All Questions
96 questions
32
votes
3
answers
2k
views
What does it mean for a category to admit direct integrals?
Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals.
Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...
- 43.2k
28
votes
0
answers
2k
views
Finite-dimensional subalgebras of $C^\star$-algebras
Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
- 25.5k
17
votes
5
answers
2k
views
If two projections are close, then they are unitarily equivalent
Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$.
The proof that immediately occurs to me uses comparison of ...
- 2,793
16
votes
4
answers
1k
views
Von Neumann algebra associated to the infinite Cuntz algebra
The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition ...
- 7,637
16
votes
3
answers
2k
views
Is the group von Neumann algebra construction functorial?
Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...
- 1,373
15
votes
2
answers
1k
views
Is a C*-algebra with an isomorphic predual a von Neumann algebra?
It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...
- 3,497
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
14
votes
2
answers
899
views
Do torsion-free groups give projectionless group ($C^\ast$) algebras?
One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
- 5,425
13
votes
1
answer
1k
views
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 ...
- 7,047
13
votes
1
answer
451
views
Factor states on C*-algebras
Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras ...
- 771
13
votes
0
answers
3k
views
Connes Embedding Conjecture is false [closed]
This preprint from yesterday claims to prove that Connes Embedding Conjecture fails.
Since the paper is from outside Operator Algebras (Computer Science/Quantum Computing) and they actually work on ...
- 2,793
12
votes
1
answer
901
views
Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem?
This post is an appendix of this one.
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is ...
10
votes
1
answer
492
views
Which W*-algebras are the duals of C*-coalgebras?
A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
- 2,754
9
votes
3
answers
568
views
Defining the abstract tensor product of W*-algebras via a universal property
I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt:
It is easy to show that such an object ...
- 175
9
votes
1
answer
372
views
Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?
Let $\Gamma$ be a discrete group whose reduced group $C^\ast$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $\mathcal{L}(G)$ is a full $\text{II}_1$-factor, ...
- 741
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
8
votes
2
answers
961
views
The monotone closure of a $C^*$-algebra
Related to Jon's question, I have two questions. Let $\mathcal{A}$ be a concrete $C^*$-algebra on a Hilbert space $\mathcal{H}$. For any selfadjoint subset $S$ of $\mathbb{B}(\mathcal{H})$, let $S^m$ ...
- 619
8
votes
1
answer
332
views
The double dual of the unitization of a $C^*$-algebra
I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the ...
- 267
8
votes
1
answer
403
views
When a $C^*$-algebra is an ideal in its second dual?
I would like to know which $C^*$-algebras are ideals in their second duals?
There is a paper by S. Watanabe that claims in introduction that it is well known that a $C^*$-algebra is an ideal in its ...
- 1,697
8
votes
3
answers
2k
views
Definition of a von Neumann algebra
Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in ...
- 37.8k
8
votes
1
answer
460
views
Finite dimensionality of certain $C^{\star}$-algebras
In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up:
Let $H$ be a Hilbert space and $a_1, \dots, a_n \in B(H)$ be self-...
- 25.5k
8
votes
0
answers
952
views
About generator and isomorphism problems for free groups operator algebras
Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
7
votes
1
answer
491
views
Projections in the tensor product of von Neumann algebras
This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...
- 619
7
votes
1
answer
433
views
How the modular theory of von Neumann algebras, deal with generating C*-algebras?
Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable ${\rm C}^*$-algebra such that $A''=M$. Suppose the existence of a ...
7
votes
1
answer
701
views
A Question About Pure States, Support Projections and Central Covers
I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as ...
user36116
7
votes
1
answer
429
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
- 619
6
votes
2
answers
248
views
Extension of a von Neumann algebra by a von Neumann algebra
I asked this question at MSE now I repeat it at MO:
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0\to A\to C\to B\...
- 356
6
votes
1
answer
287
views
Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity
Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...
- 175
6
votes
1
answer
397
views
Real rank 0 implies stable rank 1 on $C^\ast$-algebras?
A $C^\ast$ algebra has defined stable rank (https://www.univie.ac.at/nuhag-php/bibtex/open_files/2079_Rieffel-StableRank.pdf) and real rank (https://core.ac.uk/download/pdf/82123484.pdf), which are ...
- 233
6
votes
1
answer
643
views
Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?
Let $H$ be an infinite dimensional separable Hilbert space.
Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} \}...
6
votes
1
answer
680
views
Is there an operator algebraic reformulation of the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
6
votes
0
answers
233
views
Group $C^*$ vs group von-Neumann algebras
Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
- 9,853
6
votes
0
answers
168
views
Characterizing fullness of a von Neumann algebra by the topology of its bimodules
Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
- 1,090
6
votes
0
answers
132
views
Schröder–Bernstein for representations of operator algebras
This is claimed in a Wikipedia Article:
If two representations (of a $C^*$-algebra $A$) $\rho$ and $\sigma$, on Hilbert spaces $H$ and $G$ respectively, are each unitarily equivalent to a ...
- 18.7k
5
votes
2
answers
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
5
votes
2
answers
988
views
Projections in a W*-algebra as a continuous lattice?
A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...
- 85
5
votes
1
answer
394
views
Polar decomposition in abstract von Neumann algebra
Probably an easy question, but here goes:
In a concrete von Neumann algebra $M \subseteq B(H)$, every element $m \in M$ has a polar decomposition $m= p|m|$ where $p$ is a partial isometry and $|m|= \...
user167952
5
votes
1
answer
428
views
Separable C* algebras and type I states
Let $A$ be a separable $C^*$-algebra and let $\omega$ be a state on $A$.
Then there is an "orthogonal" probability measure $\mu$ on the pure state space $P(A)$ of $A$ such that $\omega(x) = \...
- 759
5
votes
1
answer
321
views
Takesaki's proof of the Kaplansky density theorem
Consider the following fragment from Takesaki's book "Theory of operator algebra I":
Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...
- 175
5
votes
1
answer
499
views
Variations on Kaplansky Density
Let $A$ be a $C^*$-algebra and $\pi:A\rightarrow B(H)$ a faithful $*$-representation, so $M=\pi(A)''$ is a von Neumann algebra and $A\rightarrow M$ is an inclusion. von Neumann's Bicommutant Theorem ...
- 18.7k
5
votes
3
answers
633
views
Ideal of "Compact Operators" in a W*-algebra which gives the sigma-strong-* topology.
In the case of bounded operators on a Hilbert space $\mathcal{H}$, $L(\mathcal{H})$, there are multiple descriptions of the $\sigma$-strong-* topology, namely:
1) As given by seminorms $p_{\phi},~p_{\...
- 83
5
votes
1
answer
208
views
Cartan subalgebras in the group algebras of virtually abelian groups
Let $G$ be a virtually abelian group. Are there any general results on the existence or non-existence of Cartan subalgebras in the generated group $C^*$-algebra or group von Neumann algebra?
- 741
4
votes
1
answer
332
views
Normal linear functionals on bicommutants of C*-algebras
I am going through the proof of the Sherman-Takeda theorem and Fillmore's book "A User's Guide on Operator Algebras" seems to have a nice approach, but something seems off to me:
We need to ...
- 267
4
votes
2
answers
298
views
Takesaki volume II chapter VII lemma 1.15
Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...
- 175
4
votes
1
answer
464
views
Kadison-Singer problem in exotic Hilbert spaces
The Kadison-Singer problem is considered in relation to the separable Hilbert space:
KS: Does every pure state on the diagonal (atomic) masa of $B(\ell_2)$ has a unique extension to $B(\ell_2)$?
...
- 387
4
votes
1
answer
135
views
Is the module action $M\times M^*\to M^*$ jointly continuous?
Let $M$ be a W*-algebra and consider the following map:
$$\gamma: M\times M^*\to M^*: (a,f)\to af$$
where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
- 4,058
4
votes
1
answer
445
views
von Neumann algebras generated by commutators
Let $A$ be a UHF-algebra of type $n^{\infty}$ and denote its unique and faithful trace by $\tau$. Let $L^2(A)$ be the Hilbert space of the GNS-representation associated to $\tau$. We have two ...
- 7,637
4
votes
1
answer
332
views
Support projection vs closed support projection of a normal state in enveloping von Neumann algebra
I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
- 135
4
votes
1
answer
1k
views
Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra
Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.
Let $p,q \in M_{\infty}(A)$ be (...
4
votes
1
answer
217
views
Noncommutative version of Littlewood's First Principle
There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle:
Every Lebesgue
...
- 175