All Questions
Tagged with oa.operator-algebras operator-norms
15 questions
3
votes
0
answers
75
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 73
13
votes
0
answers
174
views
Existence of more than two C*-norms on algebraic tensor product of C*-algebras
Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
- 536
1
vote
0
answers
54
views
Row-wise conjugation of completely bounded map by group action
Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...
- 2,998
0
votes
1
answer
174
views
Does $\{\left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right||\varphi\in\{\psi\}^{\perp}\}$ split $\mathfrak{S}_1$?
Let $\mathfrak{S}_1$ be the space of trace-class self-adjoint operators on $L^2(\mathbb{R}^n)$, and $\psi\in L^2(\mathbb{R}^n)$ such that $\int |\psi|^2 = 1$. Is there a projection from $\mathfrak{S}...
user141436
1
vote
0
answers
76
views
specific sequence of matrices making a strange ratio of matrix norms diverging
For any $t>0$ define $d_t:=\operatorname{diag}j^t=\operatorname{diag}(1^t,2^t,\ldots)$. Now pick up such a $t>0$ and an arbitrary $\theta\in\big(0,\frac12\big)$. For every $k\in\mathbb{N}$ find ...
- 375
2
votes
1
answer
109
views
Retractions for completely positive unital maps, and their effect on spectral diameter
Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf ...
- 1,414
4
votes
1
answer
204
views
Retractions for completely positive unital maps, with particularly nice norms
Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions. Does $\Psi$ admit a retraction $\Phi: \...
- 1,414
7
votes
1
answer
442
views
Completely bounded norm for unital maps with completely positive sections
Consider a completely bounded unital map $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\...
- 1,414
0
votes
1
answer
556
views
Bounds on spectral norm of difference of powers of linear operators
Given bounded linear operators $A$ and $B$, with $||A-B||<\epsilon$, what is the tightest upper bound on $||A^n-B^n||$, for positive integer $n$? Additionally, for my specific situation, $A$ is ...
0
votes
1
answer
775
views
A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n
In my research of operator algebras and their connection with machine learning I of course use the well know result:
For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...
- 221
3
votes
2
answers
3k
views
Weak convergence implies norm convergence for trace class operators?
It is known that weak convergence implies norm convergence in $\ell^1(\mathbb{N})$, see e.g. here.
Because of the typical analogies of the Schatten ideals $C_p \subset B(H)$ (where $H$ is a Hilbert ...
- 9,853
4
votes
1
answer
175
views
CB-norm of mappings from a matrix space
The following result of Roger Smith is well known to operator algebraists:
$$\| \phi: E \rightarrow M_n\|_{cb} =\| \phi^{(n)} \otimes id_{M_n}: E \otimes_{min}M_n \rightarrow M_n \otimes M_n\|,$$
...
- 769
10
votes
2
answers
926
views
Continuity of the product map
Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...
- 4,705
3
votes
1
answer
1k
views
injective tensor norms for real tensors
If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be
$$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1}
|\...
- 316
3
votes
1
answer
490
views
Bounds on operator 2-norms on partial traces of linearly related operators
Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on &#...
- 1,414