All Questions
Tagged with oa.operator-algebras fa.functional-analysis
778 questions
3
votes
2
answers
376
views
Elements of the minimal tensor product of a finite dimensional operator system and a $C^*$-algebra
I am trying to prove something that seemed simple to me at first sight but apparently it is giving me a hard time. Here is the same question on MSE.
Let $E\subset A$ be a finite dimensional operator ...
- 267
11
votes
1
answer
2k
views
Motivation for $C^*$-algebras
I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
- 111
1
vote
0
answers
313
views
Confusion: Normal homomorphism vs. weak*-continuous vs $\sigma$-weakly continuous
$\newcommand\M{\mathcal M}
\newcommand\N{\mathcal N}
\newcommand\A{\mathcal A}
\newcommand\B{\mathcal B}$
In Takesaki [1], I find the following theorem:
Proposition 5.13. Let $\M_1,\M_2,\N_1,\N_2$ be ...
- 896
5
votes
1
answer
303
views
Non-unital Russo-Dye Theorem
Let $A$ be a C$^*$-algebra and let $\phi$ be a positive linear map from $A$ to $B(H)$ (bounded linear operators on Hilbert's
space). If $A$ is unital, then the Russo-Dye Theorem implies that $\|\phi\...
- 483
1
vote
0
answers
140
views
Haagerup tensor product
Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $V\subset B(H,K)$ is called a ternary ring of operators(TRO) if $xy^*z \in V$ for all $x,y,z \in V$. Obviously a TRO is also a ...
- 1,115
6
votes
1
answer
212
views
Embedding of $C(X)$ into $B(H)$ where $H$ is separable
I would like to ask a question which may look strange at the first sight nevertheless I find it interesting. Let $H$ be a separable Hilbert space: for any separable $C^*$-algebra $A$ one can embed $A$ ...
- 9,330
4
votes
0
answers
548
views
Understanding vector-valued analytic functions vs holomorphic functional calculus
Let $A$ be a unital Banach algebra over complex numbers and call elements of $A$ "vectors". Let $\Omega$ be an open set in $\mathbb{C}$ and $H(\Omega)$ the space of analytic functions on $\...
- 83
8
votes
1
answer
174
views
Is the bitranspose continuous for the $\sigma$-strong topology?
Let $\varphi\colon A\to B$ be a bounded, linear map between C*-algebras. Is the bitranspose $\varphi^{**}\colon A^{**}\to B^{**}$ continuous when the von Neumann algebras $A^{**}$ and $B^{**}$ are ...
- 3,497
3
votes
1
answer
112
views
Is restriction to the center an open map?
Given a type one $C^*$-algebra $A$, its center $Z$ acts by scalars on each irreducible representation space. Mapping a representation to its central character yields a continuous map from the ...
user130903
12
votes
1
answer
2k
views
Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
- 6,853
1
vote
1
answer
129
views
Calculation of the norm of linear combinitation of two states on a $C^*$-algebra
Let $A$ be a unital $C^*$-algebra. Suppose $\rho_1$ and $\rho_2$ are two states on $A$. If $\rho_1=\rho_2$, we have $\|\rho_1+i\rho_2\|=\sqrt{2}$.
If we have $\|\rho_1+i\rho_2\|=\sqrt{2}$, can we ...
- 427
9
votes
1
answer
667
views
Reference for "Every compact quasinilpotent operator is the limit of nilpotent ones"
It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
- 191
7
votes
1
answer
532
views
Almost commuting matrices, one a projection, is there a nearby projection that commutes?
Suppose that $P, A, Q \in \mathbb{M}^{n \times n}(\mathbb{R})$ (I'm still interested if it must be done over $\mathbb{C}$), (EDIT:) suppose that $A$ is given, $P$ is an orthogonal projection, and $\...
- 393
4
votes
2
answers
176
views
Positive maps on finite group algebras and group homomorphisms
Let $G$, $H$ be finite groups. Consider the group algebra $\mathbb{C}G$ acting on $L^2(G)$, making $\mathbb{C}G$ into a C* algebra, and the resulting positive elements, say $P_G\subset \mathbb{C}G$. ...
- 1,143
3
votes
1
answer
221
views
Coincidence of two topology on a bounded subset of a finite von Neumann algebra
Let $M\subset B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$. There is a norm $\|.\|_\tau$ on $M$ given by $\sqrt{\tau(xx^*)}$. How to show the $\|.\|_\tau$-topology ...
- 391
2
votes
1
answer
182
views
About nuclear-by-exact extensions
I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras
$$0 \to I \to A \to B \to 0$$
such that $I$ ...
- 115
4
votes
1
answer
152
views
$\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$
Could you give an example of a unital simple $C^*$-algebra that $\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$?
- 581
4
votes
1
answer
101
views
Functional calculus for "pre-linear" regular operators on a Hilbert module
Let $E$ be a Hilbert module over a $C^*$-algebra $A$. Let $T\colon E\to E$ be a densely defined, unbounded $A$-linear operator. (In particular, the initial domain of $T$ is an $A$-submodule of $E$.) ...
- 1,903
2
votes
1
answer
448
views
The maximal tensor product is a continuous functor
I am trying to prove continuity of the maximal tensor product functor. I have a problem in the proof that I cannot see how to handle; If anyone could give me a clue on how to go on from here, I would ...
- 267
9
votes
2
answers
298
views
Two inequalities in $C^*$ algebras
Under what conditions on a $C^*$ algebra $A$ we have the following inequality:
$$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$
The second identity which I am looking for is ...
- 356
0
votes
2
answers
123
views
$(ST)_{[13]}= S_{[13]}T_{[13]}$ for $S,T \in B(\mathcal{H}\otimes \mathcal{H}).$
Let $T\in B(\mathcal{H} \otimes \mathcal{H})$ where $\mathcal{H}$ is a Hilbert space. We can define operators
$$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$
and if $\Sigma: \mathcal{H} \otimes \...
user167952
2
votes
1
answer
279
views
The algebra of continuous functions on Cantor set
Let $C(K)$ be the algebra of continuous functions on Cantor set. Is it possible to prove that $C(K)$ forms an AF-algebra without Bratteli diagram?
- 581
2
votes
2
answers
302
views
Is $x \mapsto x \otimes 1$ $\sigma$-weakly continuous?
Let $M\subseteq B(H)$ be a von Neumann algebra. Is it true that the mapping
$$\psi: M \to B(H \otimes H): m \mapsto m \otimes \text{id}_H$$
is $\sigma$-weakly continuous? Here the $\sigma$-weak ...
user167952
7
votes
0
answers
158
views
$C^*$ algebras whose nontrivial projections form a non empty compact connected set
Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set?
Is there an example of this situation such that ...
- 356
2
votes
2
answers
217
views
Kernel of intertwiner is invariant (compact quantum groups)
Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $...
user167952
7
votes
1
answer
201
views
Unitary representation is strictly continuous
Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Then is $u: G \to B(H)$ strictly continuous?
That is, give $B(H)$ the topology ...
user167952
4
votes
1
answer
510
views
Strict topology and $*$-strong toppology on $B(H)$ coincide
In the paper Woronowicz - $C^*$-algebras generated by unbounded elements, I read that the $*$-strong operator topology on $B(H)$ and the strict topology on $B(H)$ coincide. I believe this means the ...
user167952
6
votes
1
answer
316
views
Compatibility of inductive and projective limits with dualization in functional analysis
Assume $(A_i)_{i \in I}$ is a family of locally convex topological
vector spaces which are all moreover assumed to be Banach spaces.
We assume moreover that $(A_i)_{i \in I}$ has additional
structure ...
- 6,018
0
votes
0
answers
88
views
Is A an amenable $C^{*}$-algebra?
Let $A$ be $C^{*}$-algebra. Suppose that, for any $\epsilon > 0$ and finite subset $F \subset A$, there are an amenable $C^{*}$-subalgebra $B \subset A$, contractive completely positive linear maps ...
- 581
2
votes
1
answer
186
views
Von Neumann algebras with isomorphic sets of partial isometries
Given a von Neumann algebra $M$, let
$$
S(M) = \{u\in M: uu^*u=u\}
$$
be the set of partial isometries in $M$. Given $u,v\in S(M)$, it is well known that $uv \in S(M)$, provided $u^*u$ ...
- 2,263
-1
votes
1
answer
210
views
A commuting pair of isometries
Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$.
The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
- 4,058
0
votes
1
answer
275
views
comparison of two projections in a non-factor von Neumann algebra
In a factor $M$, we know that for any two projections $P$ and $Q$ in $M$, either $P\preceq Q$ or $Q\preceq P$ holds true. Here $\preceq$ denotes the Murray-von Neumann subequivalence of two ...
- 165
3
votes
1
answer
246
views
Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras
In "Weierstrass-Stone, the Theorem" by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following:
Let $\mathcal{A}$ be a subalegebra of $C(X, \mathbb{R})$ and $...
- 119
3
votes
1
answer
170
views
Integration on quasi-Banach spaces and Schatten ideals
Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its ...
- 143
4
votes
0
answers
253
views
Inflating the double dual of a C*-algebra (matrix algebra of double dual)
in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})\cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can't really see it. Actually, it is enough for me ...
- 267
4
votes
1
answer
332
views
Normal linear functionals on bicommutants of C*-algebras
I am going through the proof of the Sherman-Takeda theorem and Fillmore's book "A User's Guide on Operator Algebras" seems to have a nice approach, but something seems off to me:
We need to ...
- 267
8
votes
1
answer
332
views
The double dual of the unitization of a $C^*$-algebra
I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the ...
- 267
8
votes
2
answers
570
views
Are (completely) positive maps approximated by normal (completely) positive maps?
Let $\mathcal{H}$ denote a Hilbert space and $B(\mathcal{H})$ denote the algebra of all bounded operators on $\mathcal{H}$. By recognizing the (Banach) dual of $B(\mathcal{H})$ with the double dual of ...
- 165
4
votes
1
answer
196
views
A kind of holomorphicity of maps on Hilbert space
Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?:
1)For every open set $U\subset H$ and every ...
- 356
7
votes
0
answers
107
views
Potential p-norm on tuples of positive operators
This is a follow-up to this question on p-norms of tuples of operators.
Consider $\left[\begin{matrix} A \\ B \end{matrix}\right] \in B(H)^2_+$, meaning $A,B\geq 0$, and define
$$
\left\|\left[\begin{...
- 3,984
6
votes
1
answer
236
views
Potential p-norm on tuples of operators
Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define
$$
\left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|_p = \| |A|^p + |B|^p\|^{1/p}.
$$
Q: Is this a norm?
...
- 3,984
2
votes
2
answers
214
views
Commutative C*-rings
Let us consider the unital commutative $C^*$-algebra $C[0,1]$. We say $A\subseteq C[0,1]$ forms a C*-subring if it satisfies the following conditions:
1- $A$ is an involutive unital subring (closed ...
- 4,058
0
votes
0
answers
47
views
An algebraic property that makes a per-C*-algebra complete
Let $A$ be a normed *-algebra with $\|x^*x\|=\|x\|^2$. Suppose that for every subset S of A, the left annihilator
${\displaystyle \mathrm {Ann} _{L}(S)=\{a\in A\mid \forall s\in S,as=0\}\,}$
is ...
- 4,058
3
votes
0
answers
130
views
Question about a paper on approximate identities
I am currently reading this paper on approximate identities of ternary Banach algebras. Assume that $(A, [.,.,.])$ is a ternary Banach algebra. A bounded net $(e_{\alpha}, f_{\alpha})$ is said to be ...
- 1,115
-2
votes
1
answer
138
views
Weak center is same as center for $C^{\ast}$-Algebra? [closed]
Let $A$ be a $C^{\ast}$-algebra. We say $A$ is weakly commutative if $ab^*c=cb^*a$ for all $a,b,c \in A$ and define weak center of $A$ as $$Z_w(A)= \{ v \in A : av^*c=cv^*a \;\forall a,c \in A \}.$$
...
- 1,115
6
votes
2
answers
282
views
The Calkin representation for Banach spaces
Let $X$ be an infinite dimensional Banach space. Let $\Lambda_{0}$ be the set of all finite dimensional subspaces of $X$ directed by the inclusion $\subseteq$. For each $\alpha\in \Lambda_{0}$, let $...
- 3,343
4
votes
1
answer
200
views
Alternative proof of existence of absolute value of a functional on a C*-algebra
The usual proof of the existence of an absolute value of a functional on a C*-algebra $A$ uses the polar decomposition of normal functionals on $A^{**}$, which relies on the compactness of the unit ...
- 3,816
1
vote
0
answers
183
views
G-abelian systems
Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...
2
votes
0
answers
116
views
General construction of enveloping C*-algebra, left/right-regular representation, etc
In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...
- 3,816
8
votes
1
answer
393
views
A question about comparison of positive self-adjoint operators
I have the following question but have no idea on its proof (one direction is trivial):
Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that
$$\...
- 1,906