All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
2
votes
2
answers
1k
views
Why is this a conditional expectation into the fixed point algebra?
Let $A$ be a C*-algebra and let $\alpha$ be an action of the circle group $S_1$ on $A$ (Gauge action).
We define the following map:
$$E:A\rightarrow A;\quad E(a):=\int\alpha_t(a)\textrm{d} t.$$
My ...
- 21
3
votes
1
answer
1k
views
Self-adjoint bounded operator, resolution of the identity, def. of the diagonal
Let $A$ be a self adjoint bounded linear operator with a continuous spectrum
$\sigma(A)=[a,b]$ which acts on a separable Hilbert space. Let
$E_\lambda$ be its resolution of the identity.
For ...
3
votes
1
answer
785
views
Maximal ideals of some algebras
This question comes up after I read the chapter 7 : Banach algebras and spectrum theory from Conway's book. As we have known that if $X$ is compact space, then all the maximal ideals of $C(X)=\{ f : X\...
- 8,559
0
votes
0
answers
167
views
Primitive Ideals of $\mathcal{B}_0(\mathcal{H})^\sim$ (identity adjoined)
A note I'm studying says its primitive ideals space is $\lbrace \lbrace 0 \rbrace, \mathcal{B}_0(\mathcal{H}) \rbrace $. I think it might be just $\lbrace \lbrace 0 \rbrace \rbrace $ so I'm somewhat ...
- 1
8
votes
1
answer
813
views
Tomita-Takesaki theory for a simple class of crossed products
This question is inspired by the construction of the time evolution for endomotives as given by Connes and Marcolli in their book http://www.alainconnes.org/docs/bookwebfinal.pdf.
Let $M$ be a monoid ...
CommunityBot
- 1
10
votes
0
answers
325
views
H-space structure on the Calkin algebra
By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
- 7,637
3
votes
1
answer
338
views
Ultraproduct of n-dimensional Banach spaces and algebras
Hi, I am interested in the following question:
Fix $n$.
Let $M_n$ be matrix algebra over the field of complex numbers with normalized trace $tr_n$. Let $M_n^{\omega}$ be an ultrapover of $M_n$, ...
- 2,943
8
votes
1
answer
684
views
Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras
Let $u:A \to \prod_{\mathcal U} M_n$ be a unital completely positive map (ucp) from a unital separable $C^*$algebra into the von Neumann algebra ultraprodut $\prod_{\mathcal U} M_n$.
Here $\mathcal ...
CommunityBot
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1
vote
2
answers
177
views
Restriction on the coefficients for an operator in the free group factor $ L(\mathbb{F}_2) $
Let $\mathbb{F}_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family
$\{L_{x_g} : g \in G\}$, here, with $g \in G$...
- 5,425
1
vote
2
answers
335
views
Bounded operators on direct limit of direct sums of spaces of cusp forms
Consider $S_{2k} (\Gamma_0 (N))$ and let $S(N)$ denote the direct limit of the finite direct sums of the $S_{2k}$. Since each $S_{2k} (\Gamma_0 (N))$ is also a Hilbert space w.r.t. the Petersson inner ...
- 23k
0
votes
2
answers
492
views
trace measurable operators
Hello,
I have a question about trace measurable operators and I think it's not a hard one. However, I'm quite confused because I cannot prove it.
Let $\mathcal{M}$ be a semi-finite von Neumann ...
- 18.7k
16
votes
2
answers
1k
views
Discrete groups G whose full C*-algebra C*(G) is not quasidiagonal?
Is there a known example of a countable discrete group G whose full group C*-algebra C*(G) is not quasidiagonal?
Let us recall that a separable C*-algebra A is quasidiagonal if it admits a faithful
*-...
- 2,066
3
votes
1
answer
504
views
In quantum dynamical simulations, what is the symmetric (Riemannian) analog of a Poisson bracket?
The question narrowly posed is:
What is the accepted name of the bracket operation that is obtained by replacing the (antisymmetric) symplectic structure of the Poisson bracket with a (symmetric) ...
- 54.6k
7
votes
1
answer
811
views
Actions orbit equivalent to profinite ones
Let $G$ be a countable discrete residually finite group.
Is there a way to characterise the actions of $G$ that are orbit-equivalent to profinite ones?
Ozawa and Popa introduced the concept of ...
- 4,624
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......
- 11.1k
2
votes
1
answer
391
views
When C(K) is closed in sigma strong topology?
Fix a compact Hausdorff space $K$ and think about $C(K)$ as a C*-algebra acting on a Hilbert space $H$. Suppose that $C(K)$ is closed in $\mathcal{B}(H)$ in:
$\sigma$-strong
$\sigma$-strong*
...
- 43.2k
5
votes
0
answers
352
views
"topological" conjugacy of group automorphisms
In the paper "Orbit Equivalence and Topological Conjugacy of Affine
Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:
Theorem. Given two actions $\alpha$ and $\...
- 3,897
1
vote
0
answers
212
views
Grading on Multiplier Algebras
A graded C*-algebra A is inner-graded if there exists a self-adjoint unitary $\varepsilon$ in the multiplier algebra M(A) of A which implements the grading automorphism $\alpha$ on A: $\alpha(a)=\...
- 18.7k
4
votes
1
answer
246
views
Are all continuous linear operators on the space of entire functions "simple"?
Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions.
For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||_n : \operatorname{Ent} \to \mathbb{R}$ ...
- 56.9k
9
votes
2
answers
928
views
Property (T) for pseudogroups
Let $H$ be a Hilbert space, $S(H)$ be the inverse semigroup (pseudogroup) of linear maps between (closed) subspaces of $H$ preserving the dot product (the operation is composition of partial maps). ...
CommunityBot
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14
votes
3
answers
3k
views
The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$
Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
- 11.1k
5
votes
1
answer
528
views
Completely bounded maps on Mn
The aim of this question is to collect nice maps on $M_n(\mathbb{C})$ with the following property:
$\phi_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ with $||\phi||=1$ and $||\phi_n||_{cb}\...
CommunityBot
- 1
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 $...
- 11.1k
4
votes
2
answers
660
views
Spatial isomorphisms of tensor product of factors
Suppose $N \subset M$ are two factors, neither of them Type I, acting on a separable Hilbert space $H$. Let $\pi_1$ be a faithful normal representation of $N$ and $\pi_2$ a faithful normal ...
- 251
4
votes
1
answer
270
views
A detail in the construction of the coarse index of a Dirac operator in "Roe: An Index Theorem on Open Manifold, I"
Hi,
I'm currently wreading "Roe: An Index Theorem on Open Manifolds, I, J. Differential Geometry 27 (1988), p. 87-113" and there is a detail in the construction of the coarse index of a Dirac ...
- 76
26
votes
2
answers
4k
views
Finite subgroups of unitary groups
Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite ...
- 44.4k
5
votes
2
answers
491
views
Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
2
votes
1
answer
259
views
Nuclearity of certain semigroup crossed product C*-algebras
This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
CommunityBot
- 1
2
votes
1
answer
554
views
About properties of groupoid C*-algebras
I'm interested in the following kind of questions about groupoid $C^*$-algebras.
1) If $G_1 \times_{H} \ G_2$ is a fibre product of (nice) groupoids do we have something like $$C^\star(G_1 \times_{H} ...
- 7,057
8
votes
1
answer
2k
views
Limits of Nilpotent and Quasi-nilpotent Operators in a $\mathrm{II}_1$-factor
A bounded operator $A$ in a Hilbert space is called nilpotent if there exists $n$ such that $A^{n}=0$. An operator is called quasi-nilpotent iff
$$
\limsup_{n\to\infty}{ \|A^{n}\|^{1/n}}=0.
$$
Every ...
- 3,626
2
votes
3
answers
3k
views
Is there any conclusions generalized Singular Value Decomposition into Hilbert Space
Spectrum decomposition can be regarded as the generalizations of the following fact that:
Every Hermitian matrix $A$ can be decomposed into $A=U^{*}\Lambda U$,where $U$ is a unitary matrix
Singular ...
- 1,706
6
votes
2
answers
901
views
Universal characterization and explicit description of elements of the group von Neumann algebra and the crossed product
Group von Neumann algebras and crossed products for a locally compact group G
can be constructed in many different ways.
For example, one can take the von Neumann algebra generated
by certain ...
5
votes
1
answer
721
views
Subspaces of a Subfactor
Is the following true?
Let $\mathcal N \subset \mathcal M$ be a subfactor. There is a bijective correspondence between the ultraweakly closed subspaces of $\mathcal M$ that are bimodules over $\...
- 7,057
2
votes
0
answers
249
views
Strict complete contractions
Let $A$ be a $C^*$-algebra. The Stinespring construction shows that a completely positive contraction $T:A\rightarrow B(H)$ has the form $T(x) = U^* \pi(x) U$ where $U:H\rightarrow K$ is a ...
- 18.7k
11
votes
1
answer
1k
views
Does equality of the operator norm and the cb norm for every bimodule map over a C*-subalgebra imply that the subalgebra is matricially norming?
In this post, without further mention all C*-algebras are assumed to have an identity element and subalgebras inherit the identity.
Question: Let $\mathcal{C}$ be a C*-subalgebra of $\mathcal{B}$. ...
CommunityBot
- 1
4
votes
0
answers
417
views
Connes fusion and the composition of completely positive maps
Let $N$ be a type $II_{1}$-factor with trace $\tau$.
An $N-N$ correspondence is a Hilbert $N$-bimodule $H$ where the left and right actions are both ultraweakly continuous. Equivalently, a ...
- 7,057
12
votes
1
answer
329
views
Ideals in smooth subalgebras of C*-algebras
Let $B$ be a $C^{*}$-algebra and $\mathcal{B}$ a dense *-subalgebra stable under holomorphic functional calculus and $C^{1}$-functional calculus for selfadjoint elements. Also, $\mathcal{B}$ is a ...
- 25.8k
5
votes
1
answer
410
views
Is the unitary group of $l^2(A)$ with the strict topology contractible?
Let $A$ be a $C^*$-algebra with countable approximate unit. Let $\mathbb{K}$ denote the compact operators on a separable Hilbert space. Mingo and later Cuntz and Higson have shown that the unitary ...
- 2,203
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? ...
- 12.8k
8
votes
0
answers
339
views
Canonical Time Evolution for Type $II_{1}$-Factors?
This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor (...
CommunityBot
- 1
11
votes
2
answers
2k
views
Completely positive maps as "positive operators"
Let $A$ be a unital $C^{*}$-algebra and $\phi:A \rightarrow A$ be a completely positive map, i.e. $\phi^{(n)}:M_{n}(A) \rightarrow M_{n}(A)$ preserves positivity for any natural number $n$, where $\...
- 7,057
3
votes
2
answers
564
views
Positive extension of functionals on a subset of the state space of a $C^*$ algebra
Let $A$ be a finite dimensional $C^*$ algebra and $S(A)$ the state space. Let $K\subset A$ be an intersection of $S(A)$ with a vector subspace $J\subset A$ and let $f$ be a positive affine functional ...
- 41
1
vote
1
answer
287
views
How coarse is the coarse correspondence?
Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) \overline{\...
- 7,057
-1
votes
1
answer
382
views
derivation between two $C^{*}$ algebras
given two $C^{*}$ algebras $A\subset B $, acting on the same Hilbert space $H$, and $\delta $ is a derivation from $A$ into $B$,(in this case, a derivation is a linear mapping such that $\delta(ab)=a\...
- 1,528
13
votes
1
answer
404
views
Self map of unitary group
Let $H$ be a Hilbert space and let $u_1 \in U(H)$ be a unitary operator on $H$. Consider the self-map $w: U(H) \to U(H)$ which is given by
$$w(v) := v^2 u_1 v^{-1}.$$
Since $U(H)$ is connected, there ...
- 61.9k
18
votes
1
answer
1k
views
Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
15
votes
1
answer
686
views
Amenability of groups in terms of a perturbation condition
Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$
$$\inf \lbrace\...
- 25.5k
4
votes
0
answers
282
views
If the flip automorphism of a finite factor can be connected to the identity is it approximately inner?
This intriguing question is due to Sorin Popa, so I'm marking it community wiki. I'd like to know the status of the question, and if it is still open, perhaps gather some new ideas for attacking it.
...
- 7,057
3
votes
0
answers
178
views
One-parameter groups acting on dual Banach spaces
Let $E$ be a Banach space, and $M=E^*$ (my application has $M$ a von Neumann algebra, but this is unimportant). Let $(\sigma_t)$ be a SOT cts one-parameter group on $E$: so for $t\in\mathbb R$, we ...
- 18.7k
11
votes
1
answer
514
views
Transfinite induction, a theorem of Pedersen, and chains of subalgebras of $B(H)$
This post is closely related to this one. (In fact I copied some of its content.)
Let $H$ be an infinite dimensional separable complex Hilbert space. All $C^{\star}$-subalgebras of $B(H)$ are assumed ...
CommunityBot
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