All Questions
Tagged with c-star-algebras kt.k-theory-and-homology
78 questions
0
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44
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How to define a family of Hilbert $A-B$-bimodules $ \pi \ : \ M \to X $, parametrized by a $C^*$-algebra $X$?
Let $A$ and $B$ two $ C^* $ - algebras.
I would like to define a functor $ X \to \mathrm{Bimod}_{A,B} (X) $ which associate to any object $X$, the set of isomorphism classes of a family of Hilbert $A-...
- 595
5
votes
0
answers
114
views
Realize a $K_0$-group homomorphism by a unital $\ast$-homomorphism
This question is inspired by Exercise $7.7$ in *An Introduction to $K$-theory for $C^*$-algebras (available here). Given a unital AF-algebra $A$ and another unital $C^*$-algebra $B$ that has ...
- 307
7
votes
0
answers
160
views
Maps in the Künneth theorem for K-theory of C*-algebras
The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...
- 2,998
3
votes
0
answers
84
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Cohomological counterpart of the K-theory of the Roe C*-algebra in non-periodic systems
In crystalline insulating quantum systems the dynamics of electrons is governed by a Schrödinger operator which is periodic with respect to a Bravais lattice $\Gamma\cong \mathbb{Z}^d$ and whose ...
- 31
11
votes
0
answers
379
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Why are projectionless $C^*$-algebras important (Kadison's conjecture)
It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
- 211
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0
answers
71
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Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group
Let $A$ be a unital $C^*$-Algebra with non-trivial $K_0$ group. Define $CA = \{f\in C\big([0, 1], A\big)\,\vert\,f(0) = 0\}$. It can be shown that $CA$ is homotopic equivalent to the set $\{0\}$ and ...
- 307
2
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0
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202
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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
70
views
The $K_0$ mapping of an automorphism induced by a derivation
Let $\mathfrak{A}$ be a unital $C^*$-Algebra and let $\delta: \mathfrak{A} \rightarrow \mathfrak{A}$ be a linear map that is not constantly zero and satisfies, for every $A, B\in\mathfrak{A}$, $\delta(...
- 307
6
votes
0
answers
109
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$C(X)$-Fredholm operators and Atiyah-Jänich theorem
Let $X$ be a compact Hausdorff topological space and consider the Hilbert space $\ell^2(\mathbb N)$. As shown here, any $T\in C(X,\ell^2(\mathbb N))$ induces a $C(X)$-Fredholm operator
$$
\begin{array}...
- 61
4
votes
0
answers
107
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KK-theory for commutative $C^*$-algebras
The Gelfand--Naimark theorem tells us to regard noncommutative $C^*$-algebras as "noncommutative function spaces". In that spirit $K$-theory the Grothendieck group of "noncommutative ...
- 1,144
3
votes
1
answer
290
views
Approximation of continuous projections on a manifold by smooth idempotents
Every continuous vector bundle on a closed smooth manifold $M$ has a smooth structure. On the other hand, every vector bundle $E$ is the image of a trivial bundle $M\times\mathbb{C}^n$ under some ...
- 1,903
0
votes
0
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119
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Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant
Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$.
Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$
So $A$ is a Banach algebra.
Can we equip $A$ ...
- 356
2
votes
0
answers
108
views
Special case of Elliott's Theorem
Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...
- 581
10
votes
2
answers
689
views
Stable rank one and corners of $C^\ast$-algebras
Thanks to a result of Herman and Vaserstein in [3], Rieffel's notion of stable rank [4] coincides with the Bass stable rank [1] for every $C^\ast$-algebra $A$: we denote it by $\mathrm{sr}(A)$ and we ...
- 660
3
votes
0
answers
166
views
"Somewhat connected" spaces or algebras
Before we state our question, we give a motivational simple example:
Put $X$ for disjoint union of two circles. However $X$ is not a connected space but it has an open dense subset $U$ such that $U$ ...
- 356
2
votes
0
answers
124
views
Representation of $C^{*} (S_{\infty})$
I was wondering what is the group $C^{*}$-algebra of infinite symmetric group?
Mainly, I was trying to calculate the k-theory of $C^{*}$-algebra of infinite symmetric group and I found K-Theory of $C^{...
- 581
2
votes
1
answer
352
views
K-Theory of $C^{*}(X)$
I'm new to K-Theory for $C^{*}$-algebra and $C^{*}$-algebra of groups.
If $X$ is the group of finite support bijections of natural numbers then what is the K-Theory of $C^{*}(X)$?
I was planning to ...
- 581
0
votes
1
answer
109
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How to define an equivariant Kasparov's KK-theory map?
I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct ...
- 325
1
vote
0
answers
69
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A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov
Let $ G $ be a locally compact group.
Let $ H $ and $ K $ be two normal subgroups of $ G $.
In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$
where, $$ F(G/H,G/K) = KK^{G/H} ( ...
- 325
3
votes
0
answers
129
views
Another way for defining $K_1$ group for a C*-algebra
Thank you for answering my question.
I have another question about the $K_1$ group. As you may know, some books define the $K_1$ group like below:
Also, it defines the $K_0$ group for an arbitrary C*-...
- 581
8
votes
2
answers
246
views
Example of a C*-algebra whose $K_1$ is uncountable
We know that if $A$ is a separable $C^{*}$-algebra then $K_1(A)$ is countable.
Can anybody give an example of a C*-algebra for which $K_1(A)$ is uncountable?
- 581
5
votes
1
answer
342
views
Question on Cuntz' proof of Bott periodicity
I am reading the presentation of Cuntz' proof of Bott periodicity for $C^*$-algebras in Wegge-Olsen (Thm. 11.2.1). Here one considers the short exact sequence of $C^*$-algebras
$$0 \longrightarrow \...
- 9,853
4
votes
1
answer
214
views
What are the generator for $K_1(C(\mathbb{T})\otimes\mathbb{K})$?
I've been working computing several K-groups associated to some $C^*$-algebras involved with my master's thesis, however I've just got stucked finding some generators for $K_1(C(\mathbb{T})\otimes\...
- 143
3
votes
0
answers
156
views
Left and right topological K-theory of Banach algebras
Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette.
The definition is based on ...
- 356
4
votes
0
answers
333
views
Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
- 356
2
votes
1
answer
246
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$C^*$-algebras appearance in study of Lie groupoids and differentiable stacks
I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...
- 6,023
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
11
votes
0
answers
401
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The term "absolute geometry"
My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
- 1,338
2
votes
0
answers
116
views
Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 993
4
votes
1
answer
277
views
Producing $K$-homology cycles from $KK$-cycles
For two unital (separable) $C^*$-algebras $A$ and $B$, let $(H,\rho,F)$ be a $KK$-cycle in the sense of Kasparov, or in the sense of Wikipedia :)
I wonder if there us a natural way to "forget" the ...
- 993
8
votes
1
answer
356
views
Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra
In his paper "Comparing analytic assemby maps", J. Roe considers a proper and cocompact action of a countable group $\Gamma$ on a metric space $X$. He constructs the Hilbert $C^*_r(\Gamma)$-module $L^...
8
votes
1
answer
572
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
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
6
votes
0
answers
233
views
Group $C^*$ vs group von-Neumann algebras
Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
- 9,853
7
votes
1
answer
373
views
Generator of $K_0(C_0(\mathbb{C}))$
$\newcommand{\C}{\mathbb{C}}\newcommand{\Z}{\mathbb{Z}}$
I know from Bott-periodicity that $K_0(C_0(\mathbb{C}))\simeq \Z$, is there any easy way to compute an explicit generator of $K_0(C_0(\mathbb{C}...
- 173
5
votes
0
answers
125
views
When is K0 of a C* algebra finitely generated?
Are there workable conditions that imply that $K_0(A)$ is finitely generated, for a noncommutative unital C* algebra $A$. My actual question is in fact much more specific than this:
Is it possible ...
2
votes
2
answers
373
views
Finitely generated $K_0$ of $C^*$-algebras
Let $A$ and $B$ be $C^*$-algebras. Let $A_0$ be a dense *-subalgebra of $A$, and let $B_0$ a dense *-subalgebra of $B$. Assume that $A_0$ is isomorphic to $B_0$.
Finally, assume that $K_0(A)$ is ...
- 213
2
votes
0
answers
125
views
Computing the $K$-theory of the free inverse semigroup $C^*$-algebra
A typical example where I know the $K$-theory of the quotient, and wonder if this could help to compute the $K$-theory of the extension. (Or other way round: knowing one part out of three ones.)
I ...
- 685
8
votes
1
answer
725
views
Role of the UCT problem in classification theory for C*-algebras
Elliott's program for nuclear C*-algebras deals with the problem of classifying nuclear C*-algebras by K-theoretical invariants. A major open question in this context is the UCT problem.
A separable ...
- 741
4
votes
1
answer
157
views
Geometric Motivation for Hilbert $C^*$-Bimodules
I'm trying to get an understanding of Hilbert $C^*$-bi-modules from a geometric point of view. As is well-known, we have that
i) Commutative unital $C^*$-algebras correspond to compact Hausdorff ...
- 167
3
votes
1
answer
232
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unital embedding into the coner $C^*$-algebra
Let $A$ be a $C^*$-algebra, we denote with $V(A)$ the semigroup of Murray-von Neumann equivalence classes of projections in matrices over $A$ (as usual). In https://arxiv.org/pdf/math/0310340.pdf, ...
- 1,188
1
vote
1
answer
269
views
description of a map in KK-theory
The following situation is given: Let $A$ be a unital, separable, nuclear $C^*$-Algebra, $i:\mathbb{C}\to A$ the unital embedding. All $C^*$-algebras are considered as trivially graded. Consider the ...
user62639
3
votes
1
answer
159
views
K-group properties of quasi-diagonal $C^*$-algebras
Let $A$ be a separable unital quasidiagonal $C^*$-algebra.
What can be said about the $K$-theory of $A$, for example some properties? Especially, are there some criterions to decide whether or not $K_*...
4
votes
0
answers
389
views
Künneth formula for $C^*$ algebras, equivalent condition for full generality
I'm crrently reading the paper about the Künneth-theorem for $C^*$-algebras: http://msp.org/pjm/1982/98-2/pjm-v98-n2-p15-s.pdf and I'm trying to understand remark 4.9. I henceforth asumme that $A$ is ...
- 81
5
votes
1
answer
370
views
explicit description of the product map in K-theory
Let $A$ and $B$ be unital $C^*$-algebras. Let $u\in M_n(A)$ be a unitary representing an element $[u]\in K_1(A)$ and $p\in M_m(B)$ be a projection representing an element $[p]\in K_0(B)$. Then the ...
user62639
17
votes
1
answer
515
views
K-theory space of a C*-algebra
Let $A$ be a unital C*-algebra.
Let me define its "$K$-theory space" to be the image of its $K$-theory spectrum under the functor $\Omega^\infty:$ Spectra $\to$ Spaces.
I denote the $K$-theory space ...
- 43.2k
5
votes
1
answer
269
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Equivalence of two pictures of odd $K$-theory
One can show that two functors $K^0$ and $K_0(C(-))$ from the category of compact topological spaces to the category of abelian groups are naturally equivalent. The first one is topological $K$-theory ...
- 9,330
7
votes
1
answer
454
views
Relation between the UCT and Künneth ($C^*$-algebras)
I have a (maybe dumb) question about the relation between the Künneth theorem and the Theorem-universal coefficient theorem (UCT for short) in $KK$-theory (for the setting see "The Künneth theorem and ...
user62639
4
votes
1
answer
175
views
commutativity of a diagram in cohomology of $C^*$-algebras
The setting is the same as in my last question commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras) :
Let $A$ be in the bootstrap category (=N in the other ...
user62639
5
votes
1
answer
222
views
commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras)
I have a question about a proof in Rosenberg and Schochet's paper "the Künneth theorem and the Universal Coefficient Theorem for Kasparov's generalized K-functor", proposition 2.6. First of all, the ...
user62639