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

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What's the relation between spin model for subfactors theory and physics?

In the sense of subfactor theory, a spin model is a commuting square of the form $$\begin{matrix} \Delta &\subset & M_n(\mathbb{C})\cr \cup &\ &\cup\cr \mathbb{C} &\subset ...
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projective module over C*-algebra [closed]

Suppose we have $V$ a projective module over C*-algebra $A$. Suppose we define another projective module $W$ which is same as $V$ but the action is given by: $v.a := v.\alpha(a)$, where $\alpha$ is ...
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265 views

Langlands reciprocity for C*-algebras

I just came across this paper which, judging by what I understood, establishes the Langlands reciprocity conjecture for a certain Shimura variety. My question, regardless of the validity of the proof, ...
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119 views

A “slice-map” type problem for symmetric tensors in the square of a nuclear C*-algebra

Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras. Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...
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A question about correlations between $ C^{*} $-algebras

I was studying J. M. G. Fell’s paper The Structure of Algebras of Operator Fields when I encountered the concept of a correlation between two $ C^{*} $-algebras. Definition. Let $ A $ and $ B $ be ...
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The completely reducible bimodules coming from subfactors

This post is a sequel of: Are all the R-R-bimodules completely reducible? Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...
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Element Analytic, C*-dynamical system

good night... I was looking into the Pedersen Book, $C^{*}$-Algebras and their automorphism groups, and found the definition of analytic elements $x\in A$, where $(A,\alpha)$ is a $C^{*}-$dynamical ...
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Is the (hyperfinite) TLJ subfactor unique at fixed index (if it exists)?

Let $(N \subset M)$ be an inclusion of hyperfinite ${\rm II}_1$ factors, with the following principal graph (called TLJ) Question: Is such a subfactor unique (up to ${\rm W}^*$-isomorphism) at fixed ...
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Are the two-side TLJ subfactors maximal?

Let $(N \subset M)$ be an inclusion of ${\rm II}_1$ factors, with the following principal graph (called two-side TLJ) Question: Is $(N \subset M)$ a maximal subfactor?
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Is there no extra intermediate subfactor for the basic construction?

Let $(N \subset M)$ be an inclusion of ${\rm II}_1$ factors, the basic construction is $N \subset M \subset M_1 = \langle M , e^M_N \rangle$. Question: For any intermediate subfactor $N \subset P ...
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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes [closed]

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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When is a cycle in $KK^G(A,A)$ with zero operator the identity cycle?

Given a cycle of the form $(\pi,H,0)$ in $KK^G(A,A)$, when is it equivalent to the identity cycle $1_A=(i_A,A,0)$? The operator $T=0$, and $\pi:A \rightarrow L(H)$ may be injective. Any criterions ...
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Koopman representation, weakly compact action, Ozawa Popa

Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
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104 views

Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$. Ultimately, I'm interested in finding a ...
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217 views

Analogues of the Monster for central charges different from 24

One way to define the Monster group is to consider a conformal field theory (CFT) corresponding to central charge $c=24$ and look at the automorphism group of its vertex operator algebra. For one of ...
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104 views

Matrix norm on $L^p$ operator algebras

I have a question about the framework of $L^p$ operator algebras ($1<p<\infty$), i.e. norm-closed subalgebras of $B(L^p(X,\mu))$ for some measure space $(X,\mu)$ (see e.g. ...
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Determining the primitive ideal space of C-star algebras

Is there a general way of finding a primitive ideal space of $C^*$-algebra? For example, if $C^*$-algebra is given by the universal $C^*$-algebra generated by two self-adjoint unitary elements, how ...
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138 views

Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$

(This is a repost of a question from math.SE, http://math.stackexchange.com/questions/1240966/existence-of-state-on-a-c-algebra-satisfying-tauab-ab) Let $a,b$ be elements of a unital C*-algebra $A$ ...
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Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold

Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$. This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ...
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A $C^{*}$ algebra associated to a graded $C^{*}$ algebra

A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ ...
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For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?

I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...
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Arveson spectrum for a unitary representation of a group on a Hilbert space

Although this is not research, I think the question is a little bit too specific for math.stackexchange Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a ...
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Why does one only consider one-parameter groups in Borchers-Arveson theorem?

(question from math.stackexchange) The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter ...
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Are there infinitely many amenable Hadamard-Petrescu subfactors?

The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by ...
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When are countably generated Hilbert modules generated by c.p.c. order zero maps?

Throughout let $B$ be a stable C*-algebra, i.e. $B\cong B\otimes K$, where $K$ is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. It is well-known that any ...
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Two questions on topological and geometric structure of projections in a simple $C^{*}$ algebra

Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the ...
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Is $KK^G(\mathbb{C}^n,B)$ countably additive in $B$ and countable?

Let $G$ be a finite discrete groupoid, $A=\mathbb{C}^n$ a finite dimensional, commutative $C^*$-algebra and assume we have given a $G$-action on $A$. Note that the action of $G$ on ...
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A $C^{*}$ algebra associated to a group [closed]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...
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What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas? More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
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How to generalize multiplication and addition to cyclic subfactors?

Let $(N \subset M)$ be a finite index irreducible subfactor and $P=P(N \subset M)$ its subfactor planar algebra. Definition: $(N \subset M)$ is cyclic if its lattice of intermediate subfactors ...
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Range of a trace preserving completely positive projection

I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is $$\text{Tr}(P(A)) = ...
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Formula for the distance in noncommutative geometry

Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data: $$(1) \ \ ...
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fixpoint algebras of a permutation action

Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a ...
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Dimension of Birman-Murakami-Wenzl Algebra

I was reading the paper Braids, Link Polynomials and A New Algebra by J. S. Birman and H. Wenzl, and I was wondering is there a combinatorial way to compute the dimension of the algebras ...
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Conjugacy of circle actions on UHF C*-algebras

Consider pointwise continuous actions of the unit circle on the $2^{\infty}$-UHF C*-algebra A by *-automorphisms. Assume that two such actions have the same fixed point algebra, i.e., elements that ...
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Do c.p.c. order zero maps between local C*-algebras map C*-subalgebras to C*-subalgebras?

For the sake of this question I will assume the following Definition A pre-C*-algebra $A$ is local in the sense of [1], i.e. if there is a family of C*-subalgebras $\{A_i\}$ of $A$ with the property ...
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Question about projections in von Neumann algebras

Let $M$ be a von Neumann algebra, and let $\mathcal{P}$ be the set of nontrivial (not equal to $0$ or $e$) projections of $M$. Define $p,q \in \mathcal{P}$ to be equivalent if there exist projections ...
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Approximation in the tensor square of a weakly exact von Neumann algebra

Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...
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Does the Approximation Property (AP) pass to quotients by amenable subgroups?

Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP? In particular, does there exist a group $G$ with the AP and a surjective group ...
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Are isometric homorphisms of C* algebras *-homorphisms

Here is my precise question: Let $A$ and $B$ be two $C^*$ algebras. Let $f: A \rightarrow B$ a homomorphism of $C^*$ algebras which is isometric (on its image). Is $f$ necessary a $*$-homomorphism ? ...
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A matrix trace inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...
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Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
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Is there an irreducible subfactor with an infinite homogeneous single chain lattice?

We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here). Now ...
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A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology. Is there a continuous map ...
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Does this C*-algebra embed into a simple nuclear C*-algebra?

Let $\mathcal K$ denote the C*-algebra of compact operators, and fix an embedding $\phi_n:M_n(\mathbb C) \to \mathcal K$ for each $n\in \mathbb N$. Define the C*-algebra $A := \{(a_n)_{n=1}^\infty ...
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Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems? $$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...
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Embedding the group von Neumann algebra into an injective von Neumann algebra on the same Hilbert space

Let $\Gamma$ be a discrete group, $\newcommand{\VN}{\rm VN}$ and let $\VN(\Gamma)$ denote its von Neumann algebra, regarded as a subalgebra of ${\sf B}(\ell^2(\Gamma))$. It is well known that ...
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Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element? Let $n(A)$ be the infimum of such ...
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Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements are path connected. Is the tensor product of two path connected algebra, a path connected algebra? What ...
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A possible characterization of finite dimensional $C^{*}$ algebras

Is there an infinite dimensional unital $C^{*}$ algebra $A$ with invertible group $U$ such that for every finite dimensional subvector space $Y\subset A$, the number of connected components of ...