All Questions
Tagged with c-star-algebras oa.operator-algebras
597 questions
2
votes
0
answers
84
views
Are quasitrace extensions unique?
I'm trying to understand the basics of quasitraces on $C^*$-algebras. Using the terminology of Haagerup, given $n \geq 2$, an $n$-quasitrace $\tau$ on a $C^*$-algebra $A$ is a 1-quasitrace on $A$ ...
- 285
5
votes
1
answer
202
views
Relating different constructions of the universal compact quantum group
Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections.
...
user167952
2
votes
1
answer
316
views
Decomposition of Hilbert spaces via groups and algebras representations
Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...
- 23
3
votes
0
answers
196
views
Cuntz semigroups of basic C*-algebras
I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103):
"[...] $A_i$ is ...
- 285
2
votes
1
answer
452
views
Uniqueness of the direct sum of $C^*$ algebras as quotient of free products
Suppose that you have $A, B$ two unital $C^*$ algebras and let $A \ast B$ the reduced free product (I think that it is the reduced amalgamated product over the common $*$-subalgebra $\mathbb{C} 1$) ...
- 115
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
4
votes
1
answer
152
views
$\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$
Could you give an example of a unital simple $C^*$-algebra that $\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$?
- 581
4
votes
1
answer
101
views
Functional calculus for "pre-linear" regular operators on a Hilbert module
Let $E$ be a Hilbert module over a $C^*$-algebra $A$. Let $T\colon E\to E$ be a densely defined, unbounded $A$-linear operator. (In particular, the initial domain of $T$ is an $A$-submodule of $E$.) ...
- 1,903
2
votes
1
answer
448
views
The maximal tensor product is a continuous functor
I am trying to prove continuity of the maximal tensor product functor. I have a problem in the proof that I cannot see how to handle; If anyone could give me a clue on how to go on from here, I would ...
- 267
9
votes
1
answer
445
views
Reference request: Brown Ozawa and strong completely positive approximation property?
The notion of a $C^*$-algebra being nuclear has many equivalent characterisations. These are considered in the excellent, modern textbook $C^*$-Algebras and Finite-Dimensional Approximations by Brown ...
- 18.7k
10
votes
2
answers
688
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
9
votes
2
answers
298
views
Two inequalities in $C^*$ algebras
Under what conditions on a $C^*$ algebra $A$ we have the following inequality:
$$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$
The second identity which I am looking for is ...
- 356
2
votes
1
answer
279
views
The algebra of continuous functions on Cantor set
Let $C(K)$ be the algebra of continuous functions on Cantor set. Is it possible to prove that $C(K)$ forms an AF-algebra without Bratteli diagram?
- 581
2
votes
2
answers
302
views
Is $x \mapsto x \otimes 1$ $\sigma$-weakly continuous?
Let $M\subseteq B(H)$ be a von Neumann algebra. Is it true that the mapping
$$\psi: M \to B(H \otimes H): m \mapsto m \otimes \text{id}_H$$
is $\sigma$-weakly continuous? Here the $\sigma$-weak ...
user167952
5
votes
1
answer
417
views
Faithful traces on quasi-diagonal C*-algebras
Recall that a separable C*-algebra $A$ is quasi-diagonal if there are completely positive and contractive maps $\varphi_k \colon A \rightarrow M_{n(k)}$ such that $||\varphi_k(ab) - \varphi_k(a)\...
- 733
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
8
votes
1
answer
403
views
When a $C^*$-algebra is an ideal in its second dual?
I would like to know which $C^*$-algebras are ideals in their second duals?
There is a paper by S. Watanabe that claims in introduction that it is well known that a $C^*$-algebra is an ideal in its ...
- 1,697
0
votes
1
answer
261
views
Definition intertwiner of representations of compact quantum groups
Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $...
user167952
2
votes
2
answers
217
views
Kernel of intertwiner is invariant (compact quantum groups)
Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $...
user167952
6
votes
0
answers
108
views
Sufficient conditions for a map $\phi: M(A) \to M(B)$ to have strictly closed image
Let $A$ and $B$ be $C^*$-algebras with multiplier algebras $M(A)$ and $M(B)$. Are there any nice conditions that ensure that a strict (= norm-continuous + strictly continuous on bounded subsets of $M(...
user167952
5
votes
1
answer
245
views
Colimits of short exact sequences of C*-algebras
Assume I have an inductive system of short exact sequences of $C^{\ast}$-algebras (i.e., short exact sequences $0 \to A_n \to B_n \to C_n \to 0$ together with transformations from the $n$-th to the $(...
- 2,998
4
votes
1
answer
510
views
Strict topology and $*$-strong toppology on $B(H)$ coincide
In the paper Woronowicz - $C^*$-algebras generated by unbounded elements, I read that the $*$-strong operator topology on $B(H)$ and the strict topology on $B(H)$ coincide. I believe this means the ...
user167952
4
votes
1
answer
201
views
Is the unit ball of $A \odot B$ strictly dense in that of $M(A \otimes B)$?
Let $A$ and $B$ be $C^*$-algebras and let $A \otimes B$ their minimal tensor product and $M(A \otimes B)$ the associated multiplier algebra.
On $M(A \otimes B)$, we consider the strict topology which ...
user167952
0
votes
0
answers
88
views
Is A an amenable $C^{*}$-algebra?
Let $A$ be $C^{*}$-algebra. Suppose that, for any $\epsilon > 0$ and finite subset $F \subset A$, there are an amenable $C^{*}$-subalgebra $B \subset A$, contractive completely positive linear maps ...
- 581
-1
votes
1
answer
210
views
A commuting pair of isometries
Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$.
The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
- 4,058
6
votes
2
answers
556
views
Endomorphisms of the Cuntz algebra
Consider the Cuntz algebra $\mathcal{O}_n$ with $n \geq 2$ and let $\text{End}(\mathcal{O}_n)$ be the set of all (unital) $\ast$-endomorphisms of $\mathcal{O}_n$. I was wondering if there exists an ...
- 741
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
12
votes
1
answer
461
views
Maximal ideals of ultraproducts of full matrix algebras
Let $\mathscr U$ be a non-principal ultrafilter over the natural numbers. Let $M_{\mathscr U}$ be the ultraproduct of all full matrix algebras $M_n$ along $\mathscr U$. This is a C*-algebra that is ...
- 11.3k
5
votes
1
answer
385
views
hereditary C*-subalgebra of a non-elementary simple C*-algebra
A is said to be elementary if A is isomorphic to some $K(H)$ or $M_n$.
A C*-subalgebra $B$ is said to be hereditary if for every $0≤a≤b∈B$ we have $a∈B$.
I wanted to know that is this statement true?
...
- 581
16
votes
3
answers
2k
views
Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?
Many interesting C*-algebras can be realized as convolution algebras over a groupoid, an idea introduced in 1980 by Jean Renault (this entry in nLab provides plenty of context to the general approach ...
- 7,853
5
votes
0
answers
220
views
Pushout of $C^*$-algebras using generalised morphisms
There is a known construction of pushout of $C^*$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $*$-homomorphisms. Jensen and Thomsen in their book ...
- 35.4k
4
votes
1
answer
139
views
Characterization of simple C*-algebras via GNS representations
Let $\mathfrak{A}$ be a [separable] unital C*-algebra and let $Q$ be a dense subset of the state space of $\mathfrak{A}$. Suppose that for each $f\in Q$ the associated GNS representation is faithful. ...
- 73
3
votes
1
answer
119
views
Universal representations of quotient C*-algebras
Suppose that $\mathfrak{J}$ is a closed ideal of a C*-algebra $\mathfrak{A}$. Let $(\pi_u, H_u)$ be the universal representation of $\mathfrak{A}$. Is there a way to use these data to describe the ...
- 73
4
votes
0
answers
253
views
Inflating the double dual of a C*-algebra (matrix algebra of double dual)
in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})\cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can't really see it. Actually, it is enough for me ...
- 267
4
votes
1
answer
332
views
Normal linear functionals on bicommutants of C*-algebras
I am going through the proof of the Sherman-Takeda theorem and Fillmore's book "A User's Guide on Operator Algebras" seems to have a nice approach, but something seems off to me:
We need to ...
- 267
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
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
8
votes
1
answer
332
views
The double dual of the unitization of a $C^*$-algebra
I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the ...
- 267
7
votes
1
answer
479
views
Characterisation of finite dimensional C*-algebras?
$\DeclareMathOperator\Spec{Spec}$Let $A$ be a finite dimensional $*$-algebra over $\mathbb C$.
(Namely, an associate algebra equipped with an involution $*:A\to A$ satisfying $(ab)^*=b^*a^*$ and $(\...
- 43.2k
1
vote
1
answer
184
views
Need reference for $\phi(Z)=Z'$ if and only if $\Phi: \operatorname{Prim}(Z')\to \operatorname{Prim}(Z)$ is injective
Let $A$ and $B$ be $C^{\ast}$-algebras with centers $Z$ and $Z'$ respectively. Let $\phi:A \to B$ be surjective $C^{\ast}$-morphism. Then
$\phi(Z)=Z'$ if and only if the map $\Phi: \operatorname{Prim}...
- 1,115
2
votes
2
answers
291
views
If either $A$ is exact or $B$ is nuclear then every closed ideal of $A\otimes_{min}B$ is of the form $A \otimes _{min}J$ for some ideal $J$ of $B$
From one of the talks I attended long back, I vaguely seem to remember the following fact:
Let $A$ and $B$ be $C^{\ast}$-algebras. If either $A$ is exact or $B$ is nuclear then every closed ideal of $...
- 1,115
2
votes
0
answers
305
views
Ideals of maximal tensor product of $C^{\ast}$-algebras
Let $A$ and $B$ be $C^{\ast}$-algebras. It is well known that maximal tensor product of simple $C^{\ast}$-algebras need not be simple. So basically the ideal structure of $A\otimes_{max}B$ does not ...
- 1,115
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
3
votes
1
answer
146
views
Modular and primitive ideals of $C_{0}(X,A)$
Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\...
- 1,115
2
votes
2
answers
214
views
Commutative C*-rings
Let us consider the unital commutative $C^*$-algebra $C[0,1]$. We say $A\subseteq C[0,1]$ forms a C*-subring if it satisfies the following conditions:
1- $A$ is an involutive unital subring (closed ...
- 4,058
0
votes
0
answers
47
views
An algebraic property that makes a per-C*-algebra complete
Let $A$ be a normed *-algebra with $\|x^*x\|=\|x\|^2$. Suppose that for every subset S of A, the left annihilator
${\displaystyle \mathrm {Ann} _{L}(S)=\{a\in A\mid \forall s\in S,as=0\}\,}$
is ...
- 4,058
12
votes
0
answers
814
views
Why do some tricks in homological algebra work over the category of C*-algebras?
The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's ...
- 1,084
3
votes
0
answers
130
views
Question about a paper on approximate identities
I am currently reading this paper on approximate identities of ternary Banach algebras. Assume that $(A, [.,.,.])$ is a ternary Banach algebra. A bounded net $(e_{\alpha}, f_{\alpha})$ is said to be ...
- 1,115