Questions tagged [c-star-algebras]
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
36 questions from the last 365 days
4
votes
0
answers
121
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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 \...
- 113
19
votes
0
answers
474
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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 $\mathrm{C}^\ast$-...
- 10.1k
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, ...
- 261
-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.
- 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})...
- 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 ...
- 261
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 ...
- 191
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 $H$,...
- 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,...
- 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 ...
- 356
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 ...
- 261
0
votes
0
answers
124
views
Do the following two notions of quantum groups sometimes coincide?
On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
- 163
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(...
- 569
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 ...
- 261
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 $...
- 9,019
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^*,~ ...
- 9,019
0
votes
0
answers
43
views
How to define a family of Hilbert $A-B$-bimodules $ \pi \ : \ M \to X $, parametrized by a $C^*$-algebra $X$?
Let $A$ and $B$ two $ C^* $ - algebras.
I would like to define a functor $ X \to \mathrm{Bimod}_{A,B} (X) $ which associate to any object $X$, the set of isomorphism classes of a family of Hilbert $A-...
- 595
0
votes
0
answers
27
views
How to endow a cross normed tensor product $C^*$ - algebra with a structure of $G$ - $C^*$ - algebra?
Let $G_1$ and $G_2$ two topological groups which are locally compact, Hausdorff, and second countable.
Let $A_1$ ( resp. $A_2$ ) a $G_1$ - $C^*$ - algebra ( resp. $G_2$ - $C^*$ - algebra ).
Let $A_1 \...
- 595
2
votes
0
answers
125
views
Is there a theory of fundamental groups for $C^*$-algebras instead of topological spaces?
Is it possible to construct a theory of fundamental groups $\pi_1 (A,a_0)$ for pointed $C^*$-algebras $(A,a_0)$ instead of pointed topological spaces $(X,x_0)$ : $\pi_0 (X,x_0)$ ?
If the answer is yes,...
- 595
1
vote
0
answers
82
views
How to define explicitly the Kasparov product $ x \otimes_B y \in KK_{i+j}^G (A,C) $ of $x \in KK_i^G (A,B)$ and, $y \in KK_j^G (B,C)$?
Let $A,B,C$ be separable $G-C^*$ - algebras. Then there is a biadditive pairing for $i,j \in \mathbb{Z}_2$,
$$ KK_i^G (A,B) \times KK_j^G (B,C) \to KK_{i+j}^G (A,C) $$
If $x \in KK_i^G (A,B)$ and, $y \...
- 595
1
vote
0
answers
63
views
$\operatorname{ker}(q_I \otimes^{\text{min}} q_J) $ is a primal ideal of $\mathcal{A} \otimes^{\text{min}} \mathcal{B}$
In the proof of Theorem $4.1$ of the paper titled continuous bundles of $C^{\ast}$-algebras and tensor products following result is mention with a reference to Proposition $3.3$ of the paper "A. ...
- 1,115
0
votes
0
answers
188
views
Behavior of subtree of $\mathbb{Z}^2$ embedded in $\mathbb{C}$ under compactification of the latter to the riemann sphere
I consider a countable subtree $T$ of the integer lattice isomorphic to $\mathbb{Z}^2$ with directed edges. It shall be embedded in $\mathbb{C}$ where the edge $(u,v)$ points from $u$ to $v$ if and ...
- 91
3
votes
1
answer
181
views
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras, is there any good notion of "normal bundle of $B$ in $A$"?
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras (maybe more restricted kind of star algebra), is there any good notion of "normal bundle of $B$ in $A$"? By a "...
- 31
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 ...
- 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 ...
- 595
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 ...
- 11
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 ...
- 297
1
vote
0
answers
148
views
Vanishing (infinite) tensor products
Since the advent of free probabilities and QFT, infinite tensor products of $R$-associative algebras with units has become more familiar to the working mathematician.
Starting from the (permuting) ...
- 3,738
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-...
- 51
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^{...
- 1,485
2
votes
0
answers
145
views
About normal states in abstract von Neumann algebras
In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16
but this was state only for concrete von Neumann algebras (because ...
- 424
3
votes
0
answers
132
views
Takesaki's duality in representation theory of $C^*$-algebras
In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting.
...
- 139
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 ...
- 1,027
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 ...
- 33
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}{...
- 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{...
- 1,027