# Questions tagged [operator-spaces]

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31
questions

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### Is it true that $M_n(X)\cong X \mathbin{\hat{\otimes}}M_n(\mathbb{C})$ for the projective operator space tensor product

This question is probably obvious for experts here.
Let $X$ be an operator space. Is it true that
$$X\mathbin{\hat{\otimes}}M_n(\mathbb{C})\cong M_n(X)$$
where $\hat{\otimes}$ is the projective tensor ...

1
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1
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110
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### Do completely bounded maps on an operator space have a completely contractive Banach algebra structure?

Let $X$ be an operator space and $CB(X)$ be the set of all completely bounded linear maps $f: X \to X$. Note that $CB(X)$ becomes a Banach algebra for the composition of operators.
Is the ...

2
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1
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### Completely contractive Banach algebra structure on the dual of a Hopf $C^*$-algebra

Let $(A, \Delta)$ be a Hopf $C^*$-algebra, i.e. $A$ is a $C^*$-algebra, and $\Delta: A \to M(A\otimes A)$ is an injective non-degenerate $*$-homomorphism that is coassociative:
$$(\iota \otimes \Delta)...

1
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1
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138
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### Reference request for a preprint by Effros-Ruan

I am not sure this question is appropriate for this site, but here goes. If not, please let me know and I will delete the question.
In my literature search, I came across the following reference:
&...

3
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0
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291
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### Dunford-Pettis like properties for Banach spaces of operators

Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\left.\begin{array}{l}
(x_n)\subseteq A \textrm{ ...

0
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0
answers

50
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### Explicit description for dual to operator space of Hilbert space

Let $H$ be a separable Hilbert space, and $B := \mathcal B(H)$ be the space of bounded operators on $H$.
It is known that $B^\ast_\mathrm{strong} = B^\ast_\mathrm{weak}$ (see [Dunford, Schwartz, VI.1....

1
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0
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127
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### 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 ...

7
votes

1
answer

260
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### Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?

Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...

0
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0
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127
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### Operator space tensor products

Given two Banach algebras $A$ and $B$ with operator space structure on each of them, i.e both of them are closed subspaces of $B(H_1)$ and $B(H_2)$ respectively for some Hilbert spaces $H_1,H_2$. ...

3
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2
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393
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### Is the ideal property of $X^{**}$ inheritable to $X$?

Let $X$ be an operator space such that there is a weak$^*$-continuous complete isometry $\phi$ from its second dual $X^{**}$ into a $W^*$-algebra $M$ in which $\phi(X^{**})$ is a (necessarily weak$^*$-...

7
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1
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268
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### Completely bounded maps approximately factoring through finite matrices

Let $A$, $B$ be two $C^\ast$-algebras and $\mathcal{F}(A,B)$ be the operator ideal of all completely bounded operators $T:A \to B$ for which there are uniformly bounded nets of completely bounded maps ...

1
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1
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106
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### Norm of a cb-homomorphism restricted to a generating operator space

Let $\mathcal A \subset B(H)$ be an operator algebra and $\varphi: \mathcal A \rightarrow B(K)$ a completely bounded homomorphism. Suppose $\mathcal M \subset \mathcal A$ is an operator space such ...

4
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1
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156
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### Terminology: jointly completely bounded?

This question has a subjective component but I would like answers that try to stick to concrete observable facts, such as which papers use which terminology. However, the informed impressions of those ...

1
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0
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78
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### weak convergence in operator space structure

Let $M$ be von Neumann algebra and $B(H)$ be it's universal representation. Let $(e_i)$ be a Hilbert basis of $H$ and $\zeta_n\xrightarrow{w}\zeta $ in $H$. I know that $[w_{\zeta_n ,e_i}]_{1\times I}\...

6
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2
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522
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### "Identity tensor transpose" as a map $M_n \hat{\otimes} M_n \to M_n \overline{\otimes} M_n$

Equipping $M_n$ with its usual operator space structure,
$\newcommand{\ptp}{\widehat{\otimes}}$
we can form the projective tensor product of operator spaces $M_n\ptp M_n$. In particular this puts a ...

5
votes

1
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134
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### Operator space structures on CB(H,K) where H and K are Hilbertian operator spaces?

(I'd be grateful if anyone thinking of putting MathJax in the question title refrains from doing so.)
By consulting various standard sources (Effros-Ruan's book, Pisier's book, the lexicon of ...

5
votes

0
answers

90
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### $p$-operator space structure on Banach algebras

There is an abstract characterization of operator algebras, which says that if $A$ is an operator space that is also an approximately unital Banach algebra, then the following are equivalent:
For any ...

3
votes

1
answer

154
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### Why are the convolvers in the bicommutant of the pseudo-measures? ($CV_p(G)\subseteq PM_p(G)''$)

Let $G$ be a locally compact group. For $1<p<\infty$ let $\lambda_p:G\to\mathcal{B}(L^p(G))$ (resp. $\rho_p:G\to\mathcal{B}(L^p(G))$) be the left (resp. right) regular representation. $CV_p(G)$ ...

1
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0
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69
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### Interpolation of the row and column operator spaces

If $R$ and $C$ are respectively the row and column operator spaces, and $\theta \in (0, 1)$, we denote by $R(\theta)$ the interpolation operator space $(R, C)_{\theta}$ (with $R(0) = R$ and $R(1) = C$)...

7
votes

0
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265
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### Approximation in the tensor square of a weakly exact von Neumann algebra

Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...

12
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1
answer

376
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### Can a non-commutative C*-algebra be a minimal operator space?

By an operator space structure on a Banach space $X$ I mean a sequence of norms on spaces $M_n \otimes X$ that satisfies Ruan's axioms.
Among such admissible sequences there is always the smallest ...

3
votes

1
answer

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### What is the significance of matrix ordered algebras?

I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
...

21
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1
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808
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### On complemented von Neumann algebras

Edit: according to Narutaka Ozawa, question 3) is still open in the type $\mathrm{II}_1$ case. In other terms, it is not known whether every topologically complemented type $\mathrm{II}_1$ factor in $...

2
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1
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88
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### Isometries between Hilbertian homogeneous finite dimensional operator spaces

We know that if $i:R_n\rightarrow C_n$ is an isometry then for any $n$-dimensional operator space E, there is a factorization $i=uv$ with $v:R_n\rightarrow E$, $u:E\rightarrow C_n$ such that $\|u\|_{...

5
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0
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153
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### Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...

4
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0
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278
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### Extensions of completely positive mappings

I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...

5
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1
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482
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### Projections which are not completely bounded

There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand ...

2
votes

1
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### Decomposition of order-3 tensors over the complex numbers

This is a question about decomposition of order-3 tensors. The survey Tensor Decompositions and Applications give a good account of recent developments in this area.
Let $T$ be an order-3 tensor, i....

5
votes

1
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515
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### Completely bounded maps on Mn

The aim of this question is to collect nice maps on $M_n(\mathbb{C})$ with the following property:
$\phi_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ with $||\phi||=1$ and $||\phi_n||_{cb}\...

9
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1
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1k
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### When are certain group C*-algebras exact?

This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic.
Anyway. There has been a lot of attention given to showing that for certain ...

6
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0
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354
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### Ordering of completely bounded maps

Let A be a C*-algebra, let H be a Hilbert space, and let $T:A\rightarrow B(H)$ be a completely bounded (cb) map (that is, the dilations to maps $M_n(A)\rightarrow M_n(B(H))$ are uniformly bounded). ...