Questions tagged [c-star-algebras]
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
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Matrix coefficients of a compact quantum group
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz).
Definition: A corepresentation matrix of $(A, \Delta)$ is a matrix $a=(a_{i,j}) \in M_n(A)$ such that
$$\...
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Finite compact quantum groups
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
3
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Is restriction to the center an open map?
Given a type one $C^*$-algebra $A$, its center $Z$ acts by scalars on each irreducible representation space. Mapping a representation to its central character yields a continuous map from the ...
2
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0
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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$ ...
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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.
...
2
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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}$ ...
0
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0
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Criteria for $(V \otimes W)^* \cong V^* \otimes W^*$ in Banach spaces
Let $V$ and $W$ be Banach spaces.
$V^* \otimes W^*$ embeds into $(V \otimes W)^*$ (projective tensor product). I am looking for criteria for it to be an isomorphism.
If $V$ and $W$ are $C^*$-algebras, ...
3
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0
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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 ...
2
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1
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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$) ...
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Invertible elements of the Hopf algebra quantum $SU(2)$
Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see
https://en.wikipedia.org/wiki/Compact_quantum_group
(Note that on the ...
2
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0
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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 ...
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$\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$?
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Is there a Hilbert space approach to commutative probability theory on locally compact spaces?
I was recently made aware (thanks to the answers on Why does Riesz's Representation Theorem apply in quantum mechanics?) that the $C^*$ algebra approach and the Hilbert space approach to quantum ...
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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$.) ...
2
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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 ...
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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 ...
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2
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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 ...
9
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2
answers
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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$$
The second identity which I am looking for is ...
2
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1
answer
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If $(\text{id}_A\otimes \text{ev}_x)(z)= 0$ for all $x \in X$. Do we have $z=0$?
Let $A$ be a $C^*$-algebra (not necessarily unital). Let $X$ be a compact Hausdorff space. We can consider the minimal $C^*$-tensor product $A \otimes C(X)$. On this space, we can consider the slice ...
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Reference request for Deterministic $\subset$ Random $\subset$ Quantum
I hope this post is on topic as a reference request.
I have seen somewhere the idea of (and saw it written just like this):
$$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$
I am ...
2
votes
1
answer
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The algebra of continuous functions on Cantor set
Let $C(K)$ be the algebra of continuous functions on Cantor set. Is it possible to prove that $C(K)$ forms an AF-algebra without Bratteli diagram?
2
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2
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Is $x \mapsto x \otimes 1$ $\sigma$-weakly continuous?
Let $M\subseteq B(H)$ be a von Neumann algebra. Is it true that the mapping
$$\psi: M \to B(H \otimes H): m \mapsto m \otimes \text{id}_H$$
is $\sigma$-weakly continuous? Here the $\sigma$-weak ...
5
votes
1
answer
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Faithful traces on quasi-diagonal C*-algebras
Recall that a separable C*-algebra $A$ is quasi-diagonal if there are completely positive and contractive maps $\varphi_k \colon A \rightarrow M_{n(k)}$ such that $||\varphi_k(ab) - \varphi_k(a)\...
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$C^*$ algebras whose nontrivial projections form a non empty compact connected set
Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set?
Is there an example of this situation such that ...
8
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1
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When a $C^*$-algebra is an ideal in its second dual?
I would like to know which $C^*$-algebras are ideals in their second duals?
There is a paper by S. Watanabe that claims in introduction that it is well known that a $C^*$-algebra is an ideal in its ...
2
votes
1
answer
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Peter-Weyl theorem (compact quantum groups)
I'm reading the paper Notes on compact quantum groups. In this paper, the following theorem is proven:
Question: Why is the marked equality true?
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1
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158
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Direct sum of representations of a compact quantum group
Let $(A, \Delta)$ be a compact quantum group and $\{(H_\alpha, v_\alpha)\}$ be a collection of representations of $A$. That is,
$$v_\alpha \in M(B_0(H_\alpha) \otimes A); \quad \quad(\text{id}\otimes \...
3
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Show commutativity of a diagram involving multiplier $C^*$-algebras
Let me recall the following fact:
If $A$ is a $C^*$-algebra and $\pi: A \to \mathcal{B}(\mathcal{H})$ is a faithful non-degenerate representation, then we can explicitely realise the multiplier ...
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1
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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 $...
2
votes
2
answers
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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 $...
6
votes
0
answers
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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(...
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Constructing intertwiners between representations of compact quantum groups
Consider the following paper by Van Daele en Maes Notes on compact quantum groups. For convenience of the reader, here is a picture of the relevant section:
(1) How is compact operator defined in ...
4
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1
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Is there a C*-algebra whose Pedersen ideal is not proper?
In [1, 5.6.3] Pedersen states without proof or reference that there are non-unital C*-algebras whose Pedersen
ideal is the whole algebra.
Does anyone know where can I find such an example?
Is it ...
5
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1
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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 $(...
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0
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A noncontinous algebra map between Banach algebras
What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
4
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1
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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 ...
4
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1
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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 ...
2
votes
1
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Minimal components of the translation action on the Stone–Čech compactification
$\newcommand\Cb{C^\text b}$Let $\Cb(\mathbb R)$ be the C*-algebra formed by all bounded, continuous, complex valued functions on $\mathbb R$.
Consider the action $\tau $ of $\mathbb R$ on $\Cb(\...
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0
answers
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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 ...
3
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1
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Identifying the multiplier $C^*$-algebra $M(C_0(X) \otimes B)$
Is there an accessible proof for the following fact?
If $A=C_0(X)$ with $X$ locally compact Hausdorff and $B$ is a
$C^\ast$-algebra then $M(A\otimes B)$ is the set of bounded strictly
continuous ...
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1
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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 $...
2
votes
1
answer
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Extending $*$-morphisms to the multiplier algebras
I'm reading the following fragment in the paper "Notes on compact quantum groups":
While I'm familiar with the multiplier algebra (constructed via double centralizers) and its universal ...
1
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0
answers
99
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Classification of all groupoids $G$ whose automorphism group is in bijective correspondence the automorphism group of $C^*_\text{red}(G)$
Is there a terminology (and a classification) for all groupoids $G$ for which all automorphisms of $C^*_\text{red}G$ are induced from a groupoid automorphism of $G$. (A groupoid automorphism has ...
3
votes
2
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Convolution of functionals on compact quantum group
Let $\mathbb{G}= (A, \Delta)$ be a ($C^*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^*= B(A, \mathbb{C})$ obtains a product
$$\omega_1*\omega_2:= (\omega_1\otimes \omega_2) ...
6
votes
2
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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 ...
3
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0
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"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 $U$ ...
12
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1
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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 ...
5
votes
1
answer
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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?
...
14
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3
answers
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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 ...
5
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0
answers
201
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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 ...