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

learn more… | top users | synonyms (1)

0
votes
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
2answers
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
3answers
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
1answer
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
1answer
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
3answers
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
1answer
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
0answers
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
3answers
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
3answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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 ...