All Questions
Tagged with oa.operator-algebras c-star-algebras
597 questions
1
vote
0
answers
305
views
Is the reduced free product of C*-algebras unique
Suppose $\mathcal A_1$ and $\mathcal A_2$ are unital C$^*$-algebras with faithful states $\varphi_1,\psi_1$ of $\mathcal A_1$ and $\varphi_2, \psi_2$ of $\mathcal A_2$.
Are the reduced free ...
- 3,984
5
votes
2
answers
216
views
On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilbert $ C^{\ast} $-module by a certain linear subspace
Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let
$$
\mathcal{E}_{I}
\stackrel{\text{df}}{=}
\{ x \in \mathcal{E} \mid \...
- 1,396
4
votes
0
answers
121
views
Almost order zero approximations- separability and localizations
I'm currently reading the paper The nuclear dimension of $C^*$-algebras by Winter and Zacharias.
I'm trying to understand the proof of:
Proposition 3.2.: Let $A$ be a $C^*$-algebra with $dim_{nuc}A=...
- 81
4
votes
0
answers
389
views
Künneth formula for $C^*$ algebras, equivalent condition for full generality
I'm crrently reading the paper about the Künneth-theorem for $C^*$-algebras: http://msp.org/pjm/1982/98-2/pjm-v98-n2-p15-s.pdf and I'm trying to understand remark 4.9. I henceforth asumme that $A$ is ...
- 81
5
votes
1
answer
370
views
explicit description of the product map in K-theory
Let $A$ and $B$ be unital $C^*$-algebras. Let $u\in M_n(A)$ be a unitary representing an element $[u]\in K_1(A)$ and $p\in M_m(B)$ be a projection representing an element $[p]\in K_0(B)$. Then the ...
user62639
1
vote
1
answer
332
views
Every norm-continuous group of $C^*$-algebra automorphisms weakly inner?
Please, help out of the mind trap. In this prominent paper Kadison and Ringrose prove among other things the following
Corollary 8. Each norm-continuous representation of a connected topological
...
- 1,959
8
votes
1
answer
307
views
Induced map of an approximately inner automorphism on the multiplier algebra of $A\otimes\mathcal{K}$
Suppose that $A$ is a separable, simple, non-unital C*-algebra. Let $\varphi$ be an approximately inner automorphism on $A\otimes\cal K$, meaning that there exists a sequence of unitaries $v_n$ in the ...
- 1,023
17
votes
1
answer
514
views
K-theory space of a C*-algebra
Let $A$ be a unital C*-algebra.
Let me define its "$K$-theory space" to be the image of its $K$-theory spectrum under the functor $\Omega^\infty:$ Spectra $\to$ Spaces.
I denote the $K$-theory space ...
- 43.2k
5
votes
1
answer
283
views
Reference request quantum SU(3)
Woronowicz shows that the C*-algebras of quantum $SU(2)$ are isomorphic (only as C*-algebras, forgetting the quantum group structure). Are there similar results for quantum $SU(n)$ for $n \geq 3$?
4
votes
0
answers
185
views
ring structure of $KK_*(A,A)$ for a separable $C^*$-algebra $A$
Motivation:
For a topological space $X$ one can consider under certain circumstances the cohomology ring of suitable cohomology theories, for example:
1) The cohomology ring $H^*(X;R)=\oplus_{i\ge ...
- 1,188
3
votes
1
answer
261
views
CBAP for the full group $C^*$-algebra
Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by ...
- 99
3
votes
0
answers
148
views
Full free product of $B(\mathcal H_i)$
It struck me that I know nothing about the full (universal) free product of the $B(\mathcal H_i)$ amalgamated over $\mathbb C$ for Hilbert spaces $\mathcal H_i$ with identified unit vector $\xi_i$. So ...
- 3,984
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 ...
- 4,058
3
votes
1
answer
199
views
What can be said about the algebra of continuous functions on compact countable ordinals?
Let $X$ be a compact countable Hausdorff space. By Sierpinski-Mazurkiewicz Theorem we know that $X$ is a compact countable ordinal, i.e.
$$
X \simeq \omega ^{\alpha} \cdot n + 1
$$
where $\alpha$ is ...
- 335
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
1
vote
1
answer
109
views
Continuous factors for invertible simple tensors
Our following question is motivated by this very interesting answer
Assume that $A$ is a $C^{*}$ algebra. Put $X=\{a\otimes b \mid a,b \in G(A)\}$ where $G(A)$ is the space of all ...
- 356
5
votes
1
answer
307
views
Can an AW*-algebra be recovered from its lattice of projections?
Can an AW*-algebra be recovered (up to Jordan isomorphism) from its lattice of projections? This is possible in the commutative/Boolean case.
- 3,816
7
votes
2
answers
688
views
Which C*-algebras are complemented in their bidual?
Every von Neumann algebra is 1-complemented in its bidual, and so is every injective C*-algebra. Also, if $C_0(X)$ is infinite-dimensional and separable then it is not complemented in its bidual, and $...
- 3,816
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
5
votes
1
answer
237
views
Characterization of exact groups via the existence of amenable actions on unital C*-algebras, part 2
In this recent MOF question I asked whether exact groups could be characterized via the existence of amenable actions on unital C*-algebras. The answer, provided by Caleb Eckhardt in a comment, was ...
- 2,263
5
votes
0
answers
272
views
Maximality of the maximal tensor product of C*-algebras
Given two C*-algebras $A$ and $B$, the maximal tensor product $A\otimes_{max}B$ is bigger than the minimal tensor product $A\otimes_{min}B$ in the sense that there exists an epimorphism $$A\otimes_{...
- 2,263
5
votes
1
answer
242
views
Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group
Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I ...
- 743
5
votes
0
answers
488
views
quotients of nuclear $C^*$-algebras are nuclear
At first a definition of injective (unital) $C^*$-algebras: A unital $C^*$-algebra $C$ is called injective if for all completely positive contractive (cpc, for short) maps $\varphi:X\to C$, where $X\...
- 1,188
2
votes
1
answer
159
views
induced map on state spaces
A $*$-homomorphism $f:A\to B$ between C*-algebras is called non-degenerate if $f(A)B=B$.
I guess that I can prove that a non-degenerate *-homomorphism always induces a map on state spaces $f^\ast:S(B)...
- 23
1
vote
0
answers
182
views
Characterization of exact groups via the existence of amenable actions on unital C*-algebras
It is known that a discrete group $G$ is exact if and only if it admits an amenable action on a compact topological space. Would it also be true that $G$ is necessarily exact when it admits an ...
- 2,263
5
votes
1
answer
269
views
Equivalence of two pictures of odd $K$-theory
One can show that two functors $K^0$ and $K_0(C(-))$ from the category of compact topological spaces to the category of abelian groups are naturally equivalent. The first one is topological $K$-theory ...
- 9,330
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
3
votes
2
answers
1k
views
$C^*$ algebras and states
I have a question arising from von Neumann's C*-algebra formulation of quantum mechanics. In it, a state is a $\mathbb{C}$-linear functional on a C*-algebra $A$
$\rho:A\rightarrow\mathbb{C}$
...
- 103
2
votes
1
answer
171
views
algebraic version and polar decomposition
I have been thinking about polar decomposition of $C^*$-algebras. I could not find a proper reference where it says: Let $A$ be a $C^*$-algebra, and $b$ an invertible element of $A$ with modulus $\...
6
votes
1
answer
320
views
C*-envelope of an operator system by an action
Let $V$ be an operator System in $B(H)$. By Hamana and Ruan theorems, there is an injective envelope $I(V)$ which is minimal injective subspace of $B(H)$ contains $V$.
Thus there is a completely ...
- 81
6
votes
2
answers
711
views
maximal tensor product of simple $C^*$algebras is non-simple
Let $A$ and $B$ simple $C^*$-algebras. One can prove that the minimal tensor product $A\otimes _{min}B$ is simple. This is wrong for the maximal tensor product $A\otimes_{max}B$ .
1.Do you know an ...
- 1,188
5
votes
3
answers
292
views
$*$-representation $\pi:A\odot B\to B(H_1\otimes H_2)$ such that $\pi \neq \pi_1\otimes \pi_2$
Let $A$ and $B$ two $C^*$-algebras, $H_1$ and $H_2$ complex Hilbert spaces and $\pi_1:A\to B(H_1)$, $\pi_2:B\to B(H_2)$ two $*$-representations. Then there is a $*$-representation $\pi_1\otimes \pi_2:...
user62639
5
votes
0
answers
314
views
C$^*$-algebras in which the spectral radius is comparable to the norm
For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is:
For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \leq ...
- 3,984
3
votes
1
answer
194
views
Linear independency and compactness of the set of pure states of a $C^*$-algebra
Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states.
Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\...
- 619
15
votes
2
answers
1k
views
Is a C*-algebra with an isomorphic predual a von Neumann algebra?
It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...
- 3,497
9
votes
1
answer
596
views
Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?
Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$
Now ...
- 335
4
votes
1
answer
533
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
0
answers
606
views
Approximate unit in C*-algebra with additional properties
In the book K-Theory and $C^*$-Algebras: A Friendly Approach by Niels Wegge-Olsen I came across the following notion: in Lemma 16.4 authors assume that the $C^*$-algebra $A$ possess "(commuting) ...
- 9,330
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
- 721
6
votes
2
answers
945
views
Multiplier algebra of $A \otimes \mathcal{K}$
If $A$ is unital C$^*$-algebra, is it true that the multiplier algebra of $A \otimes \mathcal{K} $ is $ A \otimes \mathcal{B}(\mathcal{H})$? Where $\mathcal{K}$ is C$^*$-algebra of compact operators ...
- 103
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
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
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
3
votes
1
answer
375
views
Representations of Calkin algebra
Let $H$ be a separable Hilbert space and consider the Calkin algebra $C(H)=\frac{B(H)}{K(H)}$.
Q) True or false: Any representation of $C(H)$ is a direct sum of irreducible representations.
- 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
7
votes
1
answer
288
views
When does a $C^*$-algebra have no nonzero projection?
Let $A$ be a $C^*$-algebra and $\hat{A}$ its spectrum of $A$,the set of classes of non-zero irreducible representation of $A$ endowed with hull-kernel topology. suppose $\hat{A}$ is a non-compact ...
- 399
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