**0**

votes

**1**answer

167 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 ...

**3**

votes

**1**answer

103 views

### Kernel of the natural map between group $C^*$-algebras

Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...

**4**

votes

**1**answer

118 views

### An extension of $K$-theory to topological $^*$-algebras

What I have in mind is the following: a (sequence of) functor(s) $K_\bullet$ on the category of topological $^*$-algebras (with values in the category of commutative groups) that satisfies (among ...

**3**

votes

**0**answers

146 views

### An equivalent relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...

**1**

vote

**0**answers

76 views

### Fredholm subvector spaces of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$.
Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if ...

**3**

votes

**0**answers

132 views

### C$^*$-algebras isomorphic after tensoring

If $\mathfrak S$ denotes the set of all non-zero C$^*$-algebras (up to $*$-isomorphism) of some bounded cardinality, for instance separable, then $(\mathfrak S, \otimes_\textrm{min})$ and $(\mathfrak ...

**6**

votes

**1**answer

318 views

### 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) \ \ ...

**1**

vote

**1**answer

156 views

### Mysterious central projections in the full group $C^*$-algebra

Let me quote the following theorem about the structure of $C^*(G)$ for property $T$ group (the reference is Higson and Roe "Analitycal K-homology"):
Let $G$ be a property $T$ (discrete) ...

**4**

votes

**2**answers

379 views

### $C^{*}$ algebras which do not admit nontrivial idempotent morphism

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.
I ...

**11**

votes

**1**answer

426 views

### A generalization of the Powers-Stormer inequality

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

**20**

votes

**2**answers

887 views

### Separating pure states on the $2\times 2$ matrix algebra

I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help.
Let $\mathcal{A}$ be the C*-algebra of ...

**9**

votes

**2**answers

482 views

### Schur's Lemma for Hilbert spaces

Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with ...

**15**

votes

**2**answers

362 views

### C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$

In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes ...

**6**

votes

**1**answer

240 views

### Fredholm operators in $K$-theory?

Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ...

**2**

votes

**0**answers

96 views

### Rank–nullity theorem for finite von Neumann algebras

The rank-nullity theorem states that for $U, V$ finite dimensional vector spaces and $T:U \to V$ a linear map $$\dim(U) = \dim(im(T)) + \dim(ker(T)) $$
Let $M \subset B(H) $ be a finite von Neumann ...

**1**

vote

**1**answer

109 views

### Faithul map and (minimal) tensor product of $C^*$-algebras

Let $f$ be a faithful state on a $C^*$-algebra $A$, i.e. $f(a^*a)=0$ implies $a=0$. in general, call a mapping $T:A \to B$ between $C^*$-algebras faithful if $T(a^*a)=0$ implies $a=0$. How to prove ...

**1**

vote

**1**answer

77 views

### examples of completely positive order zero maps to demonstrate a theorem

I'm interested explicit examples which can be used to demonstate the theorem:
Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set ...

**3**

votes

**1**answer

139 views

### A relative property gamma and $L(\mathbb F_2)$

Given any unital non-commutative subalgebra $\mathcal M$ of $L(\mathbb F_2)$ is it true that $\mathcal M' \bigcap L(\mathbb F_2)^\mathcal U = \mathbb C I$ for any free ultrafilter $\mathcal U$?

**0**

votes

**0**answers

59 views

### An equality for the trace of the join of a non-degenerate indecomposable system of projections in a finite factor

Let $M \subset B(H)$ be a finite factor (see for example here p2, or there) with a trace $tr$.
The subset of projections of $M$ is naturally a lattice, noted $(\mathcal{P}(M), \wedge, \vee)$.
A ...

**12**

votes

**1**answer

916 views

### Matrices with entries in a $C^*$-algebra

Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution ...

**1**

vote

**3**answers

374 views

### What's the relation between fusion and coproduct?

For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X_i)_{i \in I}$ of ...

**5**

votes

**1**answer

426 views

### A problem in functional calculus

This is embarrassing, I think it must work, but I can't see how to prove it works. If anyone knows enough functional calculus of operators on a Hilbert space to tell me how to do it, I would be very ...

**5**

votes

**1**answer

250 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 $ * ...

**1**

vote

**3**answers

115 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}\ ...

**4**

votes

**1**answer

230 views

### reference request: direct product of WOT-continuous unitary representations

In an article I'm revising, I spend some time giving a self-contained proof of the following result
Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...

**0**

votes

**0**answers

80 views

### Cuntz comparison of strictly positive elements in finite C*-algebras

Let $A$ be a finite, non-unital C*-algebra, $s\in A$ a strictly positive element and $a\in A$ a positive element that is Cuntz-equivalent to $s$, i.e. there exist sequences $\{x_n\},\{y_n\}\subset A$ ...

**17**

votes

**3**answers

1k views

### Reference request for translating from Top to C*-alg

Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality-- namely, that the categories of ...

**11**

votes

**1**answer

154 views

### positive not completely positive maps

In extension to this question
Positive but not completely positive?
I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. ...

**4**

votes

**1**answer

124 views

### Stinespring's dilation without $C^{\ast}$-algebras

Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra?
I will now state the version of Stinespring's dilation ...

**5**

votes

**0**answers

142 views

### Non-linear positive map

In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...

**4**

votes

**1**answer

245 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 ...

**2**

votes

**2**answers

199 views

### Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages.
I read some pages such as
Reference for the Gelfand-Neumark theorem for commutative von Neumann ...

**1**

vote

**1**answer

94 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 ...

**9**

votes

**2**answers

549 views

### General recipe for building C*-algebras out of combinatorial object

I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...

**2**

votes

**1**answer

91 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 ...

**7**

votes

**1**answer

205 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

174 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

48 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

200 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

192 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

110 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 ...

**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

60 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

154 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

151 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

309 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

58 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 ...

**6**

votes

**0**answers

170 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

136 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 ...