Questions tagged [oa.operator-algebras]

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

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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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0answers
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
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1answer
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 ...
7
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1answer
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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2answers
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 \...
2
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1answer
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 $...
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3answers
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 ...
2
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1answer
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
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1answer
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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0answers
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\{\|...
7
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1answer
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 ...
3
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2answers
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 ...
7
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1answer
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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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
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1answer
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|= \...
2
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1answer
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 ...
4
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1answer
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\...
8
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1answer
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, ...
3
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1answer
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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0answers
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 ...
6
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1answer
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 $A$ ...
3
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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 ...
7
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1answer
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 ...
6
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1answer
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 ...
3
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1answer
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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0answers
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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1answer
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. ...
2
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1answer
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 $\mathcal{H}$ ...
10
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1answer
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 ...
1
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1answer
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 ...
8
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1answer
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" ...
3
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0answers
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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0answers
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 ...
2
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1answer
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 $\mathbb{C} 1$) ...
5
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1answer
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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0answers
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) \...
6
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1answer
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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0answers
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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0answers
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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2answers
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 $P_G\subset \mathbb{C}G$. ...
2
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0answers
65 views

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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0answers
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 ...
6
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1answer
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 ...
3
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1answer
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 ...
2
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1answer
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 $I$ ...
17
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1answer
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 ...

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