# Questions tagged [oa.operator-algebras]

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

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### 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 ...

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### 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
$...

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### 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 ($A_q$) ...

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### 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 ...

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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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### 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 $...

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### 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 ...

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### 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,...

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### 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 ...

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### 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 ...

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### $w^*$-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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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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### 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-...

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### 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
$...

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### 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^{...

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### 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 ...

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### 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 ...

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### 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 $\alpha\in\mathrm{Aut}(A)$...

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### 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-...

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### 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 $b_n$, ...

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### 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?

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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....

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### 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 ...

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### 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 \...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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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}{...

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### 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{...

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### 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 $E_i:M\to M_i$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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-...