All Questions
Tagged with fa.functional-analysis c-star-algebras
316 questions
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
9
votes
1
answer
596
views
Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?
Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$
Now ...
- 335
4
votes
1
answer
534
views
The C*-envelope of the algebra of continuous functions on a compact topological space is commutative
In my research in operator theory, specifically in C* algebras and enveloping, I came across this strange footnote in a text (locally published in non English where I study) which states the following:...
- 620
3
votes
1
answer
435
views
busby invariant of extensions of $C^*$-algebras
I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras".
Let $0\to B\to E\to A\to 0$ be a short exact ...
user62639
6
votes
2
answers
351
views
$id:A\to A^{op}$ is completely positive iff $A$ is abelian
Let $A$ be a $C^*$-algebra and $A^{op}$ it's opposite $C^*$-algebra. Let $id:A\to A^{op}$ be the identity map. $id$ is positive.
The claim is: $id$ is completely positive iff $A$ is abelian.
I need ...
user62639
6
votes
1
answer
389
views
is the maximal tensor product of compact operators an essential ideal?
I'm searching for a counterexample for $C^*$-algebras $A$ and $B$ and essential ideals (I assume an ideal to be closed and only two-sided ideals) $I\subseteq A$, $J\subseteq B$ , such that the ideal $...
- 1,188
3
votes
1
answer
166
views
is a linear map on an operator system into a $C^*$-algebra (+ extra conditions) positive?
First of all, sorry for my bad english. I tried to find out whether the following statement is true or not:
Let $X$ be a operator system, $B$ a $C^*$-algebra and $f:X\to B$ linear such that $f(1)\ge ...
user62639
2
votes
0
answers
210
views
The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras
Is there a name for the following property of a $C^{*}$ algebra $A$?
$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$
Example of this situation is $A=C(X)$ where $X$ is the ...
- 356
1
vote
0
answers
101
views
Lower bound for the $C^*$-unitisation norm?
Let $A$ be a $C^*$-algebra without unit. Its unitisation $A^+$ carries the $C^*$-norm
$$\|(a,\lambda)\|_{A^+} = \sup \{\|ax+\lambda x\|_A \;\big|\; \|x\|_A = 1 \},$$
which is the operator norm of $...
- 489
2
votes
1
answer
370
views
On the second dual of $C[0,1]$
I have two questions on the second dual of $C[0,1]$:
R. D. Mauldin ([1]) proved that: For a given bounded linear functional
$T: C[0,1]^*\to \mathbb{C}$ there is a bounded function $\psi$ defined on ...
- 4,058
1
vote
1
answer
225
views
A point-wise separation Hahn-Banach theorem in C*-algebras
Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$.
Let $E$ be a norm closed convex subset of positive operators in $K(H)$ ...
- 4,058
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
7
votes
1
answer
329
views
Is this a characterization of commutative $C^{*}$ algebras?
Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$
Is $A$ necessarily a commutative ...
- 356
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
3
votes
1
answer
184
views
Can approximately periodic functions be perturbed to periodic functions on a locally compact group?
Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...
- 1,023
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
6
votes
2
answers
690
views
Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?
Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces.
So it is ...
- 24.9k
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
1
vote
1
answer
697
views
The total variation of a complex measure
Let $\Omega$ be a locally compact and Hausdorff topological space. The Riesz representation theorem says that $C_0(\Omega)^*$ , dual of the commutative C*-algebra $C_0(\Omega)$, is just the space of ...
- 4,058
5
votes
1
answer
249
views
$C^{*}$-correspondences viewed as generalized endomorphisms
I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers ...
- 367
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
7
votes
2
answers
742
views
$H^{*}$ algebras as a generalization of $C^{*}$ algebras
Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties:
$\forall \lambda \...
- 356
2
votes
1
answer
122
views
Why is the ker-hull-topology on $Irr(A)$ is the discrete topology?
Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A}, here is $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that $V\pi(a)=\rho(a)...
user62639
1
vote
1
answer
534
views
Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane
Setting:
Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$.
Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
- 398
5
votes
1
answer
277
views
Power's Theorem for irreducible representations
Let $A_{\alpha}\subset B(H)$ be a bunch of unital C*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A_{\alpha})$'s. A very nice theorem of Stephen C. Power ...
- 83
7
votes
1
answer
572
views
What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?
Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...
- 1,188
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
13
votes
2
answers
696
views
C$^*$-algebras isomorphic after tensoring
From the negative answer to this question we know that C$^*$-algebras that are isomorphic after tensoring with $M_n$ for all $n\geq 2$ need not be isomorphic. So what happens when we strengthen this?
...
- 3,984
20
votes
2
answers
1k
views
The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
- 1,344
20
votes
2
answers
870
views
C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$
In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes M_2(\...
- 3,984
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
21
votes
3
answers
1k
views
Separating pure states on the $2\times 2$ matrix algebra
I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help.
Let $\mathcal{A}$ be the C*-algebra of $2\...
- 42.8k
4
votes
1
answer
414
views
A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $
Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...
- 1,396
3
votes
0
answers
137
views
An estimate for the maximal $C^*$-norm in the group algebra of a free group
Let $F\twoheadrightarrow G$ be an epimorphism of groups, $F$ being finitely generated and free. Let $H$ be its kernel. Consider a lifting
$i:G\hookrightarrow F$ of the epimorphism.
Every element of $...
- 401
3
votes
1
answer
230
views
questions about the proof of the theorem of completely positive order zero maps
I hope my question is ok for mathoverflow. I first asked on math.stackexchange but received no answer and then delated it.
I want to understand the proof of the theorem (which you can find in the ...
user62639
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
2
votes
1
answer
386
views
Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes [closed]
Question
I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
- 123
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
7
votes
1
answer
1k
views
Inductive/Projective Limits of Topological Algebras
It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...
- 71
1
vote
0
answers
137
views
A continuous choice of invertible elements
Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map $\alpha:A^...
- 356
2
votes
1
answer
99
views
Hilbert $C^*$-module over approximately finite $C^*$-algebra
Is there a construction of a Hilbert c*-module over an approximately finite c*-algebra using Hilbert c*-modules over finite algebras? How do we get a Hilbert c*-module over an inductive limit of such ...
- 21
8
votes
1
answer
355
views
Proving a certain $ C^{*} $-algebraic inequality
Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} \|...
- 1,396
3
votes
1
answer
274
views
Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements
Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...
- 356
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
6
votes
1
answer
576
views
Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?
Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective? This module is not dual, so standard arguments via flatness do not work.
Injectivity is understood in the ...
- 1,697
2
votes
1
answer
327
views
Integration in C^* algebra
Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that
$$
\int d s \, f(s)\, \alpha_s(A)
$$
is well defined as a ...
- 552
13
votes
1
answer
807
views
Inner and extendible automorphisms of C*-algebras
If an automorphism $\alpha$ of a C*-algebra $A$ is inner then whenever $A$ is a subalgebra of another C*-algebra $B$, $\alpha$ obviously extends to $B$.
Is the converse true: if an automorphism $\...
- 1,786
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
7
votes
1
answer
698
views
When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous
It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.
I would like to know if the weakened module version of this question is answered. More precisely: ...
- 1,697
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