All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
3
votes
1
answer
378
views
Polar decomposition
Let $x$ be a trace class operator on a Hilbert space $H$. Then $x$ induces unique normal functional on $B(H)$, which we denote it by $f_x$.
Let us consider the polar decomposition $x=u|x|$ and $f_x=...
- 4,058
2
votes
1
answer
1k
views
Coaction of a group
Suppose $G$ is a finite group which acts on a $C^*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$.
I ...
- 743
16
votes
1
answer
918
views
Explicit path in the unitary group of a $C^*$-algebra
For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...
- 11.1k
2
votes
0
answers
450
views
Separable $\sigma$-unital sub-$C^*$-Algebras
Let $A$ be a $\mathbb{Z}_2$-graded $C^*$-Algebra. Then we can take the direct limit
$$
colim_{A_\sigma} KK_*(\mathbb{C}, A_\sigma)
$$
over all $\sigma$-unital graded $C^*$-sub-algebras $A_\sigma \...
- 21
5
votes
0
answers
139
views
Is the Euler characteristic of a subfactor planar algebra, nonzero?
Let $\mathcal{P}$ be an irreducible subfactor planar algebra and $\mu$ the Möbius function of its biprojection lattice $[e_1,id]$. Then the Euler characteristic of $\mathcal{P}$ is defined as follows: ...
1
vote
1
answer
1k
views
Why does an infinite tensor product depend on some vectors for operator algebras?
I have read that, in the definition of the infinite tensor product of operator algebras such as ${\rm C}^\ast$-algebras and ${\rm W}^\ast$-algebras, every factor in the product is associated with a ...
- 403
5
votes
2
answers
337
views
When is a groupoid the path groupoid of a graph?
I am actually interested in the $C^*$-algebras, so perhaps my question should be: How can you recognize whether a $C^*$-algebra $A$ is isomorphic to $C^*(\Lambda)$ for some (higher-rank) graph $\...
- 151
5
votes
1
answer
805
views
Surjective *-homs between multiplier algebras
Let A and B be C*-algebras, and let $\phi:A\rightarrow B$ be a surjective *-homomorphism. Then $\phi$ is non-degenerate, and so we can extend it to *-homomorphism between the multiplier algebras: $\...
- 18.7k
9
votes
1
answer
675
views
Does this "jumping-ahead" ordinal function exist?
While working on a project in operator algebras with a collaborator (and fellow MO user), we are able to successfully complete a transfinite induction assuming that the following has an affirmative ...
- 5,407
1
vote
0
answers
112
views
shifts in Baer*-rings
Let $A$ be a Baer*-ring with the unit $1$. An important point that holds in every Baer*-ring is this: for every element $y\in A$ there is an smallest projection $e_y\in A$, called the left projection ...
- 4,058
8
votes
1
answer
366
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 $ * $-...
- 1,396
8
votes
1
answer
459
views
Free C^*-algebra
Let $A_0$ be a set of all polynomials with complex coefficients of infinitely many noncommuting (free) variables, denoted by $X_1,X_2,...,X_1^*,X_2^*,...$. We equip $A_0$ with the operation $*:A_0 \to ...
- 9,330
1
vote
0
answers
74
views
If $f$ takes values in $L(H,L(H,\Bbb R))$ and $μ$ is a $H\hat ⊗_πH$-valued measure, how are $\int f\:dμ$ and $\int f⊗_π\text{id}_Hdμ$ related?
Let
$H$ be a separable $\mathbb R$-Hilbert space
$H\:\hat\otimes_\pi\:H$ denote the projective tensor product of $H$ and $H$
$(\Omega,\mathcal A)$ be a measurable space
$\mu$ be a $H\:\hat\otimes_\pi\...
- 167
0
votes
0
answers
72
views
weakly amenable weighted sequence algebras
Let $v=(v_n)_{n\in\mathbb{N}}$ be a positive weight with $\inf_nv_n>0$ (for convenience we may take $v_n\geqslant1$). Then $\ell_{\infty}(v)$ is a Banach algebra with coordinate-wise multiplication....
- 375
1
vote
0
answers
76
views
specific sequence of matrices making a strange ratio of matrix norms diverging
For any $t>0$ define $d_t:=\operatorname{diag}j^t=\operatorname{diag}(1^t,2^t,\ldots)$. Now pick up such a $t>0$ and an arbitrary $\theta\in\big(0,\frac12\big)$. For every $k\in\mathbb{N}$ find ...
- 375
2
votes
2
answers
1k
views
Continuous linear functionals in strong operator and $\sigma$-strong topologies
It was mentioned in the comments to https://math.stackexchange.com/questions/517369/comparison-of-strong-operator-and-weak-topologies-on-bh that continuous linear functionals on $\mathfrak{B}(\mathbb{...
- 1,731
12
votes
1
answer
1k
views
Decomposition of positive definite matrices.
It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum_{j} B_j \otimes C_j
$$
with $B_j$ and $C_j$ positive semidefinite matrices (of ...
4
votes
0
answers
398
views
Bott-type projections in $C^*$-algebras
Let $A$ be a unital $C^*$-algebra and $a\in A$. If $aa^*+1$ is invertible in $A$ then the element
$$\beta(a)=(aa^*+I)^{-1}\left(\begin{array}{cc}aa^* & a \\a^* & I\end{array}\right)$$
is an ...
- 41
3
votes
0
answers
306
views
The Baum Connes Conjecture - the approach using spectra
In this article James Davis and Wolfgang Lück introduce a approach using spectra to formulate the Baum Connes Conjecture for a discrete group $G$. In order to do so, they construct a functor
$$\...
- 31
4
votes
1
answer
643
views
Partial order - Unbounded normal operators affiliated with von Neumann algebra.
Hello, I have a question which is related to a partial order in a set of self-adjoint operators.
Let $\mathcal{M}$ be a semifinite von Neumann algebra with a faithful semi-finite normal trace $\tau$. ...
- 85
10
votes
2
answers
794
views
What is the physical difference between states and unital completely positive maps?
Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into $\...
- 7,067
4
votes
0
answers
134
views
Idea of Dirac operator on quantum groups
This is a somewhat unexact question. I would like to now more on the principle of the Dirac operator, especially for quantum groups.
I have learned in some articles about the Dirac operator on the ...
- 685
8
votes
1
answer
340
views
characterization of commutative Banach algebras
Let $A$ be a Banach algebra with the following property:
For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges.
...
- 356
11
votes
2
answers
545
views
Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?
Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by $(zf)(g):...
- 1,550
3
votes
2
answers
153
views
$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?
I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 :
Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
Neumann ...
- 659
4
votes
0
answers
119
views
Index of a subfactor of a full $II_1$ factor
On pg. 151 of "Coxeter Graphs and Towers of Algebras" by F.M. Goodman, P. de la Harpe, and V.F.R. Jones (1989), it is stated that there is no known example of a full $II_1$ factor having a subfactor ...
- 229
3
votes
1
answer
230
views
An inequality for Fuchsian groups?
Let $G$ be a finitely generated Fuchsian group.
(i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$).
Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ?
Here, $\beta_{2}^1(G)$ stands for the ...
- 11.3k
5
votes
0
answers
272
views
Maximality of the maximal tensor product of C*-algebras
Given two C*-algebras $A$ and $B$, the maximal tensor product $A\otimes_{max}B$ is bigger than the minimal tensor product $A\otimes_{min}B$ in the sense that there exists an epimorphism $$A\otimes_{...
- 2,263
3
votes
1
answer
1k
views
injective tensor norms for real tensors
If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be
$$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1}
|\...
- 316
8
votes
1
answer
499
views
Multiplicative domains and conditional expectations
Let $M$ be an injective von Neumann subalgebra of $B(H)$. For a completely positive map $\phi:B(H)\to B(H)$, let $Mult(\phi)$ denote be the multiplicative domain of $\phi$. For any conditional ...
- 83
14
votes
1
answer
1k
views
Connes's unpublished manuscript on correspondences, anyone?
There exist unpublished notes on correspondences of von Neumann algebras due to Connes. This is often cited, but I've never seen a copy. It would be nice to have this, say, to maybe look further into ...
- 7,067
0
votes
1
answer
204
views
A $C^{*}$ algebra associated to a group [closed]
Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...
- 356
6
votes
1
answer
1k
views
Noncommutative geometry and category theory
The point in which one starts to talk about noncommutative geometry is the Gelfand Najmark theorem. It establishes an equivalence of the catgeories of commutative (non)unital $C^*$-algebras and the ...
- 9,330
4
votes
1
answer
414
views
A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $
Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...
- 1,396
8
votes
1
answer
635
views
Is the Jordan decomposition of a self-adjoint functional constructive?
Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions $\...
- 974
10
votes
1
answer
828
views
Tannaka duality for C*-algebras?
Tannaka-Krein
duality shows
how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional
complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to
\mathbf{Vect}_{\...
- 6,406
9
votes
3
answers
2k
views
Conjugacy classes and reduced group $C^*$-algebra of an amenable group
The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other ...
- 25.5k
11
votes
2
answers
2k
views
Completely positive maps as "positive operators"
Let $A$ be a unital $C^{*}$-algebra and $\phi:A \rightarrow A$ be a completely positive map, i.e. $\phi^{(n)}:M_{n}(A) \rightarrow M_{n}(A)$ preserves positivity for any natural number $n$, where $\...
- 7,067
4
votes
0
answers
95
views
K-theory of a discrete groupoid crossed product
Does there exist a method to compute the K-theory
$$K(A \rtimes G)$$
for a discrete, countable groupoid $G$ and $G$-$C^*$-algebra $A$? In good cases, say $G$ is ameanable.
Say, via Baum--Connes and a ...
- 685
3
votes
0
answers
141
views
Existence of a unique cyclic and separating vector in a *-representation
I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\...
- 31
4
votes
0
answers
205
views
Minkowski determinant inequality for the Fuglede-Kadison determinant
For positive-semidefinite matrices $A, B$ in $M_n(\mathbb{C})$, the Minkowski determinant theorem tells us that $\det(A+B)^{\frac{1}{n}} \ge \det(A)^{\frac{1}{n}} + \det(B)^{\frac{1}{n}}$. For a ...
- 41
0
votes
1
answer
152
views
Transitivity of the Cuntz sub-equivalence
Let $A$ be a $C^*$-algebra and $a,b \in A$ positive elements. We define a relation (Cuntz sub-equivalence) by saying
$$a\lesssim b: \Leftrightarrow \exists\, (r_n)_{n\in\mathbb{N}}\subset{A}\text{ ...
- 1
5
votes
1
answer
333
views
One-parameter semigroups of bimodules
Suppose M is a von Neumann algebra.
Consider a monoidal category of bimodules over M.
Here a bimodule is a Hilbert space with two normal representations of M.
The monoidal structure is given by Connes'...
- 37.8k
5
votes
1
answer
137
views
Operator space structures on CB(H,K) where H and K are Hilbertian operator spaces?
(I'd be grateful if anyone thinking of putting MathJax in the question title refrains from doing so.)
By consulting various standard sources (Effros-Ruan's book, Pisier's book, the lexicon of ...
- 25.8k
3
votes
1
answer
125
views
Classification of $2k$-vectors modulo orthogonal transformations
Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where
$$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}...
- 605
15
votes
1
answer
547
views
Denseness of inner automorphisms inside automorphisms of hyperfinite type III_1 factor
Let $R$ be the hyperfinite type $III_1$ factor,
and let $Aut(R)$ be its group of automorphisms, equipped with the $u$-topology
(topology of pointwise convergence on the predual).
An automorphism $\...
- 43.2k
5
votes
0
answers
81
views
C*-algebra of a singular surface foliation
Noncommutative geometry associates a $C^*$-algebra $C^*(S,{\cal F})$ with a foliation $\cal F$ on a manifold $S.$
Did somebody study this construction for noncompact surfaces $S$?
What I am really ...
- 2,390
5
votes
0
answers
232
views
The Segal Machine constructing spectra and topological $K$-Theory
I am currently looking into the Segal-Machine for constructing spectra. I am working with his original article . The first thing that confuses me is the spectrum that arises from a $\Gamma$-space is ...
- 51
4
votes
0
answers
185
views
ring structure of $KK_*(A,A)$ for a separable $C^*$-algebra $A$
Motivation:
For a topological space $X$ one can consider under certain circumstances the cohomology ring of suitable cohomology theories, for example:
1) The cohomology ring $H^*(X;R)=\oplus_{i\ge ...
- 1,188
6
votes
1
answer
257
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 $l^2(\...