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19 votes
0 answers
472 views

On C*-rigidity problem for torsion-free groups

I'd like to address the $\mathrm{C}^\ast$-rigidity problem for torsion-free groups (see this paper), which asks for non-isomorphic torsion-free groups with isomorphic (reduced) group $\mathrm{C}^\ast$-...
Narutaka OZAWA's user avatar
4 votes
0 answers
120 views

Projection behavior under the automorphism $\Phi$ on a diffuse semi-finite von Neumann algebra

Let $\mathcal{M}$ be a diffuse semi-finite von Neumann algebra acting on a Hilbert space $\mathcal{H}$, equipped with a faithful normal semi-finite trace $\varphi$. Let $\Phi : \mathcal{M} \to \...
DenOfZero's user avatar
  • 113
1 vote
0 answers
86 views

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
1 answer
162 views

Amenable non-Hausdorff groupoids

Is there any clear definition of amenable non Hausdorff groupoids? It should be possibly non-separable nuclear C*-algebras? Please let me know if there is any existing literature talking about this.
PKOA's user avatar
  • 15
2 votes
0 answers
232 views

$\mathcal{B}(\mathcal{H})$ as the reduced $C^*$-algebra of a groupoid

Given an infinite dimensional Hilbert Space $\mathcal{H}$ what is the underlying locally compact Hausdorff etale groupoid $G$ such that $C_r^{\ast}(G)$ is $\ast$-isomorphic to $\mathcal{B}(\mathcal{H})...
PKOA's user avatar
  • 15
1 vote
1 answer
89 views

Continuous functions on HLS groupoids

I am reading a paper about property (T) for groupoids: Topological property (T) for groupoids. In section 4.4 they discuss the HLS groupoids which I describe define here. Let $\Gamma$ be a discrete ...
Tomás Pacheco's user avatar
6 votes
1 answer
170 views

Do projections in an $AW^\ast$-algebra form an orthomodular lattice?

I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
David Gao's user avatar
  • 2,830
3 votes
1 answer
116 views

Does a bounded positive modular sesquilinear form on a $C^\ast$-algebra induces an element of its multiplier algebra?

This is a question that originates from my attempt at this question. Specifically, for a $C^\ast$-algebra $A$, I am attempting to construct a map $\phi: A \times A \to A$ s.t., $\phi$ is sesquilinear,...
David Gao's user avatar
  • 2,830
3 votes
1 answer
191 views

A possible spectral characterization of commutative $C^*$ algebras

Let $A$ be a $C^*$ algebra. Assume that the spectrum $Sp(a_1a_2\ldots a_{n-1}a_n)$ is unchanged as a set after a permutation of $a_i$'s. (unless possible emerge or removing 0 from the spectrum) Does ...
Ali Taghavi's user avatar
3 votes
0 answers
73 views

What are the Cuntz semigroups of the Cuntz algebras?

Are the Cuntz semigroups known for the Cuntz algebras $\mathcal{O}_n$ ($1<n<\infty$)? I searched the literature and couldn't find it anywhere. I'm especially looking for $W(\mathcal{O}_n)$ but ...
Panini's user avatar
  • 191
4 votes
1 answer
275 views

What are the norms of the generators of the standard Podleś sphere?

Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations \begin{equation*} \begin{split} &a=a^*,~ ...
Zhaoting Wei's user avatar
  • 9,009
4 votes
0 answers
147 views

Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups

In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
Tomás Pacheco's user avatar
3 votes
0 answers
97 views

Is a localised "restricted symmetry" automorphism implementable as a unitary operator on the GNS Hilbert space?

I have a pure state $\omega$ on a quasilocal algebra $\mathcal{A}$ on a 2d lattice $\Gamma = \mathbb{Z}^2$ with a $\mathbb{C}^d$ vector space on each site. Let there be a unitary symmetry action $U_g(...
pyroscepter's user avatar
3 votes
0 answers
109 views

Faithful traces on reduced $C^*$-algebra of a measured groupoid

Let $G$ be a measured étale groupoid with quasi-invariant measure $\mu$ (that induces the reduced $C^* $-algebra, meaning it has full support) with associated equivalent measures $\nu,\nu^{-1}$. Is ...
Tomás Pacheco's user avatar
6 votes
0 answers
126 views

How obtain the right definition of smooth elements in a $C^*$-algebra?

In Alain Connes' $C^*$-algèbres et géométrie différentielle (an English translation is here,), for a $C^*$-algebra $A$, we consider a $C^*$-dynamic system $(A,G,\alpha)$, where $G$ is a Lie group and $...
Zhaoting Wei's user avatar
  • 9,009
1 vote
1 answer
146 views

Form of a hereditary subalgebra of $C^*$-algebra $C_0(X)$

I would like to show that: "every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed ...
VvvV's user avatar
  • 11
8 votes
1 answer
389 views

Order bounded version of monotone complete $C^*$-algebras

Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
Jochen Glueck's user avatar
2 votes
1 answer
231 views

Characterization of certain subalgebras of $M_2(\mathcal{A})$ where $\mathcal{A}$ is a $C^*$-algebra

Let $\mathcal{A}$ be a $C^*$-algebra generated by a single element $a \in \mathcal{A}$. Suppose that it is also generated by another element $b \neq a$. Consider a subalgebra $\tilde{\mathcal{A}}$ of ...
plllnt's user avatar
  • 133
-1 votes
1 answer
102 views

Is an $A$-$B$—$C^*$-correspondence a representation of a $G$-$C^*$-algebra, $\rho \colon A \otimes_{ \alpha } B \to \mathcal{L} ( \mathcal{H} )$?

Let $R$ and $S$ be two rings. It is known that an $R$-$S$-bimodule is actually the same thing as a left module over the ring $R \otimes_{\mathbb{Z}} S^{\mathrm{op}}$, where $S^{\mathrm{op}}$ is the ...
Angel65's user avatar
  • 595
6 votes
1 answer
322 views

Pairwise orthogonality for partitions of unity in a *-algebra

Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
JP McCarthy's user avatar
  • 1,027
10 votes
0 answers
397 views

Is $\mathcal{B}(\mathcal{H})$ a groupoid $C^*$-algebra?

Let $\mathcal{H}$ be a complex Hilbert space, and $\mathcal{B}(\mathcal{H})$ be the $C^{\ast}$-algebra of bounded operators on $\mathcal{H}$. Is there an étale groupoid $\mathcal{G}$ such that its $C^{...
Luiz Felipe Garcia'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
9 votes
3 answers
451 views

Comparison between the operator norm and the $L^1$ norm on group algebras

Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question: The 2-norm given by $||\sum_{g \in G} c_gg||_2^...
David Gao's user avatar
  • 2,830
5 votes
0 answers
265 views

Failure of Tomiyama's property ($F$) for reduced group $C^*$-algebras

Are there known examples of discrete groups such that the minimal tensor product of their reduced group $C^\ast$-algebras does not have Tomiyama's property ($F$)? Such groups must necessarily be non-...
Are Austad's user avatar
6 votes
2 answers
477 views

Linear map between projective finitely generated Hilbert modules is adjointable

Let $A$ be a (unital) $C^*$-algebra and $X,Y$ right Hilbert $A$-modules which are finitely generated and projective. It seems to be well-known that if $T: X \to Y$ is an $A$-linear map, then $T$ is ...
Andromeda's user avatar
  • 175
2 votes
1 answer
381 views

Lattices and noncommutative algebras in noncommutative geometry

This a question that I've asked in mathematics stack exchange without having received any response : I am interested in the relation between lattices and noncommutative algebras in the context of ...
Esmond's user avatar
  • 136
0 votes
0 answers
115 views

$C^*$ algebra generated by conjugation of an element

Assume $\mathcal{A}$ is a unital $C^*$ algebra and consider some positive-definite element $\Psi\in M_n(\mathcal{A})$. Can we say something about $C^*(\langle \Psi^{-\frac{1}{2}}E_{i,i}\Psi^{\frac{1}{...
GBA's user avatar
  • 167
1 vote
1 answer
284 views

A certainty principle?

Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra with $\varphi\in\mathcal{S}(\mathcal{A})$ a state. Where $$\sigma_\varphi(a):=\sqrt{\varphi((a-\varphi(a)1_{\mathcal{A}})^2)}\qquad (a\in \mathcal{...
JP McCarthy's user avatar
  • 1,027
11 votes
0 answers
375 views

Why are projectionless $C^*$-algebras important (Kadison's conjecture)

It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
Alexandar Ruño's user avatar
2 votes
0 answers
158 views

Question about the ergodic mean

This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question. I've read a thesis where there is an example on ergodic mean, where however there is ...
MBlrd's user avatar
  • 33
4 votes
0 answers
220 views

Bochner theorem for (non-abelian) discrete groups

I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
Tomás Pacheco's user avatar
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
1 answer
244 views

inclusion of von Neumann algebras implies reversing inequality of its modular operators

I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999) Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...
Gabriel Palau's user avatar
1 vote
1 answer
256 views

Intersection of two intermediate subalgebras

Suppose $B\subset A$ is an inclusion of simple $C^*$-algebras with a conditional expectation of (Watatani) index-finite type and $B^{\prime}\cap A=\mathbb{C}$. Then we know $B^{\prime}\cap A_1$ is ...
Keshab Bakshi's user avatar
3 votes
1 answer
155 views

Is a compact set of extreme points contained in a compact face?

I have run into the following question in convex analysis, which I haven't found answered in the literature: Suppose that $K$ is a "nice-enough" non-compact convex subset of a Hausdorff ...
Sean's user avatar
  • 135
1 vote
1 answer
209 views

Borel functions in C*-algebras

Is there a way of defining representations of separable $C^*$-algebras, say $\Phi$, so that $\Phi(A)$ is faithful representation of $A$ on a separable Hilbert space. There is a closure operation $A\...
user52345435's user avatar
0 votes
1 answer
152 views

Unitary representation of a group of automorphism on an abelian algebra

Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...
MBlrd's user avatar
  • 33
4 votes
0 answers
242 views

On the Dunford-Pettis property and multiplier algebras

I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that: Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...
Alexander Dobrick's user avatar
1 vote
1 answer
286 views

A subalgebra of $B(H)$ which does not contain a commutator element

Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property: The algebra $A$ has trivial intersection with the set of commutator ...
Ali Taghavi's user avatar
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 $...
Andromeda's user avatar
  • 175
5 votes
0 answers
114 views

Realize a $K_0$-group homomorphism by a unital $\ast$-homomorphism

This question is inspired by Exercise $7.7$ in *An Introduction to $K$-theory for $C^*$-algebras (available here). Given a unital AF-algebra $A$ and another unital $C^*$-algebra $B$ that has ...
Sanae Kochiya's user avatar
4 votes
0 answers
211 views

Irreducible representations of $\mathrm{UHF}_n$

I have a question about the irreducible representations of the $C^*$-algebra $\mathrm{UHF}_n = \bigotimes_{k=1}^\infty M_n$. For every sequence of unit vectors $(\xi_k)$ in $\mathbb C^n$ there is a ...
Lau's user avatar
  • 759
7 votes
1 answer
422 views

When is the multiplication map of the algebraic tensor product of C*-algebras injective?

A classic result, of Murray and Von Neumann I believe, is that if $\mathcal M\subseteq B(H)$ is a factor then the $*$-homomorphism $\pi : \mathcal M \odot \mathcal M' \rightarrow B(H)$ given by $\pi(...
Chris Ramsey's user avatar
  • 3,984
10 votes
1 answer
518 views

For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$?

Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$. For what kind of $C^*$ algebras $A$ does the following hold: $$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\...
Ali Taghavi's user avatar
-3 votes
1 answer
325 views

Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras

A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor. They form a category with usual structures. Question. Is this category equivalent to the category of $C^*$ algebras? ...
Ali Taghavi's user avatar
7 votes
0 answers
159 views

Maps in the Künneth theorem for K-theory of C*-algebras

The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...
AlexE's user avatar
  • 2,998
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
  • 1,027
2 votes
1 answer
525 views

Difference in tracial and finite von Neumann algebras

A tracial von Neumann algebra $(M,\tau)$ is a von Neumann algebra with a faithful normal tracial state $\tau$ on $M$. That is, $\tau$ is a function from $M \to \mathbb{C}$ such that it is a faithful ...
Anupam Ah's user avatar
  • 163
10 votes
1 answer
283 views

Faithful extreme traces on group C*-algebras

Let $G$ be a discrete amenable, residually finite, ICC(i.e. each non-trivial conjugacy class is infinite) group. Let $C^*_r(G)$ be the reduced group $C^*$-algebra of $G$. Since $G$ is ICC the (...
Caleb Eckhardt's user avatar
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

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