All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
0
votes
0
answers
92
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Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation
Please tell me where can I see the proof of this well-known fact?
enter image description here
2
votes
1
answer
526
views
Difference in tracial and finite von Neumann algebras
A tracial von Neumann algebra $(M,\tau)$ is a von Neumann algebra with a faithful normal tracial state $\tau$ on $M$. That is, $\tau$ is a function from $M \to \mathbb{C}$ such that it is a faithful ...
- 42.8k
1
vote
0
answers
384
views
Densely defined and closed operator
Question: Let $N\subseteq B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$ and let $n\in N$. Let $B$ be a von Neumann subalgebra of $N$. Let $\mathbb{E}_B: N\rightarrow B$ be ...
- 85
1
vote
0
answers
92
views
Algebras sitting inside reproducing kernel Hilbert space other than multiplier algebra
Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space. If the kernel is normalized then the multiplier algebra $\mathcal{M}$ is an algebra that is sitting inside $\mathcal{H}$. Is there any ...
1
vote
0
answers
85
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Spectral projection with height less than $\lambda$
Let $x\geq 0$ be a positive element in a von Neumann algebra $\mathcal M.$ Then b y functional calculus the projection $e_\lambda=1_{[0.\lambda)}(x)$ has the property that $e_\lambda$ commutes with $x$...
- 1,879
3
votes
1
answer
192
views
Characters of algebra of Schwartz functions
Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$.
Question: Does there exist some character (non-zero multiplicative ...
CommunityBot
- 1
5
votes
1
answer
239
views
Completely isometric coaction of discrete quantum group is multiplicative?
Let $\mathbb{G}$ be a compact quantum group (in the sense of Woronowicz) with discrete dual $\widehat{\mathbb{G}}$ which we view as a von Neumann algebraic locally compact quantum group (in the sense ...
- 4,351
6
votes
1
answer
166
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Every element of a $W^*$-algebra is a linear combination of exponential unitaries?
I am trying to understand a proof in this paper (specifically theorem 5.4). In it, a fact is used that every element of the $W^*$-algebra $A$ is a linear combination of exponential unitaries.
I've ...
- 2,471
2
votes
0
answers
202
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The trigonometric $C^*$-algebra
The trigonometric $C^*$-algebra is the universal $C^*$-algebra generated by $\mathcal{G}=\{x,y,z\}$ subject to relations \begin{align}x^2=x=x^*, &\quad y^2=y=y^*\\ [x, z]=y, &\quad [y,z]=...
- 55
2
votes
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answers
70
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The $K_0$ mapping of an automorphism induced by a derivation
Let $\mathfrak{A}$ be a unital $C^*$-Algebra and let $\delta: \mathfrak{A} \rightarrow \mathfrak{A}$ be a linear map that is not constantly zero and satisfies, for every $A, B\in\mathfrak{A}$, $\delta(...
- 307
2
votes
1
answer
94
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Unitary dual of universal cover
The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...
- 63.9k
3
votes
0
answers
115
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Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
3
votes
0
answers
295
views
Dunford-Pettis like properties for Banach spaces of operators
Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\left.\begin{array}{l}
(x_n)\subseteq A \textrm{ ...
- 2,605
5
votes
1
answer
212
views
States "absorbed" by a Haar idempotent on a compact quantum group
Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ absorbs $a$? Can we say ...
- 4,351
2
votes
1
answer
221
views
Non-perfect type one C^*-algebra, and a lemma in Fourier analysis
I would like to know if the following is true :
Let $\mathcal{H}$ be the complex Hilbert space $L^2([0,1])$ for the Lebesgue measure.
Let $q$ be the orthogonal projection on the subspace of $\mathcal{...
- 63.9k
2
votes
0
answers
90
views
The group of quasi unitary elements of a (simple) Banach algebra
For a Banach algebra $A$ with invertible group $G(A)$ we define the following group:
$$QG(A)=\{u\in G(A)\mid \;\text{the mapping}\; a\mapsto u^{-1} a u \;\text{is an isometry}\}$$
What is an ...
- 356
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
2
votes
1
answer
250
views
Norm continuity of the predual of a von Neumann algebra
Let $M$ be a von Neumann algebra and let $(p_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M_{\ast} $, the predual of $M$.
It is well known that
$\| p_i f \|_{M_\ast}\to_{i} 0$
for ...
- 1,848
2
votes
0
answers
92
views
DHR superselection and DR reconstruction in low spacetime dimensions
Given a completely rational net on $\mathbb{R}$, the Doplicher-Haag-Roberts (DHR) category is a modular fusion category (MFC) identical to that associated with the corresponding vertex operator ...
- 437
10
votes
1
answer
283
views
Faithful extreme traces on group C*-algebras
Let $G$ be a discrete amenable, residually finite, ICC(i.e. each non-trivial conjugacy class is infinite) group. Let $C^*_r(G)$ be the reduced group $C^*$-algebra of $G$. Since $G$ is ICC the (...
- 4,351
1
vote
0
answers
137
views
Representation of states in $C^*$-algebras
Let $\mathfrak{A}$ be a $C^* $-algebra, let $\pi : \mathfrak{A} \to \mathcal{B}(H)$ be a representation of $\mathfrak{A}$ on the space of bounded linear operators on a Hilbert space $H$ and let $\...
- 139
1
vote
1
answer
195
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Concrete example of non-norm-attaining bounded linear operator on disc algebra
A bounded linear operator $T$ from a Banach space $X$ to a Banach space $Y$ is called norm-attaining, if there exists a vector $x\in X$ with $\|x\|=1$ such that
$$\|Tx\|=\|T\|.$$
Let $\mathbb{D}=\{z\...
- 4,461
10
votes
1
answer
518
views
For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$?
Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$.
For what kind of $C^*$ algebras $A$ does the following hold:
$$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\...
- 2,471
-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 $...
- 66
13
votes
1
answer
1k
views
Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?
A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
- 2,821
3
votes
1
answer
693
views
Universal $W^*$-algebras of locally compact groups: where is the error in this argument?
Let $G$ be a locally compact Hausdorff group.
It is known that $G$ can be topologically embedded in $W^{\ast}(G)$ , its universal $W^{\ast}$-algebra (with the $\sigma$-weak topology). An element $T \...
- 12.9k
0
votes
1
answer
190
views
Are the ideals in two $C^*$-algebras the same?
Let $V_{1}, V_{2}$ be the commuting isometries. By Wold decomposition theorem, we know that $V_{i}$ admits decomposition $$V_i \cong V^s_{i}\oplus V^{u}_{i},$$ where $V^{s}_{i}$ is the shift and $V^{u}...
- 42.8k
6
votes
1
answer
623
views
Is the conditional expectation faithful?
Let $G$ be a locally compact group and let $H$ be an open subgroup in $G$.
Then the full group $C^*$-algebra of $H$, $C^*(H)$, is a subalgebra of $C^*(G)$ and there is a conditional expectation $$E\...
- 55
4
votes
1
answer
371
views
Motivation for Heisenberg's modeling of observables
What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ...
- 433
4
votes
2
answers
416
views
Is the left-regular representation of a locally compact group a homeomorphism onto its image?
Consider the left-regular representation $\lambda : G \to B(L^2(G))$, $\lambda_g f(h) = f(g^{-1}h)$, for a locally compact group.
It is well-known that this is a unitary faithful and strongly-...
- 61
4
votes
1
answer
334
views
Support projection vs closed support projection of a normal state in enveloping von Neumann algebra
I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
- 135
6
votes
0
answers
128
views
Unitary fusion category and subfactor
From a unitary fusion category $\mathcal{C}$, there are several ways to make a (hyperfinite II$_1$) subfactor.
By [Ha] there are weak Hopf algebras $H$ such that $\mathcal{C} = Rep(H)$. By unitarity (...
0
votes
0
answers
144
views
Type III von Neumann algebra
Let $\mathcal M$ be a type $\mathrm{III}$ von Neumann algebra. Is it true that for all $n\geq 1,$ $\mathcal M$ contains a copy of $M_n$ as a von Neumann subalgebra? By Theorem 9.24 of these lecture ...
- 6,367
4
votes
1
answer
227
views
Definition of Radon measure on Takesaki's first volume
Consider the following theorem from Takesaki's first volume "Theory of operator algebras":
In $(i)$, it is claimed that $L^\infty(\Gamma,\mu)$ is an abelian von Neumann algebra. How does ...
- 37.8k
2
votes
1
answer
195
views
Another formula for the Schwinger term — problems with a calculation
$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...
- 63.9k
81
votes
3
answers
9k
views
Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
- 1,446
13
votes
3
answers
1k
views
Separable von Neumann algebra
What is the simplest argument which shows that each infinite dimensional von Neumann algebra is not separable (in the norm topology)? It seems that this is a kind of folklore: at least I never saw the ...
- 2,605
1
vote
0
answers
87
views
irreducible subfactor inclusion and commutativity of induced projections
Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$.
Let $\Omega\in H$.
Does it follow that the projections onto $[N\Omega]$ and $[M'\...
- 759
1
vote
0
answers
104
views
Kernel representation of a power of (pseudo-)differential operator
Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation:
\begin{equation}
\mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt.
\end{equation}
What can ...
- 350
1
vote
0
answers
123
views
On Riesz decomposition of Volterra operator
Let $T:L^2((0,1)) \to L^2((0,1))$ be the Volterra operator defined by
$$ Tf(x) = \int_0^x f(t)\,dt.$$
Given any $\lambda\neq 0$ it is well known that there exists some positive integer $n=n(\lambda)$ ...
- 4,135
8
votes
0
answers
411
views
Semigroups of matrices closed under conjugate transposition
An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
- 563
1
vote
0
answers
76
views
Spectral measure for a finite set of mutually commuting normal operators
The following question is from Exercise $\S 11.11$ in A Course in Operator Theory written by John B. Conway:
Suppose $\{N_1, \cdots, N_p\}$ is a finite set of mutually commuting normal operators in $...
- 307
15
votes
1
answer
474
views
Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?
The physicist Yoichiro Nambu introduced in a 1950 paper A Note on the Eigenvalue Problem in Crystal Statistics the notion of an "eigenoperator" (page 12, see Nambu and the Ising model for a ...
- 188k
63
votes
7
answers
5k
views
What well known results with countability assumptions can be naturally extended to uncountable settings?
In many of the common categories of spaces (or algebras) in mathematics, one often restricts attention to those spaces or algebras which are "countable" or "countably generated" in ...
CommunityBot
- 1
7
votes
1
answer
577
views
Are the compact and Haagerup approximation properties equivalent?
The following essentially implies the equivalence of Anantharaman-Delaroche's compact approximation property (page 337 of Link) and the Haagerup approximation property.
Let $M$ be a type ${II}_{1}$ ...
- 63.9k
2
votes
0
answers
36
views
Matrix affine functions and matrix convex sets
In this post I'm trying to understand a small but significant part of an article by Corran Webster and Soren Winkler regarding a matrix convex generalization of the Krein-Milman theorem. (see the ...
- 21
2
votes
0
answers
88
views
Non-existence of idempotent via evaluation of higher order cocycle on a tuple of idempotents
The Kaplansky conjecture and Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra $...
- 356
1
vote
0
answers
81
views
Tracial linear functionals on an amenable Banach algebra
This post is related to an earlier question about Kazhdan property (T). The purpose of the snippet below is to briefly summarize the background for the question in this post.
Question: Does there ...
- 2,605
1
vote
1
answer
129
views
Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?
Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product
$$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \...
- 18.7k
2
votes
0
answers
56
views
Existence of a suitable smooth kernel
Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
- 4,135