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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60 views

Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?

Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product $$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \...
5 votes
1 answer
305 views

Separable C* algebras and type I states

Let $A$ be a separable $C^*$-algebra and let $\omega$ be a state on $A$. Then there is an "orthogonal" probability measure $\mu$ on the pure state space $P(A)$ of $A$ such that $\omega(x) = \...
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6 votes
1 answer
168 views

Commutator ideal in nonunital C*-algebra

Let $A$ be a C*-algebra that has no one-dimensional irreducible representations, that is, there is no (closed) two-sided ideal $I\subseteq A$ such that $A/I\cong\mathbb{C}$. Let $J$ denote the (not ...
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4 votes
1 answer
109 views

A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
4 votes
0 answers
125 views

Gelfand's transform for noncommutative $C^*$-algebras

Please excuse me if this is well-known, I am not very familiar with the general theory of $C^*$-algebras. Let $A$ be a unital separable liminal $C^*$-algebra (in the case I am interested in, ...
2 votes
1 answer
122 views

If $S\subseteq A^*$ is separating, does $S$ also separate $M(A)$?

Let $A$ be a non-unital $C^*$-algebra. Let $S\subseteq A^*$ be a set of continuous functionals that separates the points of $A$. Every element $\omega \in A^*$ extends uniquely to a strictly ...
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1 vote
1 answer
144 views

Commuting and generating subfactors of $ B(H)$

I have a question on subfactors of $B(H)$ (the von Neumann algebra of bounded operators on a complex Hilbert space). Say I have a subfactor $M$ of $B(H)$. Is it true that another subfactor $N \subset ...
  • 197
1 vote
0 answers
94 views

A locally convex $C^*$ algebraic structure on the disk algebra

A locally convex $C^*$ algebra is a locally convex topological vector space $A$ whose topology is generated by a familly of complete $C^*$ semi norm and $A$ is a $*$ algebra. Moreover all algebra ...
2 votes
1 answer
90 views

External tensor product of Hilbert modules

I am reading Lance's book "Hilbert $C^*$-modules". In particular, I want to understand how to construct the (external) tensor product of Hilbert $C^*$-modules. Consider the following ...
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4 votes
0 answers
80 views

KK-theory for commutative $C^*$-algebras

The Gelfand--Naimark theorem tells us to regard noncommutative $C^*$-algebras as "noncommutative function spaces". In that spirit $K$-theory the Grothendieck group of "noncommutative ...
1 vote
0 answers
150 views

A locally convex $C^*$ algebra without zero divisor

Let we have a locally convex $C^*$ algebra $A$. That is $A$ is a TVS equipped with an algebra and an involution structure such that all operations are continuous. Moreover the topology on $A$ ...
3 votes
0 answers
142 views

Can any POVM be induced by a quantum instrument?

I suspect this is the obvious result of something in operator algebras, but that's far outside my field. Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...
4 votes
2 answers
104 views

Is a unital $*$-morphism from a unital $C^*$-algebra $A$ to $\operatorname{End}_{\mathbb{C}}(K)$ automatically contractive?

Let $A$ be a unital $C^*$-algebra and let $K$ be an inner product space (not necessarily complete!). Let $\pi: A \to \operatorname{End}_{\mathbb{C}}(K)$ be a unital algebra homomorphism such that $$\...
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3 votes
1 answer
113 views

Impact of annihilators in C*-algebras

Let $A$ be a unital C*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$ Suppose that $A$ satisfies the following property: For every $S\subseteq ...
  • 3,363
5 votes
1 answer
197 views

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 ...
3 votes
1 answer
335 views

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^*...
7 votes
0 answers
102 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, ...
4 votes
1 answer
244 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 ...
9 votes
2 answers
187 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 $a\in A$ ...
4 votes
2 answers
252 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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3 votes
1 answer
179 views

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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1 vote
0 answers
35 views

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 ...
2 votes
0 answers
149 views

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 ...
5 votes
0 answers
119 views

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 ...
2 votes
0 answers
138 views

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 ...
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1 vote
1 answer
122 views

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 \...
1 vote
0 answers
109 views

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 $M(A)$)...
  • 405
9 votes
1 answer
373 views

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 ...
-1 votes
1 answer
211 views

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 ...
1 vote
0 answers
123 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 ...
  • 1,363
2 votes
1 answer
269 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*-...
  • 3,363
0 votes
0 answers
74 views

Is it true that every closed ideal $J$ of $ A \otimes_hB$ is of the form $\epsilon^{-1}(I)$ for some closed ideal $I$ of $A \otimes_\text{min} B$

Let $A$ and $B$ be $C^{\ast}$-algebras. It is known that the map $\epsilon:A \otimes_h B \to A \otimes_\text{min} B$ is injective. Is it true that every closed ideal $J$ of $ A \otimes_hB$ is of the ...
9 votes
3 answers
435 views

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 ...
  • 405
0 votes
0 answers
69 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 ...
2 votes
2 answers
111 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}_\...
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1 vote
1 answer
84 views

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 ...
1 vote
0 answers
55 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 $\...
  • 3,363
0 votes
0 answers
75 views

A list of unital Banach *-algebras whose left annihilators are principal

Let $A$ be a unital Banach *-algebra. For a given subset $S\subseteq A$, the left annihilator of $S$, denoted by Ann($S$), is given by Ann$(S)=\{x\in A: xS=0\}$. Let us say Ann($S$) is principal if ...
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3 votes
1 answer
242 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 ...
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2 votes
1 answer
183 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 ...
1 vote
1 answer
102 views

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. ...
2 votes
1 answer
120 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_{\...
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2 votes
1 answer
256 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 ...
  • 405
3 votes
0 answers
114 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(...
3 votes
1 answer
161 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 $\sigma$-...
  • 405
1 vote
0 answers
113 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 ...
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4 votes
1 answer
142 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 ...
  • 405
2 votes
0 answers
122 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:...
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1 vote
1 answer
120 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 $...
  • 405
2 votes
0 answers
29 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 ...

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