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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An inequality in C*-algebras

Let $A$ be a non-unital C*-algebra, and let $\pi: A\to B(A)$ defined by $\pi(a)(x)=ax$ be the left representation of A. Is the following inequality correct? $$\lVert I+ \pi(a) \rVert\ge 1$$ for all $a ...
Diako paez's user avatar
3 votes
1 answer
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Completions of $C(X)$ with respect to the topologies generated by states

I have no intuition in this field so excuse me if this is trivial. Let $X$ be a compact Hausdorff space, and $C(X)$ the algebra of continuous functions on $X$ with the usual $\sup$-norm. This is a $C^*...
Sergei Akbarov's user avatar
7 votes
0 answers
130 views

Is every quasi-nilpotent element in a C$^*$-algebra a norm-limit of nilpotent elements?

Let $A$ be a C$^*$-algebra. I have seen theorems either stating or implying that if $A$ is the algebra of bounded linear operators on a separable Hilbert space (Herrero et al.), or the Calkin algebra, ...
Douglas Somerset's user avatar
3 votes
1 answer
376 views

A trace inequality between self-adjoint operators

Let $A$ and $B$ be self-adjoint operators on some Hilbert space and $B$ is postive. Suppose we have $-B\leq A\leq B$.Is it true then that $\|A\|_p\leq\|B\|_p$ where $\|.\|_p$ is the Schatten-$p$ norm ...
A beginner mathmatician's user avatar
9 votes
2 answers
229 views

Lifting quasi-nilpotent elements in C$^*$-algebras

Let $A$ be a C$^*$-algebra with closed two-sided ideal $I$. Set $B=A/I$ and let $\pi:A\to B$ be the quotient map. Suppose that $b\in B$ is quasi-nilpotent. Does there exist quasi-nilpotent $a\in A$ ...
Douglas Somerset's user avatar
4 votes
2 answers
286 views

Takesaki volume II chapter VII lemma 1.15

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...
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Relating different definitions of dual of a compact quantum group

Let $\mathbb{G}$ be a compact quantum group in the sense of Woronowicz. We can look at its associated dense Hopf$^*$-subalgebra $\mathbb{C}[\mathbb{G}]$. Hence, in the framework of multiplier Hopf $*$-...
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Examples of TRO $V $ and $C^{\ast} $-algebra $B $ for which $V\otimes^hB $ is a TRO

Let $V $ be a ternary ring of operator(TRO) and $B $ be a $\mathbb {C}^{\ast} $-algebra. Let $V \otimes^hB $ denotes the Haagerup tensor product of $V $ and $B $. Obviously if $V $ or $B $ is $\mathbb ...
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2 votes
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Irreducible group representation(algebraic and topological irreducibility)

In page 280 of "C^* algebra" by Dixmier, in the context of group representation, it is written 'We never encounter the concept of algebraic irreducibility except in finite dimensional ...
Ali Taghavi's user avatar
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Trying to prove a seemingly easy fact on ideals of ternary C*-algebras

Currently I'm reading the paper by Abadie and Ferraro titled Applications of ternary rings to $C^*$-algebras. Recall that a $C^{\ast}$-ternary ring is a complex Banach space $M$, equipped with a ...
Math Lover's user avatar
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2 votes
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Convolution of continuous compactly supported functions on étale groupoid is continuous

Let $G$ be an étale Hausdorff groupoid, i.e. a topological groupoid $G$ such that the source and range maps $s,r: G \to G$ are local homeomorphisms. Consider the complex vector space $C_c(G)$ of ...
Andromeda's user avatar
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1 answer
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What is a C*-algebra generated by a subset of a direct sum of C*-algebras equal to?

I'm studying C-algebras and I don't know how to address the following question: let $(A_k)_{k\in \mathbb{N}}$ a family of C-algebras and let $\mathcal{G}$ a subset of $\displaystyle\bigoplus_{k\in \...
Xavier González's user avatar
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Is it true that $\omega = \sum_{(k,l)\in I^2}\omega(p_k - p_l)$

Let $\{p_i\}_{i \in I}$ be a collection of projections in a $C^*$-algebra $A$ such that $\sum_{i \in I} p_i = 1$ in the strict topology (note here that $1$ is the unit of the multiplier algebra $M(A)$)...
Andromeda's user avatar
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Is a $*$-automorphism $M(A) \to M(A)$ automatically strictly continuous?

Let $A$ be a (non-unital) $C^*$-algebra with multiplier $C^*$-algebra $M(A)$. Let $\phi: M(A) \to M(A)$ be a $*$-automorphism. Is it true that $\phi$ is automatically strictly continuous (on bounded ...
J. De Ro's user avatar
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Determine whether the center of a $C^*$-algebra is 0

Let $G$ be a locally compact group and $A$ be a non-unital $C^*$-algebra. )$(A,G,\alpha)$ is a $C^*$-dynamical sysytem. The space of all continous functions from $G$ to $A$ with compact support is ...
math112358's user avatar
1 vote
0 answers
127 views

Socle of an operator algebra

Let $H, K$ be Hilbert spaces. Let $A\subseteq B(H)$ be a nonselfadjoint closed subalgebra such that the identity map is in $A$. Let $C_A$ denote the $C^*$-algebra generated by $A$. Q1: (this question ...
Onur Oktay's user avatar
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2 votes
1 answer
336 views

A C*-algebra enjoying some different C*-norms

Does there exist any C*-algebra $(A,\|\cdot\|)$ enjoying the following property? $\bullet$ There exists a norm $|\cdot|$ on $A$ with $\|\cdot\|\leq|\cdot|$ such that $(A,|\cdot|)$ is a pre C*-...
ABB's user avatar
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3 answers
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Defining the abstract tensor product of W*-algebras via a universal property

I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt: It is easy to show that such an object ...
Andromeda's user avatar
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0 votes
0 answers
83 views

$*$–homomorphisms of the center of $C^*$-algebras

Let $A$ and $B$ be $C^*$-algebras with centers $Z_A$ and $Z_B$. Suppose $\rho:A\rightarrow B$ is a surjective $*$- homomorphism. It is easy to check $\rho(Z_A)\subset Z(B)$. I wonder how to assure ...
math112358's user avatar
2 votes
2 answers
145 views

Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)

I originally asked this on MSE, but did not get an answer there. Let $M$ be a von Neumann algebra. Let $\varphi: M_+ \to [0, \infty]$ be a weight on $M$. Consider \begin{align*}&\mathfrak{p}_\...
Andromeda's user avatar
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1 answer
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The closure of selfadjoint elements of an algebra whose spectrum consist of rational numbers

Let $H$ be a seperable complex Hilbert space. What is the closure of the set of all self adjoint operators in $B(H)$ whose spectrum is a subset of the rational number $\mathbb{Q}$. Apart from finite ...
Ali Taghavi's user avatar
1 vote
0 answers
57 views

Is the universal representation an order isomorphism?

Let $A$ be a Banach *-algebra. By a *-representations of $A$, we mean a *-homomorphism $\pi:A\to B(H_\pi)$, where $B(H_\pi)$ is the space of all bounded linear maps on a Hilbert space $H_\pi$. Let $\...
ABB's user avatar
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3 votes
1 answer
275 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 ...
geometricK's user avatar
  • 1,851
2 votes
1 answer
199 views

Need reference for: $\lVert\cdot\rVert_{\text{max}} \leq \lVert\cdot\rVert_h$

Let $A$ and $B$ denote $C^{\ast}$-algebras. Let $\lVert\cdot\rVert_h$ and $\lVert\cdot\rVert_{\text {max}}$ denote the Haagerup norm and max $C^*$-norms on $ A \otimes B$, respectively. I am looking ...
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Let $V$ be a TRO such that $A(V)= \mathbb{C}$, what can we say about $V$?

Let $V$ be a TRO i.e. closed subspace of $B(H,K)$ such that $xy^*z \in V$ for all $x,y,z \in V$. Let $C(V)$ and $D(V)$ denotes the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. ...
Math Lover's user avatar
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2 votes
1 answer
184 views

Extending a $\sigma$-weakly continuous map: Takesaki IV.5.13

Consider the following fragment from chapter IV in Takesaki's book "Theory of operator algebra I": Why is the boxed line true? Takesaki argues that $$\theta_0: \mathscr{M}_1\otimes_{\...
Andromeda's user avatar
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2 votes
1 answer
289 views

Predual theorem proof in Takesaki's volume I

Consider the following fragment from Takesaki's book "Theory of operator algebra I" (Section III.3 ,p133-134). Why is the boxed line true? I can see that $\epsilon: \widetilde{A}\to A$ is ...
Andromeda's user avatar
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3 votes
0 answers
150 views

Construct a non-unital nuclear $C^*$-algebra without tracial states such that its multiplier algebra is also traceless

Let $H$ be an infinite dimensional separable Hilbert space. The set $K(H)$ of all compact operators is a non-unital nuclear $C^*$-algebra which has no tracial states and the multiplier algebra of $K(...
math112358's user avatar
3 votes
1 answer
202 views

Takesaki: Lemma about enveloping von Neumann algebra

Consider the following lemma with proof from Takesaki's book "Theory of operator algebra I" (p121): It appears to me that Takesaki claims at the end of the proof that $\pi(A)_1$ is $\sigma$-...
Andromeda's user avatar
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1 vote
0 answers
136 views

Representation of quantum groups

Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if ...
Dastan's user avatar
  • 121
5 votes
1 answer
268 views

Takesaki's proof of the Kaplansky density theorem

Consider the following fragment from Takesaki's book "Theory of operator algebra I": Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...
Andromeda's user avatar
  • 209
2 votes
0 answers
145 views

Non-commutative harmonic analysis on the discrete Heisenberg group

Question: Is there a linear map $\mathcal F$ from the Hilbert space of $\ell^2$ functions on the discrete Heisenberg group to some Hilbert space of functions $ L^2(\bigcup \{\Omega_n\}) $, such that:...
LeechLattice's user avatar
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1 vote
1 answer
152 views

Convergent bounded net of positive operators converges to a positive operator

Let $A$ be a $C^*$-algebra. Endow $A$ with the strict topology for which a net $\{a_i\}_{i \in I}$ converges to $a \in A$ if $$\|a_i b-ab\| + \|ba_i-ba\| \to 0$$ for all $b \in A$. Is it true that if $...
Andromeda's user avatar
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2 votes
0 answers
50 views

A foliation with prescribed graph of foliation

**I have already asked this question on MSE https://math.stackexchange.com/questions/4272279/1-dimensional-foliation-of-surfaces-with-prescribed-graph-of-foliation ** Definition of the graph of a ...
Ali Taghavi's user avatar
1 vote
1 answer
171 views

A completely positive equivariant map $\varphi: A \to B$ induces a map on the full crossed products

Let $G$ be a discrete group. Let $(A,\alpha)$ and $(B,\beta)$ be $G$-$C^*$-algebras and $\varphi: A \to B$ be $G$-equivariant and completely positive. All crossed products in this post are full (= ...
Andromeda's user avatar
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160 views

Stinespring's theorem: can we choose the dilation to be an isometry?

Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
Andromeda's user avatar
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2 votes
1 answer
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Action of a group $G$ induces a coaction on $C_0(G)$

In this question, I follow the book "An invitation to quantum groups and duality" by Timmerman, p259. Let $G$ be a locally compact group and $C$ be a $C^*$-algebra. Assume an action $$\alpha:...
Andromeda's user avatar
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3 votes
1 answer
141 views

Norm antipode on a Kac-type compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group. Consider the associated dense Hopf$^*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ ...
Andromeda's user avatar
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4 votes
0 answers
117 views

Can the injective envelope ever be injective for $*$-homomorphisms?

The answers to the question "Is the injective envelope functorial" resoundingly remind us that the injective envelope of a C$^*$-algebra really belongs in the category of completely positive ...
Chris Ramsey's user avatar
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6 votes
2 answers
592 views

Is the injective envelope functorial?

Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-...
Andromeda's user avatar
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3 votes
1 answer
202 views

Woronowicz characters are multiplicative

I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. Let $G$ be a $C^*$-algebraic compact quantum group with function algebra $(C(G), \Delta)...
Andromeda's user avatar
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4 votes
0 answers
158 views

Tensor product of representations on a compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$. Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
Andromeda's user avatar
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4 votes
1 answer
390 views

If a completely positive unital map admits a completely positive unital left inverse, it is a complete isometry

Let $T$ be an injective operator system and $U$ be an arbitrary operator system. Let $\varphi: T \to U$ be a unital completely positive map and $\psi: U \to T$ be a unital completely positive map with ...
Andromeda's user avatar
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2 votes
1 answer
142 views

$(\iota \otimes f)(X) = 0$ for all $f \in B^*$ implies $X=0$

Let $A$ and $B$ be $C^*$-algebras. Given $f \in B^*$, we can form the right slice map $$\iota \otimes f: A \otimes B \to A: a \otimes b \mapsto af(b)$$ which extends uniquely to a bounded linear map $$...
Andromeda's user avatar
  • 209
5 votes
1 answer
182 views

Subrepresentations of C*-algebraic compact quantum groups

Let $\mathbb{G}$ be a compact quantum group with function algebra $(C(\mathbb{G}), \Delta)$ (in the sense of Woronowicz). Let $X \in M(B_0(H) \otimes C(\mathbb{G}))$ be a (possibly infinite-...
Andromeda's user avatar
  • 209
5 votes
0 answers
127 views

C^*-algebra theory with all the Koszul signs

I was wondering if someone knows of a reference in which $\mathbb{Z}_2$-graded $C^*$-algebra theory is developed using the sign convention $(ab)^* = (-1)^{|a||b|}b^* a^*$. I would be most enthusiastic ...
Luuk Stehouwer's user avatar
14 votes
0 answers
251 views

Stable isomorphism of group C$^*$-algebras

For a discrete group $G$, let $C^*_r(G)$ be its reduced group C$^*$-algebra. Question: Do there exist discrete, torsion-free non-isomorphic groups $G,H$ such that $C^*_r(G)$ and $C^*_r(H)$ are stably ...
Caleb Eckhardt's user avatar
1 vote
1 answer
161 views

$C\lVert\sum_i a_{ii}\rVert \ge \lVert(a_{ij})\rVert$ for matrices with entries in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $(a_{ij}) \in M_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a_{ij}$) such that $$\lVert(a_{ij})\rVert \le C \Bigl\lVert\...
Andromeda's user avatar
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0 votes
1 answer
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Is $N_\phi = \{x \in E: \phi(\langle x,x\rangle)=0\}$ a Hilbert submodule of $E$?

Let $E$ be a (right) Hilbert module over the $C^*$-algebra $B$. Let $\phi$ be a state on the $C^*$-algebra $B$. Then consider $$N_\phi:= \{x \in E: \phi(\langle x,x\rangle)=0\}.$$ I want to show that $...
Andromeda's user avatar
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3 votes
1 answer
223 views

Monotone approximation of elements in AF-algebras

Suppose that we are given an AF-algebra $A$ and a sequence of finite-dimensional subalgebras $\mathbb{C}=A_0\subset A_1\subset A_2\subset\ldots$ such that $A=\overline{\bigcup\limits_{n\geq 0}A_n}$. ...
Nikita Safonkin's user avatar

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