Questions tagged [operator-spaces]

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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 ...
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7 votes
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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 $\mathcal{A}$...
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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 $H_1,H_2$. ...
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3 votes
2 answers
384 views

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$^*$-...
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7 votes
1 answer
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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 ...
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1 vote
1 answer
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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 ...
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4 votes
1 answer
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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 ...
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1 vote
0 answers
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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}\...
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6 votes
2 answers
492 views

"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 ...
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5 votes
1 answer
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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 ...
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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 ...
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3 votes
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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. $CV_p(G)$ ...
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1 vote
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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 $R(1) = C$)...
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7 votes
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175 views

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 ...
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12 votes
1 answer
356 views

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 ...
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3 votes
1 answer
994 views

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): ...
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20 votes
1 answer
749 views

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 $...
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2 votes
1 answer
86 views

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\|_{...
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5 votes
0 answers
149 views

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 ...
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4 votes
0 answers
266 views

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 $C^{\ast}$-...
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5 votes
1 answer
441 views

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 ...
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2 votes
1 answer
629 views

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....
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5 votes
1 answer
506 views

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}\...
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9 votes
1 answer
1k views

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 ...
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6 votes
0 answers
352 views

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). ...
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