All Questions
Tagged with operator-algebras or oa.operator-algebras
2,152 questions
5
votes
1
answer
604
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Classification of representations of CCR algebras?
Hi,
I'm wondering if there is a some classification of representations of CCR algebras (http://en.wikipedia.org/wiki/CCR_algebra), where say the underlying vector space is a separable Hilbert space. ...
- 53
3
votes
1
answer
530
views
Weakly solid factors?
A type $II_{1}$ factor $\mathcal{M}$ is solid if for every diffuse von Neumann subalgebra $\mathcal{A}$ of $\mathcal{M}$, the relative commutant $\mathcal{A}'\cap \mathcal{M}$ is injective. (See Ozawa'...
1
vote
0
answers
189
views
Non-invariant subspaces for subfactors.
Let $\mathcal{M}$ be a II_1 factor. If $\mathcal{N}$ is a subfactor of $\mathcal{M}$ ($\mathcal{N} \neq \mathcal{M}$), does there always exist a projection in $\mathcal{M}$ such that $(I-P)AP \neq 0$ ...
- 61
1
vote
1
answer
176
views
Does a dense submodule of a free module always contain a basis?
Let $R$ be a completed normed ring, eg Banach algebra. Suppose that $F$ is a free $R$-module of infinite rank with a norm defined by the square root of sum of all norms of its components. If $F'$ is a ...
- 1,373
1
vote
1
answer
433
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Intersection of ideals in C*-algebra or even rings in general
Dear all,
here is a question that has been bothering me. It goes without saying that I would appreciate any help in answering it.
Let {I_k} be a countable sequence of two sided closed ideals in a C*-...
0
votes
0
answers
416
views
norm one approximate identities in separable C* algebras
I'm trying to prove Corollary 1.4.9 in K. Davidson's book (Exercise 1.5):
If A is a separable C* algebra, then there is an increasing sequence
$E_i, i=1,...,\infty$ of positive norm-one elements ...
1
vote
2
answers
177
views
Restriction on the coefficients for an operator in the free group factor $ L(\mathbb{F}_2) $
Let $\mathbb{F}_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family
$\{L_{x_g} : g \in G\}$, here, with $g \in G$...
- 1,528
1
vote
1
answer
103
views
A set of vector such that any vector is orthogonal to to certain other vector (a generalization of the biorthogonal system of vectors))
We are given vectors $\mathbf a_1,\dots,\mathbf a_N\in\mathbb C^n$. Let these vectors be columns of $n$-by-N matrix $\mathbf A$. We additionally know that any $n-1$ vectors are linearly independent. ...
- 11
15
votes
0
answers
790
views
Must we close weakly to apply the spectral theorem?
Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate.
The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...
- 7,329
5
votes
0
answers
157
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 ...
- 315
4
votes
0
answers
417
views
Connes fusion and the composition of completely positive maps
Let $N$ be a type $II_{1}$-factor with trace $\tau$.
An $N-N$ correspondence is a Hilbert $N$-bimodule $H$ where the left and right actions are both ultraweakly continuous. Equivalently, a ...
- 7,047
0
votes
1
answer
135
views
Geometric approximation of projections in a finite von Neumann factor
Let $(M,\tau)$ be a finite von Neumann factor (in my case $M=R^\omega$, but I don't think this additional hypothesis might be useful for this particular problem) and fix a projection $p$. Let $\tau_p$ ...
- 6,034
3
votes
0
answers
528
views
A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede
In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values)
Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq \...
- 227
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
5
votes
0
answers
352
views
"topological" conjugacy of group automorphisms
In the paper "Orbit Equivalence and Topological Conjugacy of Affine
Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:
Theorem. Given two actions $\alpha$ and $\...
- 3,897
10
votes
0
answers
325
views
H-space structure on the Calkin algebra
By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
- 7,637
13
votes
0
answers
564
views
Symmetric (extended) Haagerup tensor product
Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum ...
- 18.7k
0
votes
0
answers
164
views
Can we separate Toeplitz matrices for negative and positive eigenvalues?
Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...
- 9
1
vote
0
answers
742
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Tensor products as isomorphic functors in category theory
An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my ...
- 283
9
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0
answers
821
views
Pimsner-Popa Bases
Let $N\subset M$ be a finite index $II_1$-subfactor. Let $B=\{b_i\}$ be a finite orthonormal (Pimsner-Popa) basis for $M$ over $N$. Let $d=[M\colon N]^{1/2}$. It is well known that $B_1=\{d b_{i_1} ...
- 5,425
2
votes
2
answers
207
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Are all automorphisms of Lin(V) given by similarity transforms?
Let $V$ be a vector space with dimension greater than 1 over the field $F$ and $Sim = \{(f\in \operatorname{Lin}(V))\mapsto gf(g^{-1}) : g\in \operatorname{GL}(V)\}$, ie $Sim$ is the set of all ...
user5810
8
votes
0
answers
339
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Canonical Time Evolution for Type $II_{1}$-Factors?
This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor (...
- 7,047
1
vote
1
answer
287
views
How coarse is the coarse correspondence?
Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) \overline{\...
- 7,047
9
votes
1
answer
395
views
Is there a coalgebraic characterisation of the hyperfinite II_1 factor?
Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...
- 5,094
1
vote
2
answers
156
views
How to study the behavior of a particular function on a Vector Space.
Let, $V$ be a vector space over a field $K.$ Let, $T$ be a function from $V$ to $V$ such that
$T(kX) = kT(X)$ for all $k \in K$ and for all $X \in V$ and also
$T(k + X) = T(X)$ for all $k \in K$ ...
- 73
1
vote
1
answer
293
views
Norm inequality for stochastic maps
I know that if $\Lambda$ is a stochastic positive linear map, i.e., $\Lambda(I) = I$, it is true that
\[ \|\Lambda(B)\| \leq \| B \| \]
For any operator $B$, ...
- 702
7
votes
0
answers
322
views
Is it known that "hyperfinite length" cannot distinguish free group factors?
Given a type $II_{1}$ factor $M$, Popa and Ge defined the hyperfinite length $l_{h}(M)$ of $M$ to be the minimum natural number $n$ such that there are hyperfinite subalgebras $R_{1}, R_{2},..., R_{n}$...
- 7,047
4
votes
1
answer
246
views
Are all continuous linear operators on the space of entire functions "simple"?
Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions.
For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||_n : \operatorname{Ent} \to \mathbb{R}$ ...
user5810
4
votes
0
answers
282
views
If the flip automorphism of a finite factor can be connected to the identity is it approximately inner?
This intriguing question is due to Sorin Popa, so I'm marking it community wiki. I'd like to know the status of the question, and if it is still open, perhaps gather some new ideas for attacking it.
...
6
votes
0
answers
354
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). ...
- 18.7k
2
votes
0
answers
249
views
Strict complete contractions
Let $A$ be a $C^*$-algebra. The Stinespring construction shows that a completely positive contraction $T:A\rightarrow B(H)$ has the form $T(x) = U^* \pi(x) U$ where $U:H\rightarrow K$ is a ...
- 18.7k
0
votes
0
answers
185
views
Characterization of Complex Group Algebras
Is there a way to characterize which complex algebras arise as the group algebra of some locally compact group? To make this more concrete, say $A$ is a sub-algebra of $\text{Mat}(n,\mathbb{C})$, $n\...
- 445
1
vote
0
answers
140
views
Diagonalizing matrices of linear forms of indeterminates
Let $B$ be a matrix with elements as linear forms of indeterminates. Is there a proper diagonalization procedure for such matrices like those of matrices with real and complex entries?
- 13.9k
1
vote
0
answers
212
views
Grading on Multiplier Algebras
A graded C*-algebra A is inner-graded if there exists a self-adjoint unitary $\varepsilon$ in the multiplier algebra M(A) of A which implements the grading automorphism $\alpha$ on A: $\alpha(a)=\...
- 1,702
4
votes
1
answer
228
views
When can closedness of the range of an operator be checked on a positive cone?
Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements ...
user2412
0
votes
0
answers
320
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A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
8
votes
0
answers
298
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Is the "Laplacian" a MASA in a Burnside Factor?
It is a basic fact that every type $II_{1}$ factor posesses a maximal abelian $*$-subalgebra (MASA). My question concerns the concrete realization of such subalgebras. For example, in the free group ...
- 7,047
4
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0
answers
256
views
A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 18.7k
0
votes
0
answers
167
views
Primitive Ideals of $\mathcal{B}_0(\mathcal{H})^\sim$ (identity adjoined)
A note I'm studying says its primitive ideals space is $\lbrace \lbrace 0 \rbrace, \mathcal{B}_0(\mathcal{H}) \rbrace $. I think it might be just $\lbrace \lbrace 0 \rbrace \rbrace $ so I'm somewhat ...
- 1
5
votes
0
answers
220
views
When can't spaces of correspondences distinguish type $II_{1}$ factors?
If $M$ is a type $II_{1}$ factor with trace $\tau$, let $Corr(M)$ denote the space of unitary equivalence classes of $M-M$ correspondences (binormal $M-M$ bimodules) equipped with Popa's analogue of ...
- 7,047
5
votes
0
answers
415
views
Direct integrals and fields of operators
Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space.
Suppose now ...
- 3,897
1
vote
0
answers
308
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Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 2,600
2
votes
0
answers
200
views
Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21
4
votes
0
answers
268
views
Can one pose a Toeplitz index problem associated to a discrete group?
Before posing my question, let me provide a little background since the Wikipedia page on this stuff is sorely lacking.
Let's start with the classical case of the Toeplitz index problem on the circle....
- 29.2k
3
votes
0
answers
383
views
Neglect of Compact Quantum Metric Spaces [closed]
Does anyone have an opinion on Rieffel's theory of compact quantum metric spaces? To me it seems to be a very interesting new area of mathematics. It shows how to generalise complicated geometric ...
- 1,462
3
votes
0
answers
134
views
Sheafification of Arens-Michael algebra-valued presheaves
Let $\mathcal A$ be the category of Arens-Michael algebras, that is, projective limits of Banach algebras. Since $\mathcal A$ is a concrete category, an $\mathcal A$-valued presheaf $A$ admits a set-...
- 393
3
votes
0
answers
178
views
One-parameter groups acting on dual Banach spaces
Let $E$ be a Banach space, and $M=E^*$ (my application has $M$ a von Neumann algebra, but this is unimportant). Let $(\sigma_t)$ be a SOT cts one-parameter group on $E$: so for $t\in\mathbb R$, we ...
- 18.7k
2
votes
0
answers
169
views
The orthogonal of $[A,B]$ in $M_n(k)$
Let ${\mathcal A}$ be the algebra spanned by the words in two letters $x$ and $y$. Its (infinite) basis is $1,x,y,x^2,xy,yx,y^2,...$
Let ${\mathcal A}_0$ be the sub-space (warning: not the sub-...
- 52.3k
3
votes
0
answers
153
views
Isomorphism of categories of rigged modules via completely bounded isomorphism of operator algebras
This question is a background for my previous question.
Suppose $A$ and $B$ are two algebras over $\mathbb{C}$ with the sequences of norms $\lbrace\|\cdot\|_{\Xi,n}\rbrace$ and on $M_n(\Xi)$, $\Xi\...
- 613
3
votes
0
answers
130
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Positive block matrices over tensor algebras
Let $A$ be a unital C*-algebra. A positive block matrix in $M_2(A)$ must have the form
$$ \begin{pmatrix} a & a^{1/2} x b^{1/2} \\ b^{1/2} x^* a^{1/2} & b \end{pmatrix}, $$
where $a,b$ are ...
- 18.7k