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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4
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0answers
89 views

Drinfeld center of a tensor category

Firstly, apologies for the imprecise language, I'm a physicist trying to understand anyonic excitations from the lens of category theory. If I have a category (say $\operatorname{Rep}(\mathbb{Z}_2)$) ...
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83 views

The topological Grothendieck group of a mixed category

I have recently become interested in the notion of mixed categories, as well as the topological Grothendieck group of their derived categories. I am still very new to the field. For that, I would like ...
4
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1answer
274 views

$\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$

Let $A$ be a $C^*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}_A(E)$ be an adjointable operator satisfying $t=t^*$. Is it true that $$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
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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 $A$ ...
0
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1answer
102 views

On ultraweak continuity [closed]

Let $A$ be a C*-algebra, $f$ be a representation of $A$, $F$ be the universal representation of $A$, and $g=f \circ F^{-1}$. For an ultraweakly continuous linear functional $w$ on $f(A)$, $w\circ g$ ...
7
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2answers
525 views

Amenable action intuition

Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
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1answer
113 views

Abelian twisted reduced group C*-algebra

Let $G$ be an abelian discrete group. Then is $C_r^*(G, \sigma)$ abelian?
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85 views

Quasidiagonal C*-algebras

Let $A$ be a nuclear $C^*$-algebra satisfying UCT condition. Then under what assumptions $A$ is quasidiagonal?
3
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1answer
99 views

Direct sum of multiplier algebras

Consider a collection of $C^*$-algebras $\{A_i\}_{i \in I}$. We can form the direct sum $$\bigoplus_{i \in I}^{c_0} A_i:= \left\{(a_i)_{i \in I} \in \prod_{i\in I} A_i: \lim_{i \in I} \|a_i\| = 0\...
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1answer
56 views

Complemented submodules of a Hilbert C*-module

Let $A$ be a $C^*$-algebra and $E$ be a (right) Hilbert $C^*$-module over $A$. Assume $F$ is a closed submodule of $E$ such that $F^\perp := \{x \in E: \langle x, F\rangle=0\}$ is orthogonally ...
0
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1answer
112 views

Why is $q(f,g) = (f-g,0)$ not adjointable?

Let $A= C([0,1])$ and $J= \{f \in A: f(0) = 0\}$. Consider the Hilbert $C^*$-module $E:= A \oplus J$ (with the obvious right $A$-action and inner product). I want to prove that $$q: E \to E: (f,g) \...
2
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1answer
101 views

Primitive ideals of minimal tensor product

Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product. Is there any classification of primitive ideals of $A \otimes B$? (I'm mainly interested in the ...
2
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105 views

Reference request: definiitions of exact C* algebra and group C* algebra

I am writing my Ph.D. thesis and I would like to cite the specific papers where the concept of exact $C^*$ algebra and group $C^*$ algebra was defined. In the book of Brown and Ozawa "$C^*$-...
3
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1answer
209 views

A $*$-homomorphism $C(X) \to C(Y)$ gives a continuous map $Y \to X$

Given a $C^*$-algebra $A$, we write $\Omega(A)$ for its space of characters, i.e. its non-zero algebra homomorphisms $A \to \mathbb{C}$. If $X$ is a compact Hausdorff space, it is well-known that $$X \...
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$C^*$-algebras over an extension of $\mathbb{Q}_p$?

I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}...
2
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1answer
144 views

Strict topology on the multiplier algebra

Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by $$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
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1answer
120 views

Regarding socle of a C* algebra

I wanted to know if the socle of a complex C*-algebra is essential? Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
3
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1answer
150 views

The inequality $a^*ca \le \|c\| a^*a$ in a pre-$C^*$-algebra

Let $A$ be a pre-$C^*$-algebra, i.e. $A$ satisfies all axioms for a $C^*$-algebra except completeness. In other words, $A$ is an involutive algebra with a $C^*$-norm. We say that $x \in A$ is positive ...
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89 views

Does this groupoid have a quasi-diagonal reduced $C^*$-algebra?

Let $H$ be a discrete group, and let $X$ be the one-point compactification of $\mathbb{N}$. Consider the étale groupoid $G = H \times \{\infty\} \sqcup \mathbb{N}$, whose unit space is $X$, and with ...
3
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Fell's absorption principle proof (for reduced crossed products)

Consider the following proof from the book '$C^*$-algebras and finite-dimensional approximations' by Brown-Ozawa. Let $\Gamma$ be a discrete group and $(A, \Gamma, \alpha)$ be a $C^*$-dynamical system....
3
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1answer
74 views

Nonstandard Podles spheres as $U_c(\frak{h})$ invariants

In this paper Podles introduced a $2$-parameter family of $q$-deformed spheres $S_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "...
7
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Does the following tracial inequality (involving certain function applications) hold for positive semi-definite matrices?

Given $n \in \mathbb{N}$ we define the function $f_{i,n}: [0,1] \rightarrow \mathbb{R}$ for $i \in \{1,..., n\}$ by $f_{i,n} = 0$ on the interval $[0,(i-1)/n]$, $f_{i,n} = 1$ on $[i/n,1]$, and $f_{i,n}...
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1answer
123 views

Dimension of commutant

Suppose that $A = M_n(\mathbb{C})$ be the algebra of $n*n$ matrices over $\mathbb{C}$. If com(A) = {$B \in M_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
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Reduced twisted $C^*$-algebra and twisted crossed product

Let $G$ be a discrete group. Is it possible to represent $C^*_r(G, \sigma)$, the reduced twisted group $C^*$-algebra as a twisted crossed product?
2
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2answers
117 views

Unconditional Convergence of Positive Terms in a $C*$-algebra

I am reading the paper Frames and Outer Frames for Hilbert $C^*$-modules by L.J. Arambasic and D. Bakic. They have mentioned in passing, the following: "...Since in each $C^*$-algebra, a ...
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0answers
88 views

Continuous fields of Hilbert spaces arising from representations of abelian C*-algebras

This is a followup to a previous question [1] on MO. Let $X$ be a second-countable, locally compact, Hausdorff space, and let $\mathscr H =\{H_x\}_{x\in X}$, be a measurable field of Hilbert spaces ...
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2answers
299 views

On solvability of equation $D(x)=1$ where $D:A\to A$ is a bounded outer derivation on a $C^*$ algebra

Let $A$ be a unital $C^*$ algebra. Assume that $D:A\to A$ is a bounded derivation. Can one say that $1$ can not be in the image of $D$? If the answer is no: What is a counter example? What kind of $...
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1answer
142 views

Density of normal elements in a C*- algebra [closed]

Let $A$ be a unital C*-algebra. I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
7
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1answer
235 views

Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?

Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...
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2answers
116 views

Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra

Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions: B is a von Neumann algebra with $A'' = B$. The inclusion $A \...
3
votes
1answer
154 views

Opposite $C^*$ algebras

$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
11
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1answer
262 views

Factor states on C*-algebras

Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras ...
3
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2answers
197 views

Elements of the minimal tensor product of a finite dimensional operator system and a $C^*$-algebra

I am trying to prove something that seemed simple to me at first sight but apparently it is giving me a hard time. Here is the same question on MSE. Let $E\subset A$ be a finite dimensional operator ...
8
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1answer
603 views

Motivation for $C^*$-algebras

I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
2
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1answer
86 views

Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid?

$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let $\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let $$ \H = \{\gamma \in \G:...
5
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1answer
152 views

Polar decomposition in abstract von Neumann algebra

Probably an easy question, but here goes: In a concrete von Neumann algebra $M \subseteq B(H)$, every element $m \in M$ has a polar decomposition $m= p|m|$ where $p$ is a partial isometry and $|m|= \...
2
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1answer
145 views

Trying to understand Haagerup tensor product $B(H)\otimes_{\rm h}B(K)$

I’m self reading Haagerup tensor product of operator spaces. Understanding it properly, I’m trying some examples. Let $H$ And $K$ be Hilbert space. Let $B(H)$ and $K(H)$ denotes the spaces of bounded ...
5
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1answer
220 views

Non-unital Russo-Dye Theorem

Let $A$ be a C$^*$-algebra and let $\phi$ be a positive linear map from $A$ to $B(H)$ (bounded linear operators on Hilbert's space). If $A$ is unital, then the Russo-Dye Theorem implies that $\|\phi\...
8
votes
1answer
175 views

Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?

Let $\Gamma$ be a discrete group whose reduced group $C^\ast$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $\mathcal{L}(G)$ is a full $\text{II}_1$-factor, ...
4
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0answers
100 views

Restricting a function defined on an étale groupoid to an isotropy group

Let $\mathcal G$ be an étale groupoid, let $x$ be a point in the unit space of $\mathcal G$, and let $\mathcal G(x)$ be the isotropy group of $x$. If $f$ is a continuous, complex valued, compactly ...
2
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0answers
67 views

Example of a ternary Lie ideal which is not a Lie ideal

Let $H$ and $K$ be Hilbert spaces and $V\subset B(H,K)$ be a ternary ring of operators i.e. $xy^*z \in V$ for all $x,y,z \in V$. Let $I$ be a closed subspace of $V$. $I$ is called a ternary Lie ideal ...
6
votes
1answer
109 views

Morphisms between compact quantum groups

Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
5
votes
1answer
105 views

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 $$\...
6
votes
1answer
306 views

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
votes
1answer
98 views

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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0answers
57 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 $A$ ...
5
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1answer
146 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. ...
2
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1answer
168 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 $\mathcal{H}$ ...
0
votes
0answers
87 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, ...
3
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0answers
96 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 ...

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