# Questions tagged [oa.operator-algebras]

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

1,694
questions

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### Completely positive maps on Coxeter groups - the general case

In the reference below Bozejko and Speicher showed the following (for the full statement see the remark on page 9):
Let $(W,S)$ be a Coxeter system, let $\mathcal{H}$ be a Hilbert space and denote ...

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62 views

### Normal states in the factor of type III $_1$

Let $M$ be a factor of type III$_1$. If $w_1$ and $w_2$ are two normal states on $M$, by Connes-Stormer transitivity theorem, for any $\delta>0$, there exists a unitary $u\in M$ such that $\|w_1–...

**2**

votes

**1**answer

97 views

### Direct integral decomposition relative to a given measure space

It is well known that a separable Hilbert space $H$ decomposes as a direct integral in the presence of an abelian von Neumann algebra $\mathscr A\subseteq B(H)$.
More precisely, and quoting from ...

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**1**answer

173 views

### Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?

Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...

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vote

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85 views

### Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra

Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions:
B is a von Neumann algebra with $A'' = B$.
The inclusion $A \...

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**1**answer

126 views

### Opposite $C^*$ algebras

$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...

**17**

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**3**answers

664 views

### Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...

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74 views

### Covariant representations and crossed products of von Neumann algebras

Let $(M,G,\alpha)$ be a $W^\ast$-dynamical system with $G$ locally compact abelian (I am mostly interested in the case $G=\mathbb{R})$. A covariant representation of $(M,G,\alpha)$ is a pair $(\pi,u)$ ...

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73 views

### 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 ...

**10**

votes

**1**answer

228 views

### Factor states on C*-algebras

Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras ...

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87 views

### relationship between two modular automorphism groups associated n.s.f weights

Let $w$ be a normal semifinite faithful weight on a von Neumann algebra $M$. If $w_n$ is a sequence of weights which converge to $w$ in norm topology.
Does there exist relationship between $\{\sigma_t^...

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125 views

### 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\{\|...

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233 views

### Induction and restriction of unitary representations

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Res{Res}$Given a locally compact group $G$ and a closed subgroup $H\subset G$,
let $\Rep(G)$ and $\Rep(H)$ denote their ...

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176 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 ...

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416 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 ...

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84 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 ...

**4**

votes

**1**answer

131 views

### Polar decomposition in abstract von Neumann algebra

Probably an easy question, but here goes:
In a concrete von Neumann algebra $M \subseteq B(H)$, every element $m \in M$ has a polar decomposition $m= p|m|$ where $p$ is a partial isometry and $|m|= \...

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**1**answer

120 views

### Trying to understand Haagerup tensor product $B(H)\otimes_{\rm h}B(K)$

I’m self reading Haagerup tensor product of operator spaces. Understanding it properly, I’m trying some examples. Let $H$ And $K$ be Hilbert space. Let $B(H)$ and $K(H)$ denotes the spaces of bounded ...

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votes

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207 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\...

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150 views

### Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?

Let $\Gamma$ be a discrete group whose reduced group $C^\ast$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $\mathcal{L}(G)$ is a full $\text{II}_1$-factor, ...

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**1**answer

162 views

### English translation of von Neumann's Algebra der Funktionaloperationen (1930)

Does anyone know if there exists an English translation of von Neumann's early work in operator theory, in particular the paper Zur Algebra der Funktionaloperationen und Theorie der normalen ...

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106 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 ...

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158 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 $A$ ...

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85 views

### When can a state on a C*-algebra descend to a quotient?

Has anything been written on the following question:
Let $(\mathfrak{A}, \phi)$ consist of a C*-algebra $\mathfrak{A}$ equipped with a tracial state $\phi$. Let $\pi : \mathfrak{A} \twoheadrightarrow ...

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204 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 $\...

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110 views

### von Neumann algebras with certain properties

Let $\mathcal M$ be a von Neumann algebra which is equipped with a normal faithful tracial state. Further assume that there exist projection $e\in\mathcal{Z}(\mathcal M)$, i.e. the center of $\mathcal ...

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94 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 ...

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280 views

### Finite compact quantum groups

Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...

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97 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 ...

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56 views

### Are quasitrace extensions unique?

I'm trying to understand the basics of quasitraces on $C^*$-algebras. Using the terminology of Haagerup, given $n \geq 2$, an $n$-quasitrace $\tau$ on a $C^*$-algebra $A$ is a 1-quasitrace on $A$ ...

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136 views

### Relating different constructions of the universal compact quantum group

Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections.
...

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164 views

### Decomposition of Hilbert spaces via groups and algebras representations

Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...

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756 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 ...

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101 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 ...

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154 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" ...

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93 views

### Cuntz semigroups of basic C*-algebras

I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103):
"[...] $A_i$ is ...

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63 views

### Showing the existence of a right-inverse in a von Neumann probability space

Disclaimer: This is my first post here on Overflow as opposed to the "normal" forum, so if this question is too elementary for this forum, I'd appreciate y'all letting me know. I posted it ...

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**1**answer

95 views

### Uniqueness of the direct sum of $C^*$ algebras as quotient of free products

Suppose that you have $A, B$ two unital $C^*$ algebras and let $A \ast B$ the reduced free product (I think that it is the reduced amalgamated product over the common $*$-subalgebra $\mathbb{C} 1$) ...

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238 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 $\...

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62 views

### Extension of states satisfying a certain inequality

Let $A$, $B$ be (unital) $C^\ast$-algebras with $B \subseteq A$, let $\phi$ be a state on $B$ and let $b \in B$ be an element with $b b^\ast, b^\ast b \geq C$ for some $C>0$ and $\phi(b^\ast ab) \...

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164 views

### Image of $L^2M$ inside $L^1M$, for $M$ a von Neumann algebra

Let $M$ be a factor (von Neumann algebra with trivial center), and let $L^1M:=M_*$ be its predual.
Let $\omega:M\to\mathbb C$ be a faithful normal state.
The Hilbert space $L^2M:=L^2(M,\omega)$ admits ...

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126 views

### Morphism of non-commutative algebras

Disclaimer: this question is a "big picture" one that comes from my personal thoughts in physics. If it doesn't fit this site, please tell me.
While having a walk, I thought a bit about what ...

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99 views

### Is there a finite depth irreducible subfactor of prime index and not group-subgroup?

Let $N \subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N \subset P \...

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165 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 $P_G\subset \mathbb{C}G$. ...

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### Construction of non-commutative torus using ergodic action of $\mathbb{T}^{n}$

It is well known that non-commutative torus can be constructed using universal C* algebra, by n unitary elements and twisted relations. It can also be constructed using ergodic action of torus group $\...

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101 views

### What are all the possible indices for the finite depth subfactors?

Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to ...

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184 views

### Amenable groupoid C*-algebras satisfy the UCT in English?

As is by now well known, Tu proved in 1998 that the C*-algebras coming from amenable groupoids satisfy the so-called UCT (universal coefficient theorem). Unfortunately, I don't speak french and I've ...

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150 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 ...

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102 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 $I$ ...

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367 views

### For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?

Title. For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
If $G$ is a subgroup of either $S^0,S^1,S^3$ or $S^7$ this induces a free action ...