All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
4
votes
1
answer
823
views
Norms on Clifford algebra (C^* norm)
Basically I'm interested in operator algebras such as $C^*$ or von Neumann algebras. However I decided to learn a bit about noncommutative geometry (in particular spectral triples). Before doing this ...
- 9,330
6
votes
2
answers
1k
views
Are the banded versions of a positive definite matrix positive definite?
Consider $M$, a positive definite matrix. Let $M^{(1)}$ be the diagonal matrix which agrees with $M$ on the diagonal ($M_{ii}=M^{(1)}_{ii}$). We have that $M^{(1)}$ is positive definite because it is ...
- 375
3
votes
1
answer
701
views
Trace vs. state in von Neumann algebras
The question may be not appropriate with the title, since I do not know how to name it. I apologize.
Let $M$ be a finite $II_1$ factor, $\tau$ be the canonical trace. Let $p, q$ be two projections ...
- 31
3
votes
1
answer
1k
views
What is the significance of matrix ordered algebras?
I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
...
3
votes
1
answer
145
views
Reference for explicit quasicentral BAI in K(H) as ideal in B(H)?
As observed by Arveson and Akemann+Pedersen, if $J$ is an ideal in a ${\rm C}^\ast$-algebra $B$, then one can always find a contractive approximate identity for $J$, call it $(e_\lambda)_{\lambda\in\...
- 25.8k
4
votes
0
answers
118
views
Smoothness in von Neumann algebra of measurable functions
Let $A=L^{\infty}(M)$ be an algebra of essentially bounded measurable function on manifold $M$. Let $D$ be a first order elliptic differential operator acting on some hermitian bundle $S$ over $M$ (...
- 9,330
4
votes
1
answer
312
views
A question on $Z^{*}$ algebras
A $Z^{*}$ algebra is a $C^{*}$ algebra which satisfies each of the following equivalent conditions:
All elements of $A$ are left zero divisor.
All elements are right zero divisor.
All elements are ...
- 356
9
votes
4
answers
513
views
Does such a subgroup exist?
I am looking for a certain masa in a $II_1$ factor which is singular and has nontrivial Takesaki invariant.
For this I am looking for an example of an inclusion of groups $H\subset G$ such that:
$G$ ...
- 363
4
votes
2
answers
562
views
Are the groups $C( \mathbb{R} ; U(n) )$ isomorphic?
Let $C(\mathbb{R};{U}(n))$ denote the topological group of continuous functions $\mathbb{R}\to {U}(n)$ with pointwise multiplication and compact-open topology. My question is:
Are these groups ...
- 1,411
6
votes
1
answer
444
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 $f\in\ell^1(H)$ ...
- 5,425
18
votes
0
answers
881
views
What is operator tmf?
One of the many wonderful things about K-theory, relative to other generalized cohomology theories, is that it can be defined for not-necessarily-commutative C*-algebras. The resulting construction, ...
- 118k
3
votes
1
answer
481
views
Question on structure of von Neumann algebras, clarification in Conway's "A course in operator theory"
I was reading the section on the structure of type I von Neumann algebras in John B. Conway's "A course in operator theory" and a few questions about certain definitions and references arose, I was ...
- 417
3
votes
0
answers
237
views
Orthogonality relations for unitary representations of infinite (finitely generated) groups
Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...
- 333
4
votes
1
answer
252
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 ...
- 417
9
votes
1
answer
2k
views
Algorithm to find exponential map of differential operators acting on function
I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon \...
- 93
2
votes
1
answer
159
views
induced map on state spaces
A $*$-homomorphism $f:A\to B$ between C*-algebras is called non-degenerate if $f(A)B=B$.
I guess that I can prove that a non-degenerate *-homomorphism always induces a map on state spaces $f^\ast:S(B)...
- 23
6
votes
1
answer
644
views
Inductive limit of C*-algebras
It is known that an intertwining (or even approximately intertwining) diagram implies the isomorphism of the limit algebras. Under what conditions the converse holds?
- 169
2
votes
0
answers
210
views
The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras
Is there a name for the following property of a $C^{*}$ algebra $A$?
$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$
Example of this situation is $A=C(X)$ where $X$ is the ...
- 356
9
votes
0
answers
268
views
Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?
Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
- 24.9k
4
votes
1
answer
300
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 ...
- 356
1
vote
1
answer
88
views
Two points concerning Baer *-rings
Let $A$ be a unital Baer *-ring.
1- Assume that $\{p_i\}$ is a family of projections in $A$. Let $x$ be an isometry in A (I mean $x^*x=1$ where $1$ is the unit of $A$). True or false: $\inf (...
- 4,058
14
votes
2
answers
926
views
"Explicit" embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras
For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with pointwise multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform ...
- 25.8k
3
votes
2
answers
361
views
${\rm II}_1$-factors with finite commutant and trivial intersection generate $B(H)$?
Let $H$ be an $\infty$-dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Let $\mathcal{A}$, $\mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}'$, $...
1
vote
0
answers
182
views
Characterization of exact groups via the existence of amenable actions on unital C*-algebras
It is known that a discrete group $G$ is exact if and only if it admits an amenable action on a compact topological space. Would it also be true that $G$ is necessarily exact when it admits an ...
- 2,263
2
votes
0
answers
238
views
(Topological) K-theory for commutative $C^*$-algebras: operator and standard approaches
Let $A$ be a commutative unital $C^*$ algebra. Then $A=C(X)$ for some compact Hausdorff space $X$. Topological $K$-theory group (namely $K_0$) is defined in terms of vector bundles as a Grothendieck ...
- 9,330
6
votes
0
answers
210
views
Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
12
votes
3
answers
1k
views
What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 3,323
13
votes
1
answer
707
views
A left inverse for the comultiplication on a Hopf von Neumann algebra
Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments.
$\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following ...
- 25.8k
5
votes
1
answer
303
views
$Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$
Let $F_{2}$ be the free group with two generators.
Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way.
...
- 356
2
votes
0
answers
187
views
Sums of hermitian squares in free abelian group algebras and real positive semidefinite matrices
A little context for the following question, first. As Noah Stein notes in a comment below, the present question is closely related to the free semialgebraic geometry studied by Helton and his ...
- 7,067
2
votes
1
answer
292
views
positive maps and bimodules
Given a positive (or completely positive map) $\phi:A\to B$ between C* algebras, is there a way to construct an $A-B$ bimodule? This would more or less generalise the following construction: If $\phi$ ...
- 1,143
2
votes
1
answer
160
views
Approximation of the central support
Let $(M,\tau)$ be a tracial von Neumann algebra, i.e.
a unital subalgebra $M=M''\subset \mathbb{B}(H)$;
a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ implies ...
- 743
2
votes
0
answers
254
views
isomorphism of Chern character in kk-theory
Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...
- 493
6
votes
1
answer
244
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
$$\begin{matrix}
\Delta &\subset & M_n(\mathbb{C})\cr
\cup &\ &\cup\cr
\mathbb{C} &\subset &w\...
7
votes
2
answers
2k
views
Simple inequality in C*-algebras
Sorry the title is a bit vague. Let A be a C*-algebra, and let x and y be positive elements in A. Is it true that $$ \|x-y\|^2 \leq \|x^2-y^2\|? $$
Well, yes. But the proof I have is a bit of a ...
- 18.7k
4
votes
2
answers
542
views
The category of subfactors extending the category of groups?
This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...
12
votes
0
answers
386
views
Does every commutative $*$-algebra of operators on a prehilbert space have a character?
My question can be equivalently stated as follows:
Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of linear ...
- 541
6
votes
1
answer
229
views
A coalgebra structure on compact operators
Is there a coalgebra structure $\Delta_{n}$ on $M_{n}(\mathbb{C})$ which is compatible with the natural embedding $i_{n:}M_{n}(\mathbb{C})\to M_{n+1}(\mathbb{C})$ with $i_{n}(A)= A\oplus 0$. ...
- 356
6
votes
2
answers
366
views
Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one
A paper I'm currently reading uses the following fact. If $A$ is a unital $C^*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^*=F^2$) and
$$P=F+FT(1-F).\...
- 1,284
1
vote
0
answers
156
views
Property T Versus Property A
Kazhdan Property T for a locally compact group is defined in terms of the unitary dual. Namely, a group has property $T$ if the identity is an isolated point in the space of ...
- 2,630
2
votes
1
answer
233
views
${\rm II}_1$-factors with finite commutant: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \Rightarrow \mathcal{A}' \cap \mathcal{B}'$ hyperfinite?
Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}', \mathcal{B}' $ are also a ${\rm II}_1$-factors.
Question: $\mathcal{A} \cap \mathcal{B} = \mathbb{...
5
votes
2
answers
585
views
"Uncertainty principle" for self-adjoint operators in a finite von Neumann algebra
Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a ...
- 3,897
0
votes
1
answer
134
views
Bounded operators $T: B(H)\to H$ whose Kernel is a Lie algebra
Assume that $H$ is an infinite dimensional Hilbert space.The space of all bounded operators on $H$ is denoted by $B(H)$.We consider the Lie algebra structure $[T,S]=TS-ST$ on $B(H)$.
Is there a ...
- 356
3
votes
1
answer
161
views
Simple $Z^{*}$ algebra
What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?
- 356
3
votes
2
answers
553
views
Tensoring with a CAR-algebra
Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$?
The answer ...
- 169
1
vote
0
answers
442
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, ...
- 6,984
18
votes
0
answers
558
views
Do quotients of amenable groups C*-algebras satisfy the UCT?
Let G be a discrete amenable group.
General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal
coefficient theorem (UCT)?
I am mainly ...
- 2,689
2
votes
0
answers
111
views
proving that $\mathcal{A}_\infty(X)$ is or is not norm-closed in $\mathcal{L}(X)$ for each Banach space $X$
Fix any $1\leq p\leq\infty$. If $X$ is a Banach space and $C\in(0,\infty)$, we say that $T\in\mathcal{A}_C(X)$ whenever, for each $(x_n)_{n=1}^\infty\subset B_X$ (where $B_X$ is the closed unit ball ...
- 1,591
5
votes
0
answers
348
views
Discrete groups G whose full C*-algebra C*(G) is not MF?
This is a cheap rip-off of this question, but I am genuinely interested in an answer.
Is there a known example of a countable discrete group G whose full group C*-algebra C*(G) is not MF?
Let us ...
- 1,786
2
votes
1
answer
240
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):
Question: ...