question on $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which satisfies each of the following equivalent conditions: All elements of $A$ are left zero divisor. All elements are right zero divisor. All elements are ...
Ali Taghavi's user avatar
4 votes
1 answer
205 views

Support projection vs closed support projection of a normal state in enveloping von Neumann algebra

I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding. Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
Sean's user avatar
  • 135
4 votes
0 answers
202 views

Extending Akemann's Non-Commutative Urysohn Lemma

Assume $A$ is a C*-algebra and $p,q\in A^{**}$ are compact projections. Can we always find $a,b\in A^1_+$ with $p\leq a$, $q\leq b$ and $||pq||=||ab||$? Note if $||pq||=1$ this is immediate, while ...
Tristan Bice's user avatar
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3 votes
0 answers
145 views

Closed containment of open projections in C*-algebras

For a C*-algebra $A$ and open projections $p,q\in A^{**}$, consider the following statements. $\overline{p}\leq q$ $p\leq q$ and there exists open $r\in A^{**}$ with $rp=0$ and $r\vee q=1$ $p\leq q$ ...
Tristan Bice's user avatar
  • 1,327
2 votes
1 answer
126 views

Is the dual of the category of unital C*-algebras concretizable?

The dual to the category of commutative unital C*-algebras is equivalent to the category of compact Hausdorff spaces, a concrete category. Can the dual to the category of unital C*-algebras also be ...
Alex Mennen's user avatar
  • 2,090
2 votes
1 answer
89 views

Approximation of unity by projectors

Let $A$ be a $\sigma$-unital $C^*$-algebra and $A_s:=A\otimes K$ its stabilization (where $K$ is the algebra of compact operators on a separable Hilbert space). Is it true that there exist an ...
user avatar
2 votes
1 answer
367 views

Homotopy groups of noncommutative spaces

In the approach to noncommutative geometry of Alain Connes any Hausdorff compact space $X$ is replaced by its algebra of complex valued continuous functions $C^0(X)$, and one regard general (that is, ...
user avatar
2 votes
0 answers
87 views

Computing norms of polynomials of operator in Hilbert space and generalized von Neumann inequality

Let $T$ be an operator $l^2({\mathbb{Z}_{\geq 0}}) \to l^2({\mathbb{Z}_{\geq 0}})$, $e_n \mapsto \sqrt{1 - q^{2(n+1)}}e_{n+1} $, where $0<q<1$. I want to compute $\|f(T,T^{*}) \|$ (operator ...
Invincible's user avatar
1 vote
0 answers
98 views

Formula for the KK-theory groups $KK(A, C(S))$

I am studying $C^*$-algebras and their KK-theory. Let $A$ be a (unital if you wish) $C^*$-algebra and $S$ be a compact Hausdorff space. I am interested in computing the KK-theory groups $KK(A, C(S))$, ...
Luiz Felipe Garcia's user avatar
1 vote
0 answers
76 views

Which groups may be obtained as $K$-homology groups?

Recently I asked the following question, about the separability of the underlying $C^*$-algebra in the definition of $K$-homology: mathoverflow.net/questions/181361 As far as I understood, in ...
truebaran's user avatar
  • 9,140
1 vote
0 answers
283 views

Bootstrap subcategory abelian?

In the book "K-Theory for Operator Algebras" by Bruce Blackadar, the exercise 23.15.8. on page 246 says: "Let KKN be the full subcategory of KK with objects in N. Show that KKN is abelian category by ...
John's user avatar
  • 11
0 votes
3 answers
361 views

Specific Reference? Noncommutative topology and C^* algebras [closed]

I stumbled across this "dictionary for noncommutative topology" http://planetmath.org/noncommutativetopology and I would be very interested in learning more on the subject, particularly I'd like to ...
Cstarg's user avatar
  • 11