Skip to main content

Questions tagged [oa.operator-algebras]

Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

Filter by
Sorted by
Tagged with
0 votes
0 answers
177 views

Where can I find upcoming conferences in functional analysis and operator theory? [closed]

When I try to search for upcoming mathematical conferences, they are all either about discrete mathematics, statistics, numerical analysis or dynamical systems. I am yet to find a list of upcoming ...
S-F's user avatar
  • 179
1 vote
1 answer
137 views

Takesaki II "Bimodule" question

Consider the following fragment from Takesaki's book "Theory of operator algebras", chapter IX Non-commutative integration, Section 3 on p187-188: I have trouble understanding the equality $...
Andromeda's user avatar
  • 169
1 vote
0 answers
61 views

Affiliating the whole algebra of 'coordinates' with a locally compact quantum group

When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$) ...
szantag's user avatar
  • 113
0 votes
0 answers
144 views

Unitary operators in $\mathcal{U}(L^2(X,\mu))$ not coming from $\operatorname{Aut}(X,\mu)$

$\DeclareMathOperator\Aut{Aut}$Let $(X,\mu)$ be a measured space with probability measure $\mu$ and $\Aut(X,\mu)$ denote the group of measure preserving bijections of $(X,\mu)$. It is easy to see ...
Kajal Das's user avatar
0 votes
0 answers
95 views

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 $\...
E. Papapetros's user avatar
0 votes
0 answers
39 views

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 $...
ReverseFlash's user avatar
6 votes
2 answers
158 views

Proof that every commutative locally compact quantum group arises from a locally compact group

It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof ...
szantag's user avatar
  • 113
3 votes
0 answers
105 views

Is every strongly singly generated type $II_1$ factor generated by a pair of hyperfinite $II_1$ subfactors with a common MASA?

A type $II_1$ factor $M$ is strongly singly generated (SSG) if every amplification $M^{t}$ of $M$ is singly generated as a von Neumann algebra. Interesting characterizations of SSG type $II_1$ factors,...
Jon Bannon's user avatar
  • 7,037
2 votes
0 answers
78 views

Amenability and the unitary group of an operator algebra

Let $M$ be a von Neumann algebra and $U(M)=\{x\in M: x^*=x^{-1}\}$ be its unitary group. In this post, we equip $U(M)$ only with the relative weak$^*$ topology $\sigma(M,M_*)$. Then, $U(M)$ is a ...
Onur Oktay's user avatar
  • 2,283
2 votes
1 answer
203 views

Characterization of certain subalgebras of $M_2(\mathcal{A})$ where $\mathcal{A}$ is a $C^*$-algebra

Let $\mathcal{A}$ be a $C^*$-algebra generated by a single element $a \in \mathcal{A}$. Suppose that it is also generated by another element $b \neq a$. Consider a subalgebra $\tilde{\mathcal{A}}$ of ...
plllnt's user avatar
  • 133
3 votes
2 answers
103 views

$w^*$-limit of projections in von Neumann algebra

Let $\mathcal M$ be a semi finite von Neumann algebra with a normal faithful semi finite trace $\tau$. Let $(e_i)_{I\in I}$ be a net of projections in the von Neumann algebra which converges to an ...
A beginner mathmatician's user avatar
-1 votes
1 answer
90 views

Is an $A$-$B$—$C^*$-correspondence a representation of a $G$-$C^*$-algebra, $\rho \colon A \otimes_{ \alpha } B \to \mathcal{L} ( \mathcal{H} )$?

Let $R$ and $S$ be two rings. It is known that an $R$-$S$-bimodule is actually the same thing as a left module over the ring $R \otimes_{\mathbb{Z}} S^{\mathrm{op}}$, where $S^{\mathrm{op}}$ is the ...
Angel65's user avatar
  • 525
1 vote
1 answer
131 views

Form of a hereditary subalgebra of $C^*$-algebra $C_0(X)$

I would like to show that: "every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed ...
VvvV's user avatar
  • 11
3 votes
0 answers
83 views

Excising the trace of a $II_1$-factor

Recall that a state $\varphi$ on a $C^*$-algebra $A$ is said to be excised by projections if there exists a net of projections $e_i \in A$ such that $\| e_i a e_i - \varphi(a) e_i\| \to_{i} 0$ for all ...
pitariver's user avatar
  • 297
4 votes
1 answer
125 views

Fixed point algebra of a non-amenable factor

Let $M$ be a non-amenable factor. Suppose $\alpha$ is an action of a group $\Bbb R$ on $M$. Define $$M^{\alpha}:=\{x\in M: \alpha_t(x)=x \text{ for any} \; t\in \Bbb R\}.$$ If we know that there exits ...
mathbeginner's user avatar
0 votes
1 answer
74 views

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.
kreitz's user avatar
  • 53
5 votes
0 answers
246 views

Failure of Tomiyama's property ($F$) for reduced group $C^*$-algebras

Are there known examples of discrete groups such that the minimal tensor product of their reduced group $C^\ast$-algebras does not have Tomiyama's property ($F$)? Such groups must necessarily be non-...
Are Austad's user avatar
0 votes
1 answer
109 views

Minimal norm problem whose unknown is an operator

Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that $$ f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2 $...
user8469759's user avatar
9 votes
0 answers
287 views

Is $\mathcal{B}(\mathcal{H})$ a groupoid $C^*$-algebra?

Let $\mathcal{H}$ be a complex Hilbert space, and $\mathcal{B}(\mathcal{H})$ be the $C^{\ast}$-algebra of bounded operators on $\mathcal{H}$. Is there an étale groupoid $\mathcal{G}$ such that its $C^{...
Luiz Felipe Garcia's user avatar
1 vote
1 answer
73 views

A subfactor of finite index

We say $N$ is a subfactor of finite index of the factor $M$ if there is a normal faithful conditional expectation of $M$ onto $N$ that has finite index. If $N$ is a subfactor of finite index of the ...
mathbeginner's user avatar
12 votes
0 answers
544 views

Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry

In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry. For graphs this had been an open ...
David Roberson's user avatar
4 votes
0 answers
127 views

Automorphism-invariant positive linear functionals on $C*$-algebras

Let $A$ be a $C^*$-algebra. Does there exist a non-trivial positive linear functional $\nu\in A^*$ which is $\mathrm{Aut}(A)$-invariant? That is, $\nu\circ\alpha=\nu$ for all $\alpha\in\mathrm{Aut}(A)$...
Bedovlat's user avatar
  • 1,939
1 vote
0 answers
84 views

positive invertible maps which are not *-automorphisms

Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...
fidaleo's user avatar
  • 41
3 votes
1 answer
144 views

Rescaling Fourier coefficients of a continuous function by a bounded sequence

This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
Logan Hyslop's user avatar
7 votes
1 answer
148 views

type II$_1$ subalgebra of type III$_1$ factor

Let $M$ be a type III$_1$ factor and $N$ be the type II$_1$ subalgebra of $M$. What is the type of $N'\cap M$? Can it be any type?
mathbeginner's user avatar
1 vote
0 answers
102 views

Infinite tensor product of Hilbert spaces [duplicate]

Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
Jake's user avatar
  • 11
0 votes
0 answers
142 views

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 ...
0x11111's user avatar
  • 493
6 votes
1 answer
290 views

Pairwise orthogonality for partitions of unity in a *-algebra

Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
JP McCarthy's user avatar
7 votes
2 answers
1k views

Is there a notion of point in noncommutative geometry?

It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
Esmond's user avatar
  • 136
2 votes
0 answers
239 views

What are alternative or equivalent definitions of a positive-definite function on a group?

The standard definition of a positive-definite function on a group goes as follows: Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
S-F's user avatar
  • 179
1 vote
0 answers
143 views

Reconstructing the manifold from space of functions in quantum mechanics

Due to Banach–Mazur, every separable Banach space is isomorphic to a subspace of $C([0,1])$. But some spaces, like $C([0,1]^n)$ and generally $C(M)$ for $M$ a manifold, allow one to reason about the ...
0x11111's user avatar
  • 493
6 votes
0 answers
94 views

Conditions for completely positive maps to act homomorphically across multiple subalgebras

For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
BiPolarBear's user avatar
4 votes
0 answers
160 views

Representations of $C\left(SO_q(n)\right)$

A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
Surajit's user avatar
  • 73
3 votes
1 answer
157 views

Unitary versus isometric operators

Let $\mathbb H$ be a Hilbert space, and let $\mathcal B(\mathbb H)$ be the space of bounded operators on $\mathbb H$, equipped with the operator-norm topology. Let $\mathbb R\ni t\mapsto A(t)\in \...
Bazin's user avatar
  • 15.5k
4 votes
2 answers
1k views

Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
Daigaku no Baku's user avatar
2 votes
0 answers
157 views

Question about the ergodic mean

This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question. I've read a thesis where there is an example on ergodic mean, where however there is ...
MBlrd's user avatar
  • 33
7 votes
0 answers
159 views

Nontrivial examples of locally compact quantum groups

What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
Cameron Zwarich's user avatar
2 votes
0 answers
134 views

What is meant by saying that the Shilov boundary of the polydisc $\mathbb D^n$ is $\mathbb T^n\ $?

Let $A$ be a complex Banach algebra and $M_A$ be the space of all non-zero multiplicative linear functionals on $A$ equipped with the weak$^*$-topology. Let $\widehat A$ be the image of $A$ under the ...
Anacardium's user avatar
0 votes
0 answers
104 views

$C^*$ 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}{...
Math101's user avatar
  • 167
1 vote
1 answer
271 views

A certainty principle?

Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra with $\varphi\in\mathcal{S}(\mathcal{A})$ a state. Where $$\sigma_\varphi(a):=\sqrt{\varphi((a-\varphi(a)1_{\mathcal{A}})^2)}\qquad (a\in \mathcal{...
JP McCarthy's user avatar
7 votes
1 answer
239 views

Approximately semifinite factors

For the sake of this question, lets call a factor $M$ approximately semifinite if there exists an increasing net of semifinite subfactors $M_i$, $i\in J$, with conditional expectations $E_i:M\to M_i$ ...
Lau's user avatar
  • 749
8 votes
0 answers
233 views

Question about the homogeneity of the state space of a type $\rm{III}_1$ factor

I'm reading the paper Homogeneity of the State Space of Factors of Type $\rm{III}_1$ by Connes and Størmer. Homogeneity of the state space means that all normal states are approximately unitarily ...
Lau's user avatar
  • 749
2 votes
0 answers
168 views

Zeta zeros and prolate wave operators

Recently, Connes, Consani and Moscovici in https://arxiv.org/abs/2310.18423 have blended two of their results on zeta zeros and the prolate wave operators, which, they say, "suggests the ...
Jon23's user avatar
  • 869
7 votes
1 answer
373 views

Positive cone in Haagerup L²-space: how much information does it contain?

Given a von Neumann algebra $A$, its Haagerup $L^2$-space $H:=L^2A$ (also known as the standard form of the Neumann algebra) comes equipped with a positive cone $P\subset H$. Question:    How much ...
André Henriques's user avatar
0 votes
0 answers
111 views

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 ...
Arbiter's user avatar
  • 151
2 votes
1 answer
203 views

On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
B.Hueber's user avatar
  • 1,077
2 votes
1 answer
240 views

Continuous path of unitary matrices with prescribed first column?

Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$. Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
ccriscitiello's user avatar
4 votes
1 answer
232 views

Strengthening the direct integral decomposition of von Neumann algebas

Let $M$ be a von Neumann with separable predual. It well known that one can write $M$ as a direct sum $M=M_I\oplus M_{II} \oplus M_{III}$ of von Neumann algebras of types $I$, $II$ and $III$. It is ...
Lau's user avatar
  • 749
1 vote
1 answer
270 views

An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible

Given: $X$ - any Banach space $F : X \to X$ (linear bounded and non-invertible) $P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$ Can you help me come ...
TorteDeline's user avatar
4 votes
0 answers
194 views

Bochner theorem for (non-abelian) discrete groups

I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
Tomás Pacheco's user avatar

1
2 3 4 5
43