**0**

votes

**0**answers

38 views

### Does the following $ C^{*} $-algebraic result have a purely algebraic proof?

While studying the proof of Bott periodicity for operator $ K $-theory in this set of notes, I learned this fact:
Theorem. Let $ A $ and $ B $ be $ C^{*} $-algebras. Let $ f,g: A \to B $ be $ * ...

**0**

votes

**0**answers

14 views

### Tomita Takesaki theory and boundeness of $S$

Let $M$ be a von Neumann algebra, $\xi$-separating and cyclic vector for $M$. Let $S$ be antilinear operator acting as $x \xi \mapsto x^* \xi$ where $x \in M$. Then one can show that $S$ is closable ...

**1**

vote

**0**answers

43 views

### closed range bounded linear operators [migrated]

Let $CL(X,Y)$ be the set of all closed range bounded linear operators from Banach space $X$ to Banach space $Y$. Is $CL(X,Y)$ an open set of $B(X,Y)$?

**2**

votes

**0**answers

57 views

### A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $

Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...

**0**

votes

**1**answer

125 views

### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

**1**

vote

**2**answers

76 views

### Extending GUE to a measure on operators?

Let $\mathcal H_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu_{GUE}$ denote the following measure on $\mathcal H_n$:
$$
\mu_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ ...

**7**

votes

**1**answer

191 views

### allowing `discontinuous functions' into a C* algebra

There follows a possible construction, and I would like to know if it or a similar construction has been done before (as I suspect), so that I can reference it, or if it obviously does not work! Any ...

**1**

vote

**1**answer

163 views

### Almost complex structure and nontrivial idempotents

Is there a compact Reiemannian manifold $M$ for which the following complex $C^{*}$ algebra does not have a nontrivial idempotent:
$A=Hom(E,E)$ where $E$ is the complexification of $TM$.
Of ...

**0**

votes

**0**answers

44 views

### A constraint on the indices of irreducible finite depth subfactors

Let $(A \subset B)$ and $(C \subset D)$ be two finite index finite depth irreducible subfactors.
Question: Is it true that $[B:A] \cdot [D:C]$ is an integer iff $[B:A]$ and $[D:C]$ are integers?
...

**7**

votes

**0**answers

188 views

### 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?(For ...

**8**

votes

**1**answer

183 views

### Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...

**5**

votes

**1**answer

102 views

### The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on ...

**1**

vote

**1**answer

77 views

### 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 ...

**5**

votes

**1**answer

407 views

### When does a matrix define a convolution operator on a hypergroup?

Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ ...

**7**

votes

**3**answers

2k views

### Zero divisor conjecture and idempotent conjecture

Let $G$ be a torsion-free group and $C$ the ring of complex numbers. The zero divisor (idempotent, resp.) conjecture is that there is no nontrivial zerodivisor (idempotent, resp.) in $CG$.
The wiki ...

**1**

vote

**0**answers

56 views

### Examples of non-extremal subfactors

Every subfactor $(N \subset M)$ in this post are supposed to be finite index inclusion of ${\rm II}_1$ factors.
Definition (here p64): Such a subfactor is called extremal if $tr_{N'} = tr_M$ on ...

**2**

votes

**1**answer

153 views

### Has a subfactor with lattice $B_3$, a singly generated identity biprojection?

Let $(N \subset M)$ be an irreducible finite index subfactor.
If its lattice of intermediate subfactors is equivalent to $B_3$ (the lattice of divisors of $n=p_1p_2p_3$ square free):
...

**5**

votes

**1**answer

144 views

### 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 ...

**1**

vote

**0**answers

300 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, ...

**0**

votes

**0**answers

57 views

### 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 ...

**5**

votes

**0**answers

154 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 $A\subset B$ ...

**4**

votes

**1**answer

126 views

### 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 ...

**0**

votes

**1**answer

75 views

### 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 ...

**1**

vote

**1**answer

45 views

### 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?

**3**

votes

**0**answers

51 views

### 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 ...

**10**

votes

**2**answers

496 views

### B(H) as a direct sum of a closed two sided ideal and a subalgebra

Let $B(H)$ is the C*-algebra of all bounded linear operators on
Hilbert space $H$. Are there a closed two-sided ideal $I$ and a
subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus A$ (direct sum I and ...

**6**

votes

**3**answers

660 views

### Generalizations and relative applications of Fekete's subadditive lemma

Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...

**3**

votes

**1**answer

158 views

### 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 ...

**0**

votes

**1**answer

199 views

### 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 ...

**0**

votes

**0**answers

37 views

### 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 ...

**1**

vote

**0**answers

226 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 ...

**5**

votes

**2**answers

223 views

### 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 ...

**9**

votes

**1**answer

282 views

### K-theory of ultrapowers

It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^*$-algebras. For instance, if $A$ is a $C^*$-algebra does $K_0(A^U)$ depend on ...

**5**

votes

**0**answers

232 views

### 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. ...

**0**

votes

**1**answer

118 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 ...

**21**

votes

**1**answer

2k views

### What is about nonassociative geometry?

At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below:
Audience: You showed ...

**1**

vote

**0**answers

114 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. ...

**1**

vote

**0**answers

101 views

### 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 ...

**3**

votes

**0**answers

100 views

### 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 ...

**3**

votes

**1**answer

146 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 $A$ ...

**33**

votes

**3**answers

3k views

### Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...

**1**

vote

**1**answer

107 views

### 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 $C^{*}$ ...

**4**

votes

**1**answer

120 views

### 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 ...

**7**

votes

**1**answer

237 views

### 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 ...

**1**

vote

**1**answer

53 views

### 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 ...

**2**

votes

**0**answers

90 views

### 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 ...

**1**

vote

**0**answers

57 views

### 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 ...

**8**

votes

**0**answers

265 views

### Are there only finitely many maximal irreducible amenable subfactors at fixed finite index?

A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
Question: Are there only finitely many maximal irreducible amenable subfactors at ...

**1**

vote

**0**answers

54 views

### 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 ...

**0**

votes

**0**answers

55 views

### 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 ...