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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8 votes
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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 ...
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3 votes
1 answer
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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 ...
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2 votes
0 answers
72 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 $A$ ...
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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. ...
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2 votes
1 answer
257 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 $\mathcal{H}$ ...
vand's user avatar
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0 answers
120 views

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, ...
Ronald J. Zallman's user avatar
3 votes
0 answers
149 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 ...
Sambo's user avatar
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2 votes
1 answer
402 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 $\mathbb{C} 1$) ...
JBrude's user avatar
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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 ...
Jake Wetlock's user avatar
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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 ...
Peg Leg Jonathan's user avatar
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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 $p \sim q$?
Peg Leg Jonathan's user avatar
1 vote
1 answer
362 views

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 ...
Andrew NC's user avatar
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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 $E$.) ...
geometricK's user avatar
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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 ...
Just dropped in's user avatar
9 votes
1 answer
406 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 ...
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10 votes
2 answers
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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 ...
Julien's user avatar
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9 votes
2 answers
295 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 ...
Ali Taghavi's user avatar
2 votes
1 answer
133 views

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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5 votes
1 answer
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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 ...
JP McCarthy's user avatar
2 votes
1 answer
256 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?
Peg Leg Jonathan's user avatar
2 votes
2 answers
273 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 ...
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5 votes
1 answer
366 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)\...
Diego Martinez's user avatar
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 ...
Ali Taghavi's user avatar
8 votes
1 answer
358 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 ...
Norbert's user avatar
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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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0 votes
1 answer
158 views

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 \...
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3 votes
1 answer
119 views

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 answer
237 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 $...
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2 votes
2 answers
206 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 $...
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6 votes
0 answers
107 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(...
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4 votes
1 answer
229 views

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 ...
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4 votes
1 answer
250 views

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 ...
Black's user avatar
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5 votes
1 answer
219 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 $(...
AlexE's user avatar
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0 answers
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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?
Dick Johnson's user avatar
4 votes
1 answer
454 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 ...
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4 votes
1 answer
200 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 ...
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2 votes
1 answer
172 views

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(\...
Black's user avatar
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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 ...
Peg Leg Jonathan's user avatar
3 votes
1 answer
249 views

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 votes
1 answer
201 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 $...
ABB's user avatar
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2 votes
1 answer
147 views

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 ...
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1 vote
0 answers
99 views

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 ...
Ali Taghavi's user avatar
3 votes
2 answers
248 views

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) ...
user avatar
6 votes
2 answers
494 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 ...
worldreporter's user avatar
3 votes
0 answers
165 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 $U$ ...
Ali Taghavi's user avatar
12 votes
1 answer
436 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 ...
Tomasz Kania's user avatar
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5 votes
1 answer
319 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? ...
Peg Leg Jonathan's user avatar
14 votes
3 answers
1k 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 ...
Mirco A. Mannucci's user avatar
5 votes
0 answers
201 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 ...
David Roberts's user avatar
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