All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
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crossed product by compact groups
Do we need the ambient measure on G to be a Haar measure in order to form the crossed product by a compact group of a von Neumann algebra M? If the measure is indeed Haar, then we can obtain the ...
- 15
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35
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Operator-form correspondence without lower semiboundedness
When dealing with a normal unbounded operator $A$, it is often useful to frame questions about the operator in terms of questions about the associated form $\omega,$ which has domain $D(|A|^{1/2})$ ...
- 109
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57
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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
- 63
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111
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Integral decomposition
Let $\mathcal{A}$ be a separably acting von Neumann algebra and let $$\mathcal{A}\cong \int_{\Gamma}^{\oplus} \mathcal{A}_{\gamma}\,d\mu(\gamma)$$ be its direct integral decomposition into factors $\...
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55
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Status of generalization of timelike tube theorem to algebras of causal completions
The timelike tube theorem states that the additive algebra $A_{\text{add}}(U)$ of operators in a spacetime region $U$ is equal to the additive algebra $A_{\text{add}}(E(U))$ of the timelike envelope $...
user528837
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Reference for the G-equivariant Stinespring dilation theorem
Is there a good reference for the G-equivariant Stinespring dilation theorem? I can't find the theorem stated anywhere.
Thanks in advance.
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157
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Dependence of functional integral on the function space
In physics, the following functional integral is considered
\begin{gather}
Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf ))
\end{gather}
It is usually said that the integration is performed ...
- 593
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115
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$C^*$ algebra generated by conjugation of an element
Assume $\mathcal{A}$ is a unital $C^*$ algebra and consider some positive-definite element $\Psi\in M_n(\mathcal{A})$. Can we say something about $C^*(\langle \Psi^{-\frac{1}{2}}E_{i,i}\Psi^{\frac{1}{...
- 167
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131
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Can a non-separable C$^*$ algebra have separable GNS Hilbert space
Suppose we have a $C^*$ algebra $\mathfrak{U}$ that is non-separable. Consider a state $ω$ of $\mathfrak{U}$ and the GNS representation $(H_ω,π_ω,Ω)$. Is it possible for $H_ω$ to be separable, and if ...
- 151
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79
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Projections to orthogonal complements of conditional expectations
For a conditional expectation from a C^* algebra A to a subalgebra B, we can form a positive projection $P:A\to A$ with image equal to $B$. Question: is $Id - P:A\to A$ a positive map?
- 1,143
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110
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Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?
Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two ...
- 165
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104
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Intersection of type-I von-Neumann algebra factors
Is the intersection of a (possibly infinite) family $\{\mathcal M_i\}$ of type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$) again a type-I von-Neumann algebra factor?
- 896
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71
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Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group
Let $A$ be a unital $C^*$-Algebra with non-trivial $K_0$ group. Define $CA = \{f\in C\big([0, 1], A\big)\,\vert\,f(0) = 0\}$. It can be shown that $CA$ is homotopic equivalent to the set $\{0\}$ and ...
- 307
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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
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144
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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 ...
- 1,879
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Are banach space representations of commutative $C^*$ algebras decomposable?
It is well known that, if $\pi:A\to \mathbb B(\mathcal H)$ is a $^*$-representation of a type I $C^*$-algebra, then $\pi$ is unitarily equivalent to a direct integral of irreducible representations.
...
user490261
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390
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Monotone convergence theorem for increasing net of positive functions
Suppose that we have $(\Omega,\mu)$ a $\sigma$-finite measure space. I have the following question.
(Assume that $(f_i)_{i\in I}$ be an increasing net of positive measurable functions such that $f_i\...
- 1,879
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100
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Essential numerical range of an idempotent
The following screenshot is from J. C. Bourin and E. Y. Lee's paper "Pinchings and positive linear maps", J. Funct. Anal. 270, No. 1, 359-374 (2016), MR3419765, Zbl 1345.46050. When reading ...
- 427
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105
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Unitarily equivalent matrices that are also unitarily equivalent on orthogonal subspaces
Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$.
Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...
- 31
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88
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$*$–homomorphisms of the center of $C^*$-algebras
Let $A$ and $B$ be $C^*$-algebras with centers $Z_A$ and $Z_B$. Suppose $\rho：A\rightarrow B$ is a surjective $*$- homomorphism. It is easy to check $\rho(Z_A)\subset Z(B)$.
I wonder how to assure ...
- 451
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109
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Generator problem for reduced group C*-algebra
(Not sure if it is appropriate or not, if no I will delete the post)
Recently I am concerned about the number of generator of $C^{*}_{r}(\mathbb{F}_{k})$, the reduced group algebra of the free group, ...
- 523
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72
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Does $L^{\infty}[0,1]$ admits infinitely many densely defined derivations in weak* topology?
To clarify the question. First we define what is densely defined derivation. A densely defined derivation $\delta:D(\delta):\rightarrow L^{\infty}[0,1]$ where $D(\delta)$ is a dense subalgebra( in ...
- 523
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85
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How can we define $\chi_{\Omega}(A)$?
I was reading Spectrum and dynamics where Paolo Facchi discusses projection-valued measures and integrals. The discussion and constructions are all based on the fact that one can define characteristic ...
- 1,305
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119
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Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant
Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$.
Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$
So $A$ is a Banach algebra.
Can we equip $A$ ...
- 356
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85
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An amenable operator algebra has the total reduction property
This is from
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB20539885C03522D141C34024707702/S1446788700014026a.pdf/div-class-title-operator-algebras-with-a-reduction-property-...
- 101
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109
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Operator algebra on an invariant subset
In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^...
- 109
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120
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Are bounded maps determined by their images on the algebraic tensor product?
Let $\mathcal A,\mathcal B,\mathcal C$ be von Neumann algebras. Let $F:\mathcal A\otimes\mathcal B\to\mathcal C$ be a bounded linear map. Assume $F(a\otimes b)=0$ for all $a,b\in\mathcal A,\mathcal B$....
- 896
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101
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Questions on unbounded derivations of C* algebra
In Sakai note, on the fourth part differentiation. Sakai stated the following:
It is an open question whether the result can be extended to $n=2,3,...$
What $n$ he is referring to? Also Sakai stated ...
- 523
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144
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Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
- 1,879
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88
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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
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47
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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
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127
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classification of simple nuclear $C^*$-algebras
Can we classify the simple nuclear $C^*$-algebras completely? Can anyone recommend some papers or books concerning the calssification of nuclear $C^*$-algebras
- 427
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57
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Monotone series of projections converging to 1 in von Neumann algebra
The following statement is being used a lot in the literature, and I wonder how to prove it.
Let $M$ be an infinite-dimensional von Neumann algebra (with unit element), show that there is an ...
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58
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Lower bounds in the space of compact operators
Let $H$ be a separable Hilbert space, and $K(H)$ the corresponding space of compact operators. Consider the "unit sphere" $S:=\{T\in K(H)|T\geq 0\text{ and }||T||=1\}$. Is it true that, given any pair ...
- 1
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132
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Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]
Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)?
Wikipedia says this is true.
However, it seems to me that ...
- 253
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153
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Representations of Banach algebras
If $A$ is a Banach algebra and $L$ a left ideal of $A$, consider the representation $T_{L}$ of $A$ into the algebra $B(A/L)$ of bounded linear operators on the quotient space $A/L$ defined by $T_{L}(a)...
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70
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Limit of spectral projection of increasing sequence of positive operators
Let $\mathcal M$ be a von Neumann algebra. Suppose $(x_\alpha)\subset \mathcal M$ is bounded increasing net of positive operators converging to a positive operator $x\in\mathcal M.$ Is it true that $\...
- 1,879
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147
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Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$
The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas.
Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
- 307
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97
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Smooth sections of finite dimensional bundle and covering space
Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with ...
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89
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On mixing and weak mixing subalgebras of finite von Neumann algebras
Let $M$ be a full $\mathrm{II}_1$ factor. Consider mixing and weak mixing subfactors $B$ and $C$ of $M$. Are $B$ and $C$ full?
- 181
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90
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Special kind of translation and rotational invariance of the numerical range
Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Is the following statement true?
If $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B\in M_n$ then $W(B\otimes (X+I_n))\subseteq W(B\...
- 231
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3k
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On prime factors
Let $M$ be a prime $\mathrm{II}_1$ factor. Let $N$ be a non hyperfinite finite index subfactor $N$, is $N$ prime factor?
- 181
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131
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Operator space tensor products
Given two Banach algebras $A$ and $B$ with operator space structure on each of them, i.e both of them are closed subspaces of $B(H_1)$ and $B(H_2)$ respectively for some Hilbert spaces $H_1,H_2$. ...
- 243
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110
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On an application dominated convergence theorem in vN algebras
$M$ be a $\mathrm{II}_{1}$ factor equipped with the faithful normal trace $\tau$ in the standard form. Let $\tau(Jx'J\eta)=0,\forall x' \in M'$ and fixed $\eta$ in $L^{1}(M,\tau)$. Is it true $\eta=0$?...
- 677
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88
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On invertibility of ergodic averages
Let $x$ be invertible unbounded operator affiliated operator to the $\mathrm{II_{1}}$ factor $(M,\tau)$. Under which condition on $x$, the iterates also $1+\sigma(x)+\cdots+\sigma^{n}(x)$ are ...
- 181
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127
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On examples of action of C-star simple group on von Neumann algebra
Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
- 677
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86
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Characterzing compact actions on von Neumann algebra
Suppose $G$ is a countable discrete group acting on vN algebra $M$, the action is compact. Can we have a topology on Aut$(M)$ such that $\{\sigma_{g}\in \text{Aut}(M):g \in G\}$ form a compact subset ...
- 677
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54
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On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
- 677
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68
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On existence of sequence of unitaries in $II_{1}$ factor $M$
Let $M$ be a $\mathrm{II}_{1}$ factor acting on $L^{2}(M, \tau)$ in standard form, let $\{e_{n}:n \in \mathbb{N}\}$ be fixed orthonormal basis of $L^{2}(M, \tau)$, does there exist sequence of ...
- 677
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0
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77
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On cyclicity of fixed point algebra of flip automorphism
Let $M$ be a von Neumann algebra having a cyclic vector in $\mathcal{H}$, is the fixed point subalgebra under the flip automorphism on $M\otimes M$ has a cyclic vector in $\mathcal{H}\otimes \mathcal{...
- 677