All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
7
votes
0
answers
292
views
What morphisms / Morita equivalences induce the 2-periodicity isomorphisms of $KK$-theory?
In Kasparov's paper, the canonical isomorphisms $KK_* \rightarrow KK_{*+2k}$ are defined rather implicitely (by tensoring and stabilization).
Are there morphisms of $C^*$-algebras which induce them (...
- 435
4
votes
1
answer
392
views
Examples of special isometries
Are there examples of (distinct) Hilbert spaces $H_1$=$(H,\langle\cdot,\cdot\rangle_1)$, $H_2 $=$(H,\langle\cdot,\cdot\rangle_2)$ and a linear operator $V: H_1\to H_2$ such that $V^n: H_1\to H_2$ is ...
- 155
6
votes
2
answers
314
views
What is the subfactor planar algebra of type $\tilde{A}_n$, of index 4?
As I understand it, there is a subfactor whose principal graph is the affine Dynkin diagram $\tilde{A}_n$. Since every vertex has two neighbors, does that mean the space of 1-boxes is two dimensional? ...
- 425
3
votes
0
answers
146
views
Closed containment of open projections in C*-algebras
For a C*-algebra $A$ and open projections $p,q\in A^{**}$, consider the following statements.
$\overline{p}\leq q$
$p\leq q$ and there exists open $r\in A^{**}$ with $rp=0$ and $r\vee q=1$
$p\leq q$ ...
- 1,307
1
vote
1
answer
108
views
Comparison between spectra
Let $G$ be a normal operator with compact resolvent on a Hilbert space $H$ such that ${\rm ker}(G) \neq {0}$. Further let $P$ be the orthogonal projection onto ${\rm ker}(G)$, and let $G_{0}:=G+P$.
...
- 111
1
vote
1
answer
190
views
Infinite amenable group subfactors
Let amenable groups $\Gamma$ and $\Gamma'$. They act outerly of only one manner on the hyperfinite ${\rm II}_1$-factor $\mathcal{R}$.
Question: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) ...
2
votes
1
answer
612
views
A norm one projection
Let $\mathcal{H}$ be Hilbert space and $\mathfrak{B(}\mathcal{H}\mathcal{)}$ of all bounded linear operators on $\mathcal{H}$. Let $\mathcal{A}$ be a maximal commutative sub-algebra of $\mathfrak{B(}\...
- 43
3
votes
0
answers
144
views
Deformation and Representations
Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...
5
votes
0
answers
321
views
Unitary representations of Tarski Monsters and other beasts
Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
- 647
1
vote
2
answers
2k
views
Quantum channels, question 2: tensor products and composition of functions
Please be kind. I've been working on this for a long time and can't find an answer. Feel free to edit for clarity if you think the question can be better worded.
Background
It may help to see a ...
- 283
0
votes
0
answers
145
views
Group which is not MF or AF
Does someone know example of group (countable, discrete) which can not be embedded (monomorphism) into
$$ U(\prod M_n/\oplus M_n)$$
unitary group of universal MF-algebra? Or example of group which can ...
3
votes
1
answer
301
views
‘Non-Induced’ Left Regular Representations of $ C^{*} $-Dynamical Systems
In what follows, a ‘$ * $-representation’ always means a non-degenerate $ * $-representation.
Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and let $ \pi: \mathscr{A} \to B(\mathcal{...
user36116
5
votes
1
answer
296
views
Number of II${}_1$ factors
McDuff proved that there exist continuum many non-isomorphic (separable) II${}_1$ factors. I would like to politely ask whether it is known/open if one can find $2^{\mathfrak{c}}$ (or at least $\...
- 387
2
votes
0
answers
165
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
0
answers
174
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 $\{x_n\},\{y_n\}\subset A$ ...
- 417
7
votes
1
answer
592
views
topologies on U(H)
There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:
the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some more......
- 43.2k
7
votes
2
answers
525
views
Integrality of the canonical trace and topology
Let $G$ be a discrete group and consider the reduced group C* algebra $C_r^\ast(G)$, viewed as an algebra of bounded operators on $\ell^2(G)$ by the regular representation. The canonical trace on $...
- 29.2k
2
votes
0
answers
209
views
Number of connected components of a $C^{*}$ algebra
Inspired by the concept in the following post
What are these compact sets called?
We introduce the following concept:
Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent ...
- 356
1
vote
1
answer
491
views
Is this result of Spain correct?
Let us have a look on the proof of Theorem 2 in [P. G. Spain, Boolean algebras of projections, Proceedings of the Edinburgh Mathematical Society (Series 2) 19, 03, March 1975, 287-289]
The author ...
- 203
10
votes
0
answers
255
views
Commutative spectral triples not coming from manifolds
There is a very deep and remarkable theorem by Connes (the so called reconstruction theorem) which states that from a commutative spectral triple obeying certain axioms one can reconstruct a smooth ...
- 9,330
5
votes
1
answer
177
views
Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?
A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$.
Is there an infinite depth irreducible finite index maximal subfactor (other than ...
2
votes
0
answers
66
views
Exchanging coordinates in $K(\ell^2(G)) \rtimes G$ by a homotopy
I tried to prove the Green-Julg isomorphism in some notion of KK-theory, but came to this basic problem:
Let $G$ a finite group. Equip $K:=K(\ell^2(G))$ (compact operators) with the left translation ...
- 685
2
votes
1
answer
149
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 ...
- 21
2
votes
0
answers
234
views
The kernel of $C^{*}(G)\to C_{r}^{*}(G)$
Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism.
What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some ...
- 356
3
votes
0
answers
211
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 ...
- 189
6
votes
1
answer
378
views
Crossed product of a C*-algebra by a subgroup
Let $A$ be a unital $C^*$-algebra, let $G$ be a compact group, let $\alpha:G\to\mbox{Aut}(A)$ be a continuous action, and let $H$ be a closed subgroup of $G$. Is there any relationship between the ...
- 717
1
vote
0
answers
308
views
Is a finite depth-index irreducible subfactor, intermediate of a depth ≤ 3 one?
Let $(N \subset M)$ be a finite depth-index irreducible subfactor.
Main question: Is $(N \subset M)$ the intermediate of a finite index depth $\le 3$ irreducible subfactor?
(In others words, is ...
1
vote
1
answer
251
views
On the relation between the set of extreme points of the unit ball of $M(X)$ and $M(X)^{**}$
Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C_0(X)$ is a commutative $C^*-$algebra. ...
- 306
13
votes
0
answers
474
views
Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?
(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard bold.)$\newcommand{\FA}{{...
- 25.8k
0
votes
0
answers
410
views
A noncommutative vector bundle
We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative ...
- 356
6
votes
1
answer
363
views
von Neumann automorphisms: does convergence on a dense algebra imply $u$-convergence?
Let $M$ be a separable von Neumann algebra and let $A$ be a (von Neumann-)dense *-subalgebra.
Suppose that $\alpha,\alpha_1,\alpha_2,\dots$ are automorphisms of $M$, such that for every $a \in A$,
$$ \...
- 1,786
7
votes
2
answers
862
views
Vanishing Trace
Let $\mathcal H$ be a Hilbert space, and let $a \in \mathcal B(\mathcal H)$ satisfy $\mathrm{Tr}(a)=0$. If $a$ is self-adjoint, then we can find a vector $\xi \in \mathcal H$ such that $\langle \xi | ...
- 1,199
1
vote
0
answers
80
views
weak convergence in operator space structure
Let $M$ be von Neumann algebra and $B(H)$ be it's universal representation. Let $(e_i)$ be a Hilbert basis of $H$ and $\zeta_n\xrightarrow{w}\zeta $ in $H$. I know that $[w_{\zeta_n ,e_i}]_{1\times I}\...
- 11
0
votes
0
answers
201
views
Range of a trace preserving completely positive projection
I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...
- 515
8
votes
1
answer
1k
views
Why is it called *spectral* triple?
I know the definition a spectral triple and that it is some kind of non-commutative generalisation of (the ring of functions on) a compact spin manifold.
But, why is it called spectral triple?
- 3,174
5
votes
1
answer
254
views
Well defined Tensoring of spectral triples
Hi,
I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it.
Question: In connes standard model he takes ...
- 133
0
votes
2
answers
225
views
Isomorphism theorem for subfactors?
It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors :
Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
6
votes
1
answer
250
views
Hyperfiniteness of CCR algebra
Hi, It is known that the double commutant of the CCR algebra in it's GNS space
with respect to some quasi-free states are always type III factors.
My question is; Will some of them be hyperfinite ...
- 61
5
votes
0
answers
101
views
$p$-operator space structure on Banach algebras
There is an abstract characterization of operator algebras, which says that if $A$ is an operator space that is also an approximately unital Banach algebra, then the following are equivalent:
For any ...
- 251
10
votes
1
answer
492
views
Which W*-algebras are the duals of C*-coalgebras?
A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
- 2,754
9
votes
0
answers
351
views
How many ideals are there in $B(H)^{**}$?
It is well-known (and easy to prove) that the only closed ideals of $B(\ell_2)$ are $\{0\}$, $B(\ell_2)$ and $K(\ell_2)$, the ideal of compact operators on $\ell_2$. I am curious whether we know what ...
- 91
7
votes
1
answer
317
views
Is there an upper bound on the dimension for irreducible representations of a continuous trace $C^{*} $-algebra?
The following are questions of Don Hadwin:
If $A$ is a unital continuous trace C*-algebra, is there an upper bound on the dimension of all the irreducible representations?
It is known that all ...
- 7,057
7
votes
0
answers
269
views
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 ...
- 25.8k
0
votes
0
answers
255
views
Bounded operators with infinite matrix representations
I asked this question on StackExchange originally, but I'm giving it a go here as well.
Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and ...
- 123
1
vote
1
answer
259
views
Existence of homogeneous single chain compositions of a given maximal subfactor?
All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a single ...
1
vote
1
answer
232
views
Injective element of a commutative Banach algebra
A revision:
According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$.
Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal ...
- 356
0
votes
0
answers
134
views
semifinite projection
Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$.
( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
- 101
1
vote
2
answers
828
views
Positive operators - norm equality
I hope that somebody can help me with the following problem:
Let $A$ be a positive operator on $\mathbf{B}(\mathcal{H})$, ( $\mathcal{H}$ is a Hilbert space) with its spectral measure $E$. Show that ...
- 85
2
votes
1
answer
180
views
Are every finitely generated planar algebras, also singly generated?
Let $\mathcal{P}$ be a finitely generated planar algebra.
Question : Is it also singly generated ?
I ask this question, because, on one hand I've read on this paper of V. Jones and D. Bisch :
"...
3
votes
0
answers
291
views
Morita Equivalence of Full Corners in $C^*$-algebras
Suppose $\mathcal{A}$ is a $C^*$-algebra with a unique normalized trace and $p \in \mathcal{A}$ is a projection so that $\mathcal{B} = p\mathcal{A}p$ is a full corner.
Does $\mathcal{B}$ have a ...
- 1,010