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Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group

I am posting my question of mathstack exchange here. (see: My post on MSE) Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, ...
Tomás Pacheco's user avatar
3 votes
0 answers
96 views

Excising the trace of a $II_1$-factor

Recall that a state $\varphi$ on a $C^*$-algebra $A$ is said to be excised by projections if there exists a net of projections $e_i \in A$ such that $\| e_i a e_i - \varphi(a) e_i\| \to_{i} 0$ for all ...
pitariver's user avatar
  • 297
1 vote
1 answer
211 views

Tensor product of faithful normal states is faithful

I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful. I also ...
J_P's user avatar
  • 439
3 votes
0 answers
253 views

Two more topologies on unitary groups

Let $H$ be a separable Hilbert space and let $\operatorname{U}(H)$ be the group of unitary transformations of $H$. It is well known that the weak, strong and compact-open topologies on $\operatorname{...
Matthias Ludewig's user avatar
1 vote
1 answer
180 views

Conditioning a $\mathrm{C}^*$-algebra state with infinite precision

This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success. Let $\mathcal{A}$ be a unital $\...
JP McCarthy's user avatar
  • 1,037
3 votes
1 answer
244 views

Takesaki: question about lemma in section "Left Hilbert algebras and weights"

To make this question relatively self-contained, this post is quite long, but the question itself is rather short. Consider the following fragments in Takesaki's second volume "Theory of operator ...
Andromeda's user avatar
  • 175
6 votes
1 answer
288 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 $...
Andromeda's user avatar
  • 175
4 votes
1 answer
334 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 ...
Sean's user avatar
  • 135
3 votes
1 answer
121 views

Impact of annihilators in C*-algebras

Let $A$ be a unital C*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$ Suppose that $A$ satisfies the following property: For every $S\subseteq ...
ABB's user avatar
  • 4,058
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 $\...
Andromeda's user avatar
  • 175
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 ...
Andromeda's user avatar
  • 175
2 votes
1 answer
297 views

Predual theorem proof in Takesaki's volume I

Consider the following fragment from Takesaki's book "Theory of operator algebra I" (Section III.3 ,p133-134). Why is the boxed line true? I can see that $\epsilon: \widetilde{A}\to A$ is ...
Andromeda's user avatar
  • 175
3 votes
1 answer
255 views

Takesaki: Lemma about enveloping von Neumann algebra

Consider the following lemma with proof from Takesaki's book "Theory of operator algebra I" (p121): It appears to me that Takesaki claims at the end of the proof that $\pi(A)_1$ is $\sigma$-...
Andromeda's user avatar
  • 175
1 vote
2 answers
148 views

Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra

Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions: B is a von Neumann algebra with $A'' = B$. The inclusion $A \...
Andromeda's user avatar
  • 175
3 votes
1 answer
306 views

Opposite $C^*$ algebras

$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
A beginner mathmatician's user avatar
2 votes
2 answers
302 views

Is $x \mapsto x \otimes 1$ $\sigma$-weakly continuous?

Let $M\subseteq B(H)$ be a von Neumann algebra. Is it true that the mapping $$\psi: M \to B(H \otimes H): m \mapsto m \otimes \text{id}_H$$ is $\sigma$-weakly continuous? Here the $\sigma$-weak ...
user avatar
4 votes
0 answers
254 views

Inflating the double dual of a C*-algebra (matrix algebra of double dual)

in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})\cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can't really see it. Actually, it is enough for me ...
Just dropped in's user avatar
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 ...
Just dropped in's user avatar
4 votes
1 answer
196 views

When a normal functional is restricted to a vn Neumann sub-algebra

I have already asked this question and no comment(s) received up to now. I am so curious to get feedback concerning the problem. Let $M$ be a vn Neumann subalgebra in $B(H)$. Let $f$ and $g$ be ...
ABB's user avatar
  • 4,058
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 ...
Just dropped in's user avatar
7 votes
3 answers
676 views

Noncommutative torus as a von Neumann algebra

Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...
truebaran's user avatar
  • 9,330
0 votes
0 answers
127 views

On examples of action of C-star simple group on von Neumann algebra

Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
user136400's user avatar
0 votes
1 answer
204 views

A certain class of representations

Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$? (The word "finite-dimensional" was ...
MSMalekan's user avatar
  • 2,118
2 votes
1 answer
116 views

Extending $C^*$-norms from $*$-subalgebras

Let $A$ be a unital $*$-algebra, and $B$ a unital $*$-subalgebra of $A$. In addition, assume that there exists a $B$-$B$-sub-bimodule $C \subset A$, such that $$ A \simeq B \oplus C, $$ where $\...
Dave Shulman's user avatar
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 ...
Matthew Daws's user avatar
  • 18.7k
0 votes
0 answers
54 views

On cyclicity of a module

Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
user136400's user avatar
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 ...
Masayoshi Kaneda's user avatar
1 vote
0 answers
95 views

An example of a sequence of finite projections

Let $A$ be a vn-algebra. Suppose that $x$ is an isometry with $\inf_{n\geq1} x^nx^{*n}=0$ (Note that $x^nx^{*n}$ are all projections). Let $e$ be a (non-zero) finite projection and put $q_n$ to be ...
ABB's user avatar
  • 4,058
3 votes
1 answer
309 views

How rich the group of unitary elements in a von Neumann algebra to get "Murray-von Neumann" equivalence?

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Neumann algebra. It is well known that in a finite von Neumann algebra the equivalence of projections can be ...
MSMalekan's user avatar
  • 2,118
3 votes
2 answers
397 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$^*$-...
Masayoshi Kaneda's user avatar
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 $\...
Masayoshi Kaneda's user avatar
2 votes
0 answers
164 views

An operator valued Egoroff's theorem

The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
ABB's user avatar
  • 4,058
3 votes
0 answers
178 views

A point concerning absolute value of functionals

Let $M$ be a von Neumann sub-algebra in $B(H)$. Let $\phi$ be a normal functional on $M$. Assume $\psi$ is a normal functional on $B(H)$ with $\psi_{|_M}=\phi$ (note that $\phi$ and $\psi$ may have ...
ABB's user avatar
  • 4,058
18 votes
3 answers
1k views

In which sense the GNS-construction is a functor?

I asked this at mathstackexchange a week ago, without success. I think the Gelfand–Naimark–Segal construction must be a functor in some sense, but I can't find an explicit statement anywhere. Can ...
Sergei Akbarov's user avatar
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 ...
Hannes Thiel's user avatar
  • 3,497
4 votes
0 answers
187 views

The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment. Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...
ABB's user avatar
  • 4,058
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 ...
ABB's user avatar
  • 4,058
6 votes
2 answers
548 views

When is it $C(X)$?

Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=...
Bob's user avatar
  • 306
1 vote
1 answer
251 views

On the relation between the set of extreme points of the unit ball of $M(X)$ and $M(X)^{**}$

Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C_0(X)$ is a commutative $C^*-$algebra. ...
Bob's user avatar
  • 306
0 votes
1 answer
151 views

Checking complete positivity of maps between C* algebras

Let $\phi$ : $A \rightarrow A$ be a positive map, where $A$ is a (unital) C* algebra. Suppose we are given that $\phi$ is n positive whenever n= $2^k$ for some $k \in \mathbb{N}$. Can we conclude that ...
voldemort's user avatar
  • 181
7 votes
1 answer
702 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 ...
user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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} \}...
Sebastien Palcoux's user avatar
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 ...
Toby Bartels's user avatar
  • 2,754
3 votes
1 answer
503 views

When an AW*-algebra is a W*-algebra

In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence: It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$. ...
Jan Veselý's user avatar
5 votes
2 answers
1k views

Polar decomposition in C*-algebras

A very nice feature of W*-algebras is the following: once you have an element $a$ of a W*-algebra $M$, and $a=u|a|$ (the polar decomposition), then $u\in M$. It seems that it carries over to AW*-...
Jan Veselý's user avatar
4 votes
1 answer
276 views

Abelian sub-W*-algebras

Let $M$ be a von Neumann algebra which acts faithfully on a Hilbert space of density character $\kappa$ but does not on a Hilbert space of density character $\lambda<\kappa$ (that is, the density ...
J. Grass's user avatar
3 votes
1 answer
740 views

Centralizers in C*-algebra

Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$? For ...
spelas's user avatar
  • 179