All Questions
Tagged with oa.operator-algebras c-star-algebras
597 questions
5
votes
1
answer
254
views
A nilpotency question on $C^{*}$ algebras
What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$.
To what extent ...
- 356
7
votes
1
answer
682
views
$c_0$-direct sum of $\mathcal{K}(\mathcal{H})$
Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many ...
- 113
9
votes
0
answers
284
views
Verifying Woronowicz’s proof that all $ {\text{SU}_{q}}(2) $’s are isomorphic as $ C^{*} $-algebras, where $ -1 < q < 1 $
This question is related to one that I asked some time ago.
Definition 1. Let $ q \in (-1,1) $. Let $ A $ be a unital $ C^{*} $-algebra and $ (x,y) $ a pair of elements of $ A $. Then define the $ ...
- 1,396
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 ...
16
votes
0
answers
542
views
$C^*$-algebra generated by those operators that are bounded on every $\ell_p$
Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
- 25.8k
3
votes
0
answers
203
views
Uniqueness of the reduced free product of unital completely positive maps
For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = *...
- 3,984
7
votes
0
answers
244
views
Commutation preserving operators
Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...
- 633
3
votes
2
answers
1k
views
Normal operators and it's spectrum in C*-algebras
If $A$ is a C*-algebra and $n$ is a normal element of $A$, then we have: (By Gelfand duality for example.)
$\operatorname{spec}( |N| ) = | \operatorname{spec}(N) | := \left\{ | \lambda | ; \lambda \...
- 83
3
votes
1
answer
173
views
Obstructions for $C^\star$ algebras to contain a $Z^\star$ algebra
As the comment of Andreas Thom indicated here, a separable $C^\star$ algebra $A$ can not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is a $C^\star$ algebra which all elements are zero ...
- 356
2
votes
1
answer
320
views
Totally non hereditary $C^{*}$-subalgebras
Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any ...
- 356
2
votes
0
answers
100
views
Amenability for Actions twited with 2-cocycles
Let $A \subset B(H)$ be a unital $C^\ast$-algebra and $\theta: G \rightarrow \mathrm{Aut}(A)$ an action and let $\omega: G \times G \rightarrow U(\mathcal{Z}(A))$ be a $2$-cocyle with respect to $\...
- 1,090
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 ...
- 4,058
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
2
votes
0
answers
157
views
Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?
Recall that a locally compact group $G$ is said to be
an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure;
an $[SIN]$ group, if each neighborhood of the identity includes a ...
- 399
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
1
vote
1
answer
331
views
An unconventional definition of the $ C^{*} $-algebraic reduced crossed product
Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system, i.e., $ A $ is a $ C^{*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by ...
- 307
8
votes
3
answers
1k
views
Gelfand duality in NCG
In non-commutative geometry, Gelfand duality is the construction of multiplicative linear functionals of a commutative C*-algebra, which can be viewed as the space of all its irreducible complex ...
- 83
1
vote
0
answers
81
views
A consecutive resolution of continum algebras to a simple continum algebra
Motivated by classical Gelfand Naimark duality, the correspondence between the category of commutative $C^{*}$ algebras and the category of locally compact Hausdorff spaces, we ...
- 356
7
votes
1
answer
301
views
allowing `discontinuous functions' into a C* algebra
There follows a possible construction, and I would like to know if it or a similar construction has been done before (as I suspect), so that I can reference it, or if it obviously does not work! Any ...
- 1,143
3
votes
1
answer
187
views
Conjugacy of circle actions on UHF C*-algebras
Consider pointwise continuous actions of the unit circle on the $2^{\infty}$-UHF C*-algebra A by *-automorphisms. Assume that two such actions have the same fixed point algebra, i.e., elements that ...
- 31
3
votes
1
answer
427
views
Inner automorphisms and $K$-theory
It is known that any inner automorphism of a unital $C^{\ast}$-algebra $A$ induces the identity map on $K_{0}(A)$ because unitary equivalence implies Murray-von Neumann equivalence. What is known ...
- 169
4
votes
0
answers
202
views
Connectivity of the group of invertible elements of $C(S^{2})\otimes A$
For what type of $C^{*}$ algebras $A$, the group of invertible elements of $C(S^{2}) \otimes A$ is a connected group?
All finite dimensional $A$ satisfy this property.
Is it true to say ...
- 356
4
votes
0
answers
207
views
Extending Akemann's Non-Commutative Urysohn Lemma
Assume $A$ is a C*-algebra and $p,q\in A^{**}$ are compact projections.
Can we always find $a,b\in A^1_+$ with $p\leq a$, $q\leq b$ and $||pq||=||ab||$?
Note if $||pq||=1$ this is immediate, while ...
- 1,307
1
vote
0
answers
356
views
Determining the primitive ideal space of C-star algebras
Is there a general way of finding a primitive ideal space of $C^*$-algebra?
For example, if $C^*$-algebra is given by the universal $C^*$-algebra generated by two self-adjoint unitary elements, how ...
- 11
4
votes
1
answer
183
views
A question about correlations between $ C^{*} $-algebras
I was studying J. M. G. Fell’s paper The Structure of Algebras of Operator Fields when I encountered the concept of a correlation between two $ C^{*} $-algebras.
Definition. Let $ A $ and $ B $ be $...
- 1,396
2
votes
1
answer
299
views
Is this left ideal of C*-algebra principal?
This is a follow up of this question. Let $I$ be closed left ideal of $C^*$-algebra $A$.
Assume we are given a sequence of left $A$-module morphisms $R_n:I\to A$ with $\sum_n \Vert R_n\Vert<\infty$...
- 1,697
2
votes
1
answer
327
views
Integration in C^* algebra
Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that
$$
\int d s \, f(s)\, \alpha_s(A)
$$
is well defined as a ...
- 552
8
votes
1
answer
904
views
The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$
Let $Q\in GL_n(\mathbb{C})$. The free unitary quantum group is the universal $C^*$-algebra $A_u(Q)$ with generators $u_{ij},1\leq i,j\leq n$ and relations making $u=(u_{ij})$ as well as $Q\bar{u}Q^{-1}...
- 1,702
1
vote
1
answer
273
views
examples of completely positive order zero maps to demonstrate a theorem
I'm interested explicit examples which can be used to demonstate the theorem:
Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set $C:=C^*(\Phi(A))\...
user62639
2
votes
0
answers
88
views
When is the multiplication map bounded? [duplicate]
Forgive me if my question is too elementary however I haven't found this issue discussed anywhere. Suppose that $A$ is arbitrary $C^*$-algebra and consider the multiplication map $A \overline{\otimes} ...
- 243
1
vote
2
answers
407
views
Questions about special $C^*$-subalgebras and ideals.
Let $A$ be a $C^*$-algebra and $I$ be a two side closed (essential) ideal of $A$. Suppose that $p \in A\backslash I$ is a non trivial projection. Let $B=pIp$. My questions are:
(1) Is $B$ a $C^*$-...
- 147
2
votes
1
answer
182
views
A Possible characterization of F.D or AF commutative $C^{*}$ algebras
By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra.
Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...
- 356
4
votes
1
answer
775
views
Algebraically simple Banach algebras
There are plenty of semi-simple Banach algebras - this broad class includes C*-algebras and algebras of bounded operators on a given Banach space. On the other hand, it seems unlikely to me that there ...
- 113
2
votes
1
answer
244
views
$R$ is a right multiplier and $R(a)b=a\overset{?}{\implies} A$ is unital
Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that
$$
\exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad
$$
implies $A$ is unital. I know this is true if A is a weak$...
- 1,697
1
vote
0
answers
131
views
Ideal structure of group $C^*$-agebras [closed]
Let $G$ be a locally compact groups and $C_r^*(G)$ be a reduce group $C^*$-algebra.
$\ Question:$What is the ideal structure of reduce group $C_r^*(G)$?
- 399
4
votes
0
answers
135
views
References for a lemma about compact operators on a Hilbert module
I am looking for a reference for the following result:
If $A$ and $B$ are C* algebras, $H$ is a right Hilbert $A$-modules, $\phi :A \rightarrow B$ is a morphism, and assume that there is a map $\eta :...
- 42.4k
7
votes
0
answers
241
views
A "slice-map" type problem for symmetric tensors in the square of a nuclear C*-algebra
Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras.
Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...
- 25.8k
1
vote
0
answers
118
views
Finding the infimum of the range of a certain non-negative function associated to a $ C^{*} $-algebra
Let $ A $ be a non-trivial $ C^{*} $-algebra and $ n \in \mathbb{N} $. Setting $ \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} $, we can define a function $ f: \...
- 1,396
10
votes
1
answer
1k
views
When are certain group C*-algebras exact?
This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic.
Anyway. There has been a lot of attention given to showing that for certain ...
- 25.8k
1
vote
1
answer
611
views
Commutant of a von Neumann algebra as the linear span of unitaries.
I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ ...
7
votes
0
answers
602
views
Unique maximal ideal in group C*-algebras
Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$
...
- 2,689
7
votes
0
answers
292
views
What morphisms / Morita equivalences induce the 2-periodicity isomorphisms of $KK$-theory?
In Kasparov's paper, the canonical isomorphisms $KK_* \rightarrow KK_{*+2k}$ are defined rather implicitely (by tensoring and stabilization).
Are there morphisms of $C^*$-algebras which induce them (...
- 435
3
votes
0
answers
146
views
Closed containment of open projections in C*-algebras
For a C*-algebra $A$ and open projections $p,q\in A^{**}$, consider the following statements.
$\overline{p}\leq q$
$p\leq q$ and there exists open $r\in A^{**}$ with $rp=0$ and $r\vee q=1$
$p\leq q$ ...
- 1,307
3
votes
1
answer
301
views
‘Non-Induced’ Left Regular Representations of $ C^{*} $-Dynamical Systems
In what follows, a ‘$ * $-representation’ always means a non-degenerate $ * $-representation.
Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and let $ \pi: \mathscr{A} \to B(\mathcal{...
user36116
2
votes
0
answers
209
views
Number of connected components of a $C^{*}$ algebra
Inspired by the concept in the following post
What are these compact sets called?
We introduce the following concept:
Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent ...
- 356
2
votes
0
answers
234
views
The kernel of $C^{*}(G)\to C_{r}^{*}(G)$
Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism.
What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some ...
- 356
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. ...
- 306
0
votes
0
answers
410
views
A noncommutative vector bundle
We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative ...
- 356
0
votes
0
answers
201
views
Range of a trace preserving completely positive projection
I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...
- 515
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