All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
5
votes
1
answer
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definition of operator valued integral with spectral measure
I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011).
There, they work on a Hilbert space $H$ and on the ...
- 342
11
votes
2
answers
576
views
Seeing topological (geom.) properties of the space via corresponding C^*-algebra
Compact Hausdorff spaces bijectively correspond to C^*-algebras with identity. One needs to consider the algebra of continuous functions C(X) to go in one direction and spectrum to go in the other. (...
- 24.9k
5
votes
1
answer
284
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A perturbation question for the intersection of C*-subalgebras
This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C* algebras".
Let M be a unital C*-algebra and let ...
- 25.8k
3
votes
0
answers
148
views
Full free product of $B(\mathcal H_i)$
It struck me that I know nothing about the full (universal) free product of the $B(\mathcal H_i)$ amalgamated over $\mathbb C$ for Hilbert spaces $\mathcal H_i$ with identified unit vector $\xi_i$. So ...
- 3,984
8
votes
1
answer
222
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Hopf Galois extensions and conditional expectations for C* algebras
Suppose that $H$ is a Hopf algebra with normalised invariant integral (appropriate side) $\int:H\to \mathbb{C}$. The $H$ right comodule algebra $P$ is a Hopf Galois extension, so the canonical map $P\...
- 1,143
21
votes
1
answer
1k
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Banach spaces with few linear operators ?
Sometimes, dealing with the concrete and familiar Banach spaces of everyday life in maths, I happen nevertheless to ask myself about the generality of certain constructions. But, as I try to abstract ...
- 60.5k
1
vote
1
answer
187
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On the intersection of index 2 subfactors
Let $H_1$ and $H_2$ be two distinct index $2$ subgroups of a finite group $G$.
We can deduce several properties about the intersection $H_1 \cap H_2$:
$H_1$ and $H_2$ are normal subgroups of $G$. ...
10
votes
1
answer
533
views
Who first identified the universal $C^*$-algebra generated by an idempotent of norm at most $C$?
So much is known about hermitian and non-hermitian idempotents in a $C^*$-algebra, that someone must have written down the following.
Theorem The universal $C^*$-algebra generated by one element $x$...
- 1,749
3
votes
2
answers
576
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A version of the spectral theorem for group actions
Suppose $G$ is a sufficiently nice (maybe locally compact and abelian) group which acts on the separable Hilbert space $\mathcal{H}$ by unitary transformations. Is there a generalization of the ...
- 437
2
votes
1
answer
386
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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes [closed]
Question
I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
- 123
4
votes
2
answers
670
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Lower bounds for norms of commutators
For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound ...
- 777
4
votes
0
answers
132
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Faithfulness of the state associated with a quasiregular representation
Let $\Gamma$ be a discrete group and let $\Lambda$ be its subgroup. The function $\mathbf{1}_{\Lambda}$ is positive definite and therefore gives rise to a unitary representation $\pi: \Gamma \to \...
- 5,005
9
votes
1
answer
338
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Commuting nets for commuting projections
I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there ...
- 633
1
vote
1
answer
384
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A question on K- theory of non commutative $C^\star$ algebra
Edit: According to the comment of Andre Henriques I revise the question:
What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative ...
- 356
7
votes
3
answers
557
views
Relative Bicommutant
If $A \subseteq \mathcal B(\mathcal H)$ is an algebra of operators that is closed under adjoint, then its bicommutant $A''$ is a von Neumann algebra, and is the ultraweak closure of $A$; this is one ...
- 1,199
1
vote
0
answers
110
views
Are almost positive functionals close to positive functionals?
This is a bit of an open-ended question... Let $S$ be an operator algebra (or an operator system) and consider a functional $\nu:M\to \mathbb{C}$
that satisfies
$$\vert \nu(a)\vert \ge -\varepsilon \...
- 19
7
votes
2
answers
784
views
subfactor of finite rank but infinite index: is this possible?
A subfactor $N\subset M$ is essentially the same thing as an $N$-$M$-bimodule.
I'll recall the basic definitions in the language of bimodules, and I hope that subfactor people will excuse me.
...
- 43.2k
1
vote
1
answer
262
views
Almost complex structure and nontrivial idempotents
Is there a compact Reiemannian manifold $M$ for which the following complex $C^{*}$ algebra does not have a nontrivial idempotent:
$A=Hom(E,E)$ where $E$ is the complexification of $TM$.
Of ...
- 356
0
votes
1
answer
386
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The functor of continuous functions from compact CW-spaces to the reals
The contravariant functor $C(-)$ given by
$$
\hom_{Top}(-,\mathbb{R}):cCW\to Rng
$$
where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, ...
- 2,782
4
votes
0
answers
96
views
Approximate unit of specific form in a crossed product by $\Bbb {R} $ algebra
The following lemma appears in the paper "Rokhlin dimension for flows"- by Winter, Hirshberg, Szábo, Wu.
Lemma 6.5: Let $A $ be a $\sigma $-unital $C^*$-algebra with a flow $\alpha:\Bbb {R}\to Aut (A)...
- 41
13
votes
3
answers
686
views
Does every Frobenius algebra in a monoidal *-category give a Q-system?
Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the ...
- 28.1k
5
votes
1
answer
274
views
When are countably generated Hilbert modules generated by c.p.c. order zero maps?
Throughout let $B$ be a stable C*-algebra, i.e. $B\cong B\otimes K$, where $K$ is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. It is well-known that any ...
- 417
20
votes
0
answers
827
views
Can we define spectral triples using the language of rigged Hilbert spaces?
The traditional mathematical approach to quantum mechanics,
as developed by von Neumann, is based on Hilbert spaces and unbounded self-adjoint operators.
Another approach, which more closely resembles ...
- 37.8k
2
votes
1
answer
12k
views
Equality for matrix norm product of matrix * it's transpose, and square of the norm of the matrix [closed]
So, I know that $||AB|| \leq ||A||\cdot||B||$ (2-norm)
I'm doing a work on matrix algorithms and i seem to get as a result that $||A^TA|| = ||A||^2$
Does this always apply, or when and why does it ...
- 21
11
votes
1
answer
1k
views
Strong Atiyah conjecture
Who introduced the Strong Atiyah Conjecture?
Recall that the conjecture says the following. Let $G$ be a group, $A$ a $n\times n$-matrix over ${\mathbb Z}G$. We view $A$ as a bounded operator $l^2(...
user6976
4
votes
0
answers
290
views
C*-algebras and bounded relations
I'm trying to get used to the language of generators and relations for C*-algebras through Loring's "Lifting Solutions to Perturbing Problems in C*-Algebras". So far this is what I got from the first ...
- 417
4
votes
1
answer
386
views
Invertible unbounded linear maps defined on a Hilbert space
It is well-known that, assuming the axiom of choice, there are unbounded linear maps defined not only on a dense subset but on all of Hilbert space. Is it possible that such a map is invertible?
- 3,873
11
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0
answers
410
views
Sums of squares via semidefinite programming for the complex free group algebra
In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...
- 7,067
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). ...
user6976
5
votes
1
answer
443
views
When is the corner algebra $PM_n(A)P$ isomorphic to $A$?
In the algebra of matrices $M_n(A)$ over a $C^*$ algebra $A$, consider the corner algebra $PM_n(A)P$ for a Hermitian projection $P\in M_n(A)$. Is there any condition known for $P$ to make $PM_n(A)P$ ...
- 1,143
2
votes
0
answers
160
views
Hopf algebra translations of relations in operational calculus
Three particularly important reps of the exponential formula (cf. MO-Q) are the
refined Lah polynomials (OEIS A130561): Exp[o.g.f.] = Exp[formal power series]$\; =\exp[\frac{1}{(1-a.x)}]$, umbrally ...
- 10.5k
1
vote
0
answers
148
views
Fourier–Stieltjes as the dual space of the full group algebra
I know that this fact is classical, but I can't find the proof of it. How to proof that $B(G)=(C^*(G))^*$? As I understood, I can take a functional $F: \ell_1(G) \to \mathbb{C}$, and there is one-to-...
- 631
4
votes
1
answer
267
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 ...
- 25.8k
3
votes
0
answers
152
views
Whether a projection can "overlap" certain projections yet not commute with them
Question here about how the projections of a von Neumann algebra $\mathcal{R}$ might be arranged, relative to a projection that is not in $\mathcal{R}$.
Stipulate the following:
$H$ is ...
- 271
7
votes
0
answers
437
views
K theory as the fundamental group
There are several ways in which one can define $K$-theory for $C^*$-algebras: for $K_0(A)$ group two aproaches: algebraic (using idempotents) and topological (using projections, i.e. self-adjoint ...
- 9,330
4
votes
1
answer
161
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Commutator representation of certain smoothing operators
I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...
- 313
4
votes
1
answer
384
views
A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra
Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-...
- 1,396
3
votes
0
answers
129
views
Equivariant $K$-homology with $G$-compact support
Let $G$ be a discrete countable group and let $A$ be $\sigma$-unital $G$-$C^*$-Algebra. For a proper locally compact Hausdorff $G$-space $X$ the equivariant $K$-homology with $G$ compact support and ...
- 31
3
votes
0
answers
128
views
Stable homotopy equivalence
Let $\alpha:A \rightarrow B$ be a *-homomorphism of $C^*$-algebras. Then $\alpha$ ist a stable homotopy equivalence if there exists a $*$-homomorphism $\beta: B \otimes \mathcal{K} \rightarrow A \...
- 31
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
*-...
- 163
8
votes
1
answer
749
views
Is $SU(\infty)$ amenable?
We can write the finitary special unitary group $SU(\infty)$ as the direct limit
$\varinjlim SU(n)$ of ordinary special unitary groups. These groups $SU(n)$ are compact, thus amenable. In other ...
- 606
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
7
votes
1
answer
412
views
Morita equivalence for operator algebras and tensor products, question about proof
This is a bit of a dumb question I know, but I was reading "Morita equivalence for C*-algebras and W*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming ...
- 335
8
votes
1
answer
483
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Eigenvalues of the free sphere
Consider the usual sphere $S^{n-1}\subset\mathbb R^n$. By Stone-Weierstrass $C(S^{n-1})$ is generated by the standard coordinates $x_1,\ldots,x_n:\mathbb R^n\to\mathbb R$, and in fact we have the ...
- 1,363
3
votes
1
answer
689
views
K-homology of Cantor set and abelian AF-algebras
This may be a standard question answered in a book, or article. I don't know. I know that there exist related results with $\lim^1$-sequences (Rosenberg and Schochet).
What is
$KK(C_0(X),\mathbb{C})$...
- 58
7
votes
1
answer
731
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Formal series convergence in deformation quantization and $C^*$-condition
A link between formal series convergence in deformation quantization (strict deformation quantization) and producing $C^*$-algebras instead of mere $*$-algebras (which $(\mathcal{C}^{\infty}(M)[[t]],\...
- 585
6
votes
0
answers
441
views
Infinite number of non-isomorphic von Neumann algebras with property Gamma?
A II$_1$ factor $\mathcal M$ with trace $\tau$ has property Gamma if for every $\epsilon > 0$ and finite set $\{x_1,\cdots, x_n\} \subset \mathcal M$ there exists a trace 0 unitary element $y\in\...
- 3,984
6
votes
1
answer
449
views
Weakly amenability and exactness for discrete groups
A countable discrete group $\Gamma$ is said to be weakly amenable with Cowling-Haagerup constant 1 if there exists a sequence of finitely supported functions $(\phi_n)$ on $\Gamma$ such that $\phi_n\...
- 1,702
4
votes
0
answers
338
views
Quantization of $S^2$ as $C^*$-algebra?
The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...
- 24.9k
3
votes
1
answer
621
views
Injective von Neumann algebra
Let $G$ be a non-amenable countable discrete group. How can I show that the group von Neumann algebra $L(G)$ has no injective direct summand?
- 1,702