All Questions
Tagged with c-star-algebras oa.operator-algebras
597 questions
0
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71
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Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group
Let $A$ be a unital $C^*$-Algebra with non-trivial $K_0$ group. Define $CA = \{f\in C\big([0, 1], A\big)\,\vert\,f(0) = 0\}$. It can be shown that $CA$ is homotopic equivalent to the set $\{0\}$ and ...
- 307
0
votes
0
answers
91
views
Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation
Please tell me where can I see the proof of this well-known fact?
enter image description here
2
votes
1
answer
525
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Difference in tracial and finite von Neumann algebras
A tracial von Neumann algebra $(M,\tau)$ is a von Neumann algebra with a faithful normal tracial state $\tau$ on $M$. That is, $\tau$ is a function from $M \to \mathbb{C}$ such that it is a faithful ...
- 163
2
votes
0
answers
202
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The trigonometric $C^*$-algebra
The trigonometric $C^*$-algebra is the universal $C^*$-algebra generated by $\mathcal{G}=\{x,y,z\}$ subject to relations \begin{align}x^2=x=x^*, &\quad y^2=y=y^*\\ [x, z]=y, &\quad [y,z]=...
- 356
2
votes
0
answers
70
views
The $K_0$ mapping of an automorphism induced by a derivation
Let $\mathfrak{A}$ be a unital $C^*$-Algebra and let $\delta: \mathfrak{A} \rightarrow \mathfrak{A}$ be a linear map that is not constantly zero and satisfies, for every $A, B\in\mathfrak{A}$, $\delta(...
- 307
5
votes
1
answer
212
views
States "absorbed" by a Haar idempotent on a compact quantum group
Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ absorbs $a$? Can we say ...
- 1,027
10
votes
1
answer
283
views
Faithful extreme traces on group C*-algebras
Let $G$ be a discrete amenable, residually finite, ICC(i.e. each non-trivial conjugacy class is infinite) group. Let $C^*_r(G)$ be the reduced group $C^*$-algebra of $G$. Since $G$ is ICC the (...
- 2,689
1
vote
0
answers
137
views
Representation of states in $C^*$-algebras
Let $\mathfrak{A}$ be a $C^* $-algebra, let $\pi : \mathfrak{A} \to \mathcal{B}(H)$ be a representation of $\mathfrak{A}$ on the space of bounded linear operators on a Hilbert space $H$ and let $\...
- 139
6
votes
2
answers
477
views
Linear map between projective finitely generated Hilbert modules is adjointable
Let $A$ be a (unital) $C^*$-algebra and $X,Y$ right Hilbert $A$-modules which are finitely generated and projective. It seems to be well-known that if $T: X \to Y$ is an $A$-linear map, then $T$ is ...
- 175
10
votes
1
answer
518
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For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$?
Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$.
For what kind of $C^*$ algebras $A$ does the following hold:
$$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\...
- 356
0
votes
1
answer
190
views
Are the ideals in two $C^*$-algebras the same?
Let $V_{1}, V_{2}$ be the commuting isometries. By Wold decomposition theorem, we know that $V_{i}$ admits decomposition $$V_i \cong V^s_{i}\oplus V^{u}_{i},$$ where $V^{s}_{i}$ is the shift and $V^{u}...
- 139
4
votes
1
answer
332
views
Support projection vs closed support projection of a normal state in enveloping von Neumann algebra
I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
- 135
1
vote
0
answers
87
views
irreducible subfactor inclusion and commutativity of induced projections
Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$.
Let $\Omega\in H$.
Does it follow that the projections onto $[N\Omega]$ and $[M'\...
- 759
1
vote
1
answer
129
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 \...
- 525
5
votes
1
answer
428
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) = \...
- 759
8
votes
1
answer
286
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 ...
- 3,497
4
votes
1
answer
133
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 $...
- 356
1
vote
1
answer
177
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 ...
- 759
1
vote
0
answers
106
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 ...
- 356
2
votes
1
answer
205
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 ...
- 175
1
vote
0
answers
178
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$ ...
- 356
3
votes
0
answers
219
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 ...
- 854
4
votes
2
answers
157
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
$$\...
- 175
3
votes
1
answer
121
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 ...
- 4,058
5
votes
1
answer
219
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 ...
- 51
3
votes
1
answer
743
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 $C^*...
- 7,426
7
votes
0
answers
135
views
Is every quasi-nilpotent element in a C$^*$-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, ...
- 573
3
votes
1
answer
424
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 ...
- 1,879
9
votes
2
answers
243
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 $a\in A$ ...
- 573
4
votes
2
answers
298
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 $\...
- 175
1
vote
0
answers
45
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 ...
- 1,115
2
votes
0
answers
228
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 ...
- 356
5
votes
0
answers
137
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 ...
- 1,115
1
vote
1
answer
172
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
113
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 $M(A)$)...
- 175
9
votes
1
answer
460
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 ...
- 525
-1
votes
1
answer
230
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 ...
- 451
1
vote
0
answers
129
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 ...
- 2,605
2
votes
1
answer
374
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*-...
- 4,058
9
votes
3
answers
568
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 ...
- 175
0
votes
0
answers
88
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 ...
- 451
2
votes
2
answers
158
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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}_\...
- 175
1
vote
1
answer
136
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 ...
- 356
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 $\...
- 4,058
3
votes
1
answer
290
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 ...
- 1,903
1
vote
1
answer
118
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. ...
- 1,115
2
votes
1
answer
218
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_{\...
- 175
2
votes
1
answer
297
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 ...
- 175
3
votes
0
answers
160
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(...
- 451
3
votes
1
answer
255
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 $\sigma$-...
- 175