All Questions
Tagged with operator-theory c-star-algebras
862
questions
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Unique extension of $*$-representation into an abstract multiplier algebra
I'm trying to find a proof of the following fact:
Let $A,B$ be $C^{*}$-algebras and $\pi: A \longrightarrow M(B)$ be a non-degenerate homomorphism in the sense that $\pi(A)B$ densely spans $B$. Then ...
- 1,399
2
votes
1
answer
59
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Map defined by matrix units. [closed]
If $\{|\alpha_i\rangle\}_{i=1}^m$ and $\{| \beta_k \rangle\}_{k=1}^n$ are orthonormal bases of subsystems $A$ and $B$ respectively of a Hilbert space $H = A \otimes B$ and
$$P_{kl} = |\alpha_k\rangle\...
- 95
0
votes
0
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26
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stability under holomorphic calculus
I have two questions:
I would like to know the exact meaning of "stability under holomorphic calculus" in the context of $C^*$-algebras
Is all unital noncommutative $C^*$--algebra stable ...
- 139
1
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2
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81
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Convergence of a Sequence in the GNS Space of a von Neumann Algebra with Semi-finite Trace under the $\sigma$-Weak Topology
Let $ M $ be a von Neumann algebra. A Semi-finite trace on $ M^+ $ is a function $\phi$ on $ M^+ $, taking non-negative, possibly infinite, real values, possessing the following properties:
Linearity:...
- 2,007
0
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1
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57
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Question about hyperstandard von Neumann algebras and selfpolar cones
Consider the following fragment from the book "Lectures on von Neumann algebras" by Stratila and Zsido, second edition:
I have two questions:
(1) Why is $\mathscr{M}_q= q\mathscr{M}q \...
- 840
1
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1
answer
39
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Is the cut-down of a positive invertible element postive and invertible?
Let $\mathfrak A$ be a unital C*-algebra; let $A \in \mathfrak A$ be such that $\sigma_{\mathfrak A}(A) \subset [a,\infty)$ for some $a > 0$; let $p \in \mathfrak A$ be a (self-adjoint) projection.
...
2
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1
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43
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Property of triple systems
A triple system is a subspace $T\subseteq B(H)$ for a Hilbert space $H$ such that $xy^* z\in T$ for all $x,y,z \in T$.
Consider the $C^*$-algebra $B$ which consists of the closed linear span of the ...
- 840
3
votes
0
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77
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Extensions of non-degenerate representations of C*-Algebras
I'm aware of the result that if $A$ is a $C^{*}$-algebra and $\pi: A \longrightarrow \mathcal{B}(H)$ is a non-degenerate representation, then $\pi$ uniquely extends to a representation $\overline{\pi}:...
- 1,399
2
votes
1
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44
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Spectral Permanence Remark in Murphy's C*-algebras
In Murphy's $C^{*}$-algebras book, he states theorem $2.1.11$ which is that if $\mathfrak{B} \subset \mathfrak{A}$ are $C^{*}$-algebras with $\mathfrak{A}$ unital such that $1_{\mathfrak{A}} \in \...
- 1,399
3
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1
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45
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Basic lemma on normality of map between von Neumann algebras
Let $f: M\to N$ be a bounded linear map between von Neumann algebras.
Assume that for every net $0 \le x_i\nearrow x$, we have that $f(x_i)\to f(x)$ $\sigma$-strongly. Is it true that $f$ is normal, i....
- 840
1
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1
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34
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$C^{*}$-Algebras Generated by Sets & other C*-Algebras?
I'm reading about Cuntz-Pimsner algebras at the moment, and something simple albeit annoying has been bothering me.
Given a $C^{*}$-Correspondence $(\sf{X},\mathfrak{A})$, Pimsner defines an '...
- 1,399
0
votes
1
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44
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Sufficient condition for linear map between $ C^* $ algebras to preserve the identity
Let $ f: A \to B $ be a linear map between two $ C^* $ algebras.
What is a sufficient condition to guarantee that the linear map $ f $ takes the identity to the identity $ f(1_A)=1_B $?
For example, ...
- 8,542
2
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1
answer
76
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$\sigma$-weakly closed subalgebra of direct product of matrix algebras is again a direct product of matrix algebras
Let $A$ be a $\sigma$-weakly closed $*$-subalgebra of the $W^*$-algebra $\prod_{i\in I}^{\ell^\infty} M_{n_i}(\mathbb{C})$. I believe that we must have $A\cong \prod_{j\in J}^{\ell^\infty} M_{m_j}(\...
- 840
1
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0
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62
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$C^{*}$-algebras generated by commuting normal elements?
A person in my class asked today if $\mathfrak{A}$ is a unital $C^{*}$-algebra and $x,y \in \mathfrak{A}$ commute and are normal, then is it always the case there exists $\alpha \in C^{*}(x,1)$ and $\...
- 1,399
1
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1
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66
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Image of single GNS representation and kernel vs nullspace
I have a very basic question about GNS construction and null-spaces vs kernels of linear functionals.
Let $A$ be a unital $\mathrm{C}^*$-algebra and $\rho$ a state on $A$. Consider the GNS ...
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