All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
1
vote
2
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126
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Upper triangular $2\times2$-matrices over a Baer *-ring
Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices
$\left(\begin{array}{cc}
a_1& a_2 \\
0 & a_4
\end{array}\right)$ where $a_i$'s are in $A$. Is $B$ ...
- 4,058
1
vote
1
answer
158
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Is any ultraproduct of $\mathcal{O}_2$ still nuclear?
Is it known wether any ultraproduct of $\mathcal{O}_{2}$, regarding an ultrafilter on $\mathbb{N}$, is still nuclear or not ?
Thank you all !
(references for this would be very much appreciated)
- 11
7
votes
2
answers
658
views
Finding the commutant of a von Neumann algebra
Suppose you have a von Neumann algebra $A$ of operators on $H$ and would like to compute its commutant. You have constructed a collection $B\subset A'$ which you suspect generates it (i.e. you think $\...
- 1,010
14
votes
0
answers
647
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Countably decomposable von Neumann algebras
A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...
- 42.8k
7
votes
1
answer
378
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The quantum group SUq(n) as von Neumann algebra
i have a question about a "presentation" of the quantum $SU(n)$. Here presentation means the following. Let $(M,\Delta)$ be a quantum group in the sense of Kustermanns and Vaes. One can show that the ...
13
votes
1
answer
1k
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Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?
Regarding the hyperfinite $II_{1}$ factor $R$ as $C^{*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of ...
- 7,067
8
votes
1
answer
422
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K theory for pre $C^*$-algebras
In noncommutative geometry when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the ...
- 9,330
6
votes
1
answer
760
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Example of an infinite dimensional reflexive Banach algebra
If a $C^\ast$-algebra is reflexive (as a Banach space) then it is finite dimensional. Can anyone provide (or give a reference to) a nice example of an infinite dimensional non-commutative Banach ...
- 777
4
votes
1
answer
140
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Finding a special Banach algebra and a net of homomorphisms
If $A$ is a Banach agebra and $M$ is a Banach $A$-bimodule then a linear map $T:A\to M$ is called an $A$-module homomorphism if $$T(ab)=aT(b),\quad T(ab)=T(a)b,\qquad a,b\in A.$$ Also $A\hat{\otimes} ...
- 231
3
votes
1
answer
157
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Self adjoint operators in Kasparov-Modules
In Blackadars book in 17.4.2 it says that for each element $x \in KK(A,B)$ there is a Kasparov module $(E,\pi ,T)$ such that $T=T^*$. Now, the argument for that is that if $(E,\pi, T)$ is any Kasparov-...
- 31
9
votes
1
answer
459
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is this von Neumann algebra a tensor product?
Let $H_1$ and $H_2$ be Hilbert spaces. Let $A\subset B(H_1)$ be a factor and $A'$ its commutant.
If a von Neumann algebra $M\subset B(H_1\otimes H_2)$ contains $A\otimes 1$ and commutes with $A'\...
- 43.2k
9
votes
3
answers
409
views
What are the intermediate subfactors of the tensor product of two maximal subfactors?
Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors.
Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...
4
votes
2
answers
957
views
Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
- 619
8
votes
0
answers
493
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Two pictures of K-theory and Bott periodicity
Let me recall the definition of the Bott periodicity isomorphism in the context of $C^*$-algebras. We take a (class of) projection $p \in M_n(A^+)$ and map it to the class of $M_n(A)$ valued loop $f_p$...
- 9,330
17
votes
1
answer
807
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Operator Valued Kadison--Singer Problem
The Paving Conjecture, which is equivalent to the famous Kadison--Singer Problem, was spectacularly settled in the affirmative by Marcus--Spielman--Srivastava (arxiv:1306.3969). Let $E$ denote the ...
- 10.1k
3
votes
1
answer
166
views
is a linear map on an operator system into a $C^*$-algebra (+ extra conditions) positive?
First of all, sorry for my bad english. I tried to find out whether the following statement is true or not:
Let $X$ be a operator system, $B$ a $C^*$-algebra and $f:X\to B$ linear such that $f(1)\ge ...
user62639
5
votes
0
answers
490
views
quotients of nuclear $C^*$-algebras are nuclear
At first a definition of injective (unital) $C^*$-algebras: A unital $C^*$-algebra $C$ is called injective if for all completely positive contractive (cpc, for short) maps $\varphi:X\to C$, where $X\...
- 1,188
6
votes
1
answer
389
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is the maximal tensor product of compact operators an essential ideal?
I'm searching for a counterexample for $C^*$-algebras $A$ and $B$ and essential ideals (I assume an ideal to be closed and only two-sided ideals) $I\subseteq A$, $J\subseteq B$ , such that the ideal $...
- 1,188
5
votes
1
answer
245
views
Convergence of functionals on compact projections on a separable Hilbert space
Let $H$ be a separable Hilbert space over $\mathbb{C}$, say $\ell_2$ for simplicity. Let $\mathcal{K}(H)$ denote the space of all compact operators on $H$ and $\mathcal{P}(H)$ the set of all finite ...
- 1,151
3
votes
0
answers
269
views
Finite dimensional representation of tensor product
Let $A$ and $B$ be $C^*$ algebras, and let $\pi:A \odot B \to B(H)$ be a $*$-representation of the algebraic tensor product on a finite dimensional Hilbert space $H$. Let $x \in A \odot B$. Since $H$ ...
- 1,769
5
votes
5
answers
2k
views
Measurable functions and unbounded operators in von Neumann algebras
How do you define unbounded measurable functions for a general von Neumann algebra?
For the commutative algebra $L^\infty(X,\mu)$, we can consider the space of all measurable functions that are ...
- 228
26
votes
2
answers
4k
views
Finite subgroups of unitary groups
Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite ...
- 25.5k
1
vote
0
answers
89
views
Projections in properly infinite factor
Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let $e$ be a projection in a properly infinite factor. Is it always true that $e\sim 1$ or $1-e\...
- 2,118
5
votes
1
answer
1k
views
When is a Banach Algebra $C^\star$
I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of ...
- 53
7
votes
1
answer
477
views
positive elements in tensor product
Let x be a positive element in the spatial tensor product of two non unital C* algebras
A and B. Is there a single element $a \otimes b \geq x$?
How can we noncommutativize the following proof, in ...
- 356
2
votes
3
answers
515
views
What's the relation between fusion and coproduct?
For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X_i)_{i \in I}$ of ...
6
votes
1
answer
260
views
Fundamental class in $KO[1/2]$
Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...
- 61
3
votes
0
answers
97
views
Why c.p.c order zero maps induce morphism between cuntz semigroups
Maybe a naive question:
Right now I am reading the paper ``Completely positive maps of order zero'' written by Wilhelm Winter and Joachim Zacharias. I do not quite understand the last corollary when ...
- 181
1
vote
1
answer
401
views
Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group
I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$.
Ultimately, I'm interested in finding a ...
- 11
2
votes
0
answers
715
views
Universal $C^*$ algebra generated by two self adjoint elements with $x^2+y^2=1+{(xy-yx)}^2$
Is there universal $C^*$ algebra generated by self adjoint elements $x,y$ subject to relation $x^2+y^2=1+{(xy-yx)}^2$?
What is a precise description of such algebra?
- 356
3
votes
1
answer
352
views
Generators K-theory of Cuntz algebras
The Cuntz algebra $O_n$ is the C*-algebra generated by n isometries $S_1$, ..., $S_n$ such that $S_i^* S_j=\delta_{i,j}$ and $\sum_{i=1}^nS_iS_i^*=1$. Cuntz proved that this algebra has the following ...
- 743
1
vote
1
answer
225
views
A point-wise separation Hahn-Banach theorem in C*-algebras
Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$.
Let $E$ be a norm closed convex subset of positive operators in $K(H)$ ...
- 4,058
3
votes
0
answers
274
views
Decomposition of a von Neumann Algebra into Factors
I know that a von Neumann algebra on a separable Hilbert space can be (uniquely) decomposed into factors. But is there any non-trivial example showing how we can explicitly compute it?
Non-trivial: ...
- 241
4
votes
0
answers
389
views
Künneth formula for $C^*$ algebras, equivalent condition for full generality
I'm crrently reading the paper about the Künneth-theorem for $C^*$-algebras: http://msp.org/pjm/1982/98-2/pjm-v98-n2-p15-s.pdf and I'm trying to understand remark 4.9. I henceforth asumme that $A$ is ...
- 81
4
votes
1
answer
135
views
Strong additivity of the conformal net of an affine simple Lie algebra
Let $\mathfrak g$ be a complex simple Lie algebra, $l$ be a natural number, and $V=V^l(\hat g)$ be the vertex operator algebra of the affine Lie algebra $\hat{\mathfrak g}$ at level $l$. We know that $...
- 585
2
votes
0
answers
116
views
Factorization of the Fokker-Planck semigroup
I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
4
votes
0
answers
298
views
Operator topologies
Let $L(H)$ be the space of bounded operators on some Hilbert space.
We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT).
...
- 41
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0
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85
views
Injectivity of Fermion algebras
Is the Fermion algebra or $-1$-Fock space as defined in https://arxiv.org/pdf/math/0303045.pdf hyperfinite? Any references?
- 455
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votes
2
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925
views
Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
- 221
5
votes
2
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208
views
Extension of $C^*$ isomorphism to $W^*$ isomorphism
Let $\mathfrak{A}$ be $C^*$algebra, and $\pi$ its faithful representation on Hilbert space $\mathcal{H}$. Bicommutant $\mathfrak{B}=\pi(\mathfrak{A})''$ is the von Neumann algebra generated by $\pi(\...
- 552
9
votes
2
answers
883
views
Are the reduced group Von Neumann algebra/ Group $C^{\ast}$ algebra functorial in the case of LCH groups
Let $G$ be a LCH group and $\mu$ be its left Haar measure. Call $\lambda_G : G \to U(L_2(G,\mu))$ the left regular representation. We can define the reduced $C^{\ast}$ algebra and reduced Von Neumann ...
- 1,090
10
votes
3
answers
1k
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Subfactor summer reading list
Many people I talk to lament the nonexistence of a coherent source for learning the theory of subfactors.
Could someone suggest a nice (ordered) list of books/papers to work through to obtain a ...
- 7,067
6
votes
1
answer
632
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Are isometric homorphisms of C* algebras *-homorphisms
Here is my precise question:
Let $A$ and $B$ be two $C^*$ algebras. Let $f: A \rightarrow B$ a homomorphism of $C^*$ algebras which is isometric (on its image). Is $f$ necessary a $*$-homomorphism ?
...
- 42.4k
2
votes
1
answer
330
views
Faithul map and (minimal) tensor product of $C^*$-algebras
Let $f$ be a faithful state on a $C^*$-algebra $A$, i.e. $f(a^*a)=0$ implies $a=0$. in general, call a mapping $T:A \to B$ between $C^*$-algebras faithful if $T(a^*a)=0$ implies $a=0$. How to prove ...
- 9,330
11
votes
1
answer
512
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Decomposability of positive maps
By results of Størmer and Woronowicz, every positive map $\Phi \colon \mathcal{M}_{d \times d} \rightarrow \mathcal{M}_{d' \times d'}$ for $dd' \leq 6$ can be decomposed as a convex combination
$$\...
- 435
8
votes
1
answer
180
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$\mathcal{O}_{\infty}$ and $\mathcal{Z}$ stable isomorphism as equivalence
If $O$ is a (strongly ?) self absorbing $C^*$-algebra, one has an equivalence relation on (separable) $C^*$-algebras: "being $O$-stably isomorphic" i.e. $A$ and $B$ are $O$-stably isomorphic if and ...
- 42.4k
4
votes
1
answer
507
views
A question on complex line bundle over $S^{2}$
Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$.
Assume that $\ell$ is a sub line bundle of ...
- 356
0
votes
2
answers
158
views
Recover hermitian matrices out of the knowledge of their trace
Let $M_{1},\ldots, M_{m}$ be $m$ hermitian $N\times N$ matrices, and $tr=\frac{1}{N}Tr$ the normalised trace on the algebras of such matrices. If you know the quantities $tr(M_{i_{1}}\ldots M_{i_{...
- 137
22
votes
1
answer
745
views
The Mackey Topology on a Von Neumann Algebra
Every von Neumann algebra $\mathcal M$ is the dual of a unique Banach space $\mathcal M_* $. The Mackey topology on $\mathcal M$ is the topology of uniform convergence on weakly compact subsets of $\...
- 1,199
21
votes
0
answers
869
views
Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
- 28.6k