All Questions
Tagged with oa.operator-algebras fa.functional-analysis
778 questions
3
votes
0
answers
128
views
Stable homotopy equivalence
Let $\alpha:A \rightarrow B$ be a *-homomorphism of $C^*$-algebras. Then $\alpha$ ist a stable homotopy equivalence if there exists a $*$-homomorphism $\beta: B \otimes \mathcal{K} \rightarrow A \...
- 31
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
4
votes
0
answers
298
views
Operator topologies
Let $L(H)$ be the space of bounded operators on some Hilbert space.
We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT).
...
- 41
1
vote
0
answers
148
views
Fourier–Stieltjes as the dual space of the full group algebra
I know that this fact is classical, but I can't find the proof of it. How to proof that $B(G)=(C^*(G))^*$? As I understood, I can take a functional $F: \ell_1(G) \to \mathbb{C}$, and there is one-to-...
- 631
4
votes
1
answer
2k
views
Operator topologies on $L^{\infty}(X,\mu )$
Let $(X,\mu )$ be a measure space. Then, $L^2(X):=L^2(X,\mu )$ is a Hilbert space in the usual way and we may view $L^{\infty}(X):=L^{\infty}(X,\mu )$ as a subalgebra of bounded operators on $L^2(X)$ ...
- 1,270
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
4
votes
1
answer
212
views
Kernels of representations of $C^*(G)$
Let G be a discrete group. I am interested in the following: let $\pi$ and $\rho$ be two representations of $G$. Denote by $C^*Ker\pi$ and $C^* Ker \rho$ the kernels of the corresponding ...
- 631
0
votes
1
answer
111
views
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
We don't know that is an open problem or it has counterexample...
- 19
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
689
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
111
views
proving that $\mathcal{A}_\infty(X)$ is or is not norm-closed in $\mathcal{L}(X)$ for each Banach space $X$
Fix any $1\leq p\leq\infty$. If $X$ is a Banach space and $C\in(0,\infty)$, we say that $T\in\mathcal{A}_C(X)$ whenever, for each $(x_n)_{n=1}^\infty\subset B_X$ (where $B_X$ is the closed unit ball ...
- 1,591
8
votes
3
answers
1k
views
Conceptually, what does unitization do?
Let $(\mathcal A,||\cdot||)$ be a normed algebra (with or without a unit). The unitization of $\mathcal A$ is the space $\mathcal A_+:=\mathcal A\oplus \Bbb C$ where the multiplication operation $\...
- 1,245
0
votes
3
answers
291
views
Smallest norms on crossed product $C^*$-algebras
Let $A$ be a commutative $C^*$-algebra with a discrete group $G$ acting on it. The reduced crossed product is the completion of the algebraic crossed product $C_c(G,A)$ in the reduced norm $\Vert \...
5
votes
1
answer
242
views
Spectral decomposition of a C$^*$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
1
vote
0
answers
154
views
Kasparov's descent homomorphism for higher KK groups
I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism
$$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$
but what still ...
- 19
0
votes
1
answer
152
views
Transitivity of the Cuntz sub-equivalence
Let $A$ be a $C^*$-algebra and $a,b \in A$ positive elements. We define a relation (Cuntz sub-equivalence) by saying
$$a\lesssim b: \Leftrightarrow \exists\, (r_n)_{n\in\mathbb{N}}\subset{A}\text{ ...
- 1
1
vote
1
answer
286
views
Constructing $C^*$-Categories from Groupoids
Let $G$ be a discrete group and $\mathcal{G}$ a groupoid, that is, a small category in which every arrow is an isomorphism. Wolfgang Lück explains how we can construct a $C^*$-category from $\mathcal{...
- 11
4
votes
1
answer
140
views
Finding a special Banach algebra and a net of homomorphisms
If $A$ is a Banach agebra and $M$ is a Banach $A$-bimodule then a linear map $T:A\to M$ is called an $A$-module homomorphism if $$T(ab)=aT(b),\quad T(ab)=T(a)b,\qquad a,b\in A.$$ Also $A\hat{\otimes} ...
- 231
1
vote
1
answer
106
views
Norm of a cb-homomorphism restricted to a generating operator space
Let $\mathcal A \subset B(H)$ be an operator algebra and $\varphi: \mathcal A \rightarrow B(K)$ a completely bounded homomorphism. Suppose $\mathcal M \subset \mathcal A$ is an operator space such ...
- 3,984
6
votes
2
answers
318
views
Natural map $C^*(G) \to M(A\rtimes G)$
Let $A$ be a C*-algebra, let $G$ be a locally compact group, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be a (strongly) continuous action. It is well known that there is a natural map $\iota_G\colon C^...
- 717
6
votes
1
answer
320
views
C*-envelope of an operator system by an action
Let $V$ be an operator System in $B(H)$. By Hamana and Ruan theorems, there is an injective envelope $I(V)$ which is minimal injective subspace of $B(H)$ contains $V$.
Thus there is a completely ...
- 81
5
votes
3
answers
292
views
$*$-representation $\pi:A\odot B\to B(H_1\otimes H_2)$ such that $\pi \neq \pi_1\otimes \pi_2$
Let $A$ and $B$ two $C^*$-algebras, $H_1$ and $H_2$ complex Hilbert spaces and $\pi_1:A\to B(H_1)$, $\pi_2:B\to B(H_2)$ two $*$-representations. Then there is a $*$-representation $\pi_1\otimes \pi_2:...
user62639
3
votes
1
answer
145
views
Reference for explicit quasicentral BAI in K(H) as ideal in B(H)?
As observed by Arveson and Akemann+Pedersen, if $J$ is an ideal in a ${\rm C}^\ast$-algebra $B$, then one can always find a contractive approximate identity for $J$, call it $(e_\lambda)_{\lambda\in\...
- 25.8k
5
votes
0
answers
314
views
C$^*$-algebras in which the spectral radius is comparable to the norm
For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is:
For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \leq ...
- 3,984
4
votes
0
answers
114
views
Coming up with a represenation for sum of functions in the Fourier algebra
This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com.
Let $G$ be a discrete group.
Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...
- 161
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
4
votes
0
answers
333
views
Baum Connes Conjecture [closed]
I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as it is formulated in topology to the conjecture as it is formulated in functional analysis. I ...
- 49
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
5
votes
1
answer
219
views
Equivalence of questions regarding restrictions of pure states
In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer ...
- 3,115
5
votes
1
answer
245
views
Convergence of functionals on compact projections on a separable Hilbert space
Let $H$ be a separable Hilbert space over $\mathbb{C}$, say $\ell_2$ for simplicity. Let $\mathcal{K}(H)$ denote the space of all compact operators on $H$ and $\mathcal{P}(H)$ the set of all finite ...
- 1,151
6
votes
0
answers
210
views
Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
2
votes
1
answer
122
views
Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries
Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$
Let $\phi: S \rightarrow B_2.$
Given that the ...
- 1,429
3
votes
0
answers
237
views
Orthogonality relations for unitary representations of infinite (finitely generated) groups
Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...
- 333
4
votes
1
answer
250
views
Can we solve the FGF problem by finding an appropriate action?
If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
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
3
votes
0
answers
61
views
Isometry from a representation to the representation tensored with itself
Suppose, the group $ G=S(2^{\infty})$ has a unitary representation $ \pi $ on a separable infinite dimensional Hilbert space $ H $.
(The group $ S(2^{\infty}) $ is the direct limit of the following ...
- 817
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
3
answers
866
views
The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant
In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on:
If $ \mathbb{H} $ is an RKHS and we denote the ...
- 221
5
votes
3
answers
632
views
What is the group of automorphisms of $l^{\infty}$?
What is the group of automorphisms of $l^{\infty}$?
I think it would be the permutations of the integers. Is this right?
- 367
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
6
votes
2
answers
549
views
"Identity tensor transpose" as a map $M_n \hat{\otimes} M_n \to M_n \overline{\otimes} M_n$
Equipping $M_n$ with its usual operator space structure,
$\newcommand{\ptp}{\widehat{\otimes}}$
we can form the projective tensor product of operator spaces $M_n\ptp M_n$. In particular this puts a ...
- 25.8k
3
votes
1
answer
6k
views
About eigen-functions of the Gaussian kernel
If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
- 2,246
5
votes
0
answers
184
views
Automorphisms of Cuntz algebra
Suppose, $ O_{\infty} $ is the cuntz algebra generated by the orthogonal isometries $ \{S_i\}_{i\in \mathbb{N}} $,i.e. $ S_i^*S_j=\delta_{ij}$ and $ O_{\infty}=C^*(\{S_i\}_{i\in \mathbb{N}}) $.
Then ...
- 817
5
votes
1
answer
137
views
Operator space structures on CB(H,K) where H and K are Hilbertian operator spaces?
(I'd be grateful if anyone thinking of putting MathJax in the question title refrains from doing so.)
By consulting various standard sources (Effros-Ruan's book, Pisier's book, the lexicon of ...
- 25.8k
4
votes
0
answers
207
views
A continuous functional calculus on/positive elements in a Fréchet algebra?
I am trying to understand what (minimal) conditions one would need in order to obtain a functional calculus on a Fréchet algebra, which we demand to be equipped with an involution that leaves all semi-...
- 465
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
4
votes
0
answers
185
views
A strongly open set which is not measurable in the weak operator topology
Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
- 4,058