All Questions
Tagged with operator-algebras or oa.operator-algebras
2,152 questions
11
votes
1
answer
310
views
Are free positive operators equivalent to almost-commuting operators?
Set $A:=C_0((0,1]) * C_0((0,1])$ (the free product C*-algebra), with canonical generators $a,b$ (positive contractions). Does there exists some $\gamma>0$ such that, for any $x,y \in A$ if $x^*x=a$ ...
- 1,786
11
votes
1
answer
678
views
For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?
I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...
- 25.8k
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
11
votes
1
answer
397
views
Is there a Dedekind-Frobenius group determinant for infinite groups?
If $G$ is a finite group and $\lbrace x_{g} \rbrace_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x_{gh^{-1}}$. The determinant of this matrix is a ...
- 7,047
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 ...
- 25.5k
11
votes
1
answer
8k
views
Double Orthogonal Complement
Let $V$ be a complex inner product space. If $W$ is a closed subspace of $V$, we may define $W^\perp$ to be the subspace of all vectors $v \in V$ such that $\langle v | w\rangle =0$ for all $w \in W$....
- 1,199
11
votes
2
answers
2k
views
How "generalized eigenvalues" combine into producing the spectral measure?
Hi... I am wondering how 'eigenvalues' that don't lie in my Hilbert space combine into producing the spectral measure. I study probability and I am quite ignorant in the field of spectral analysis of ...
- 333
11
votes
1
answer
381
views
Are groups with the Haagerup property hyperlinear?
In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been ...
- 1,027
11
votes
1
answer
2k
views
Motivation for $C^*$-algebras
I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
- 111
11
votes
1
answer
1k
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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 $\mathcal{B}$. ...
- 7,329
11
votes
0
answers
375
views
Why are projectionless $C^*$-algebras important (Kadison's conjecture)
It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
- 211
11
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0
answers
388
views
Von Neumann Inequality in Banach spaces
It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space:
Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
- 5,529
11
votes
0
answers
401
views
The term "absolute geometry"
My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
- 1,338
11
votes
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 ...
- 7,047
11
votes
0
answers
259
views
Status of the analog of the Haar measure on quantum groups
In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
- 561
11
votes
0
answers
494
views
Groups with reduced C*-algebras of stable rank 1
Let $G$ be a countable discrete group, $C_r^*(G)$ its reduced $C^*$-algebra. We say that $G$ has stable rank 1 if $C_r^*(G)$ has stable rank one, that is, the set of invertible elements is dense in $...
- 2,648
10
votes
2
answers
1k
views
B(H) as a direct sum of a closed two sided ideal and a subalgebra
Let $B(H)$ is the C*-algebra of all bounded linear operators on
Hilbert space $H$. Are there a closed two-sided ideal $I$ and a
subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus A$ (direct sum I and ...
- 109
10
votes
2
answers
803
views
General recipe for building C*-algebras out of combinatorial object
I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...
- 493
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
- 721
10
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2
answers
475
views
Is this a functor on the category of $C^{*}$ algebras?
The category of $C^{*}$ algebras is denoted by $\mathcal{A}$.
Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, ...
- 356
10
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5
answers
580
views
Observables and dimensional analysis
Here is a simple question about physical units that I hope has a simple satisfying answer. In mathematically sophisticated treatments of both quantum and classical physics one often speaks of an ...
- 101
10
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4
answers
903
views
Relationship between free probability and deterministic graphs?
Consider the $N\times N$ matrix $$
M = \left(\begin{array} \\
0 & 1 & & 0 \\
1 & \ddots & \ddots & \\
& \ddots & \ddots & 1 \\
0 & & 1 & 0 \\
\end{...
- 1,890
10
votes
3
answers
1k
views
Subfactor summer reading list
Many people I talk to lament the nonexistence of a coherent source for learning the theory of subfactors.
Could someone suggest a nice (ordered) list of books/papers to work through to obtain a ...
- 7,047
10
votes
2
answers
941
views
Induced representations of topological groups
Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups)
If $G$ is a group ...
- 18.7k
10
votes
3
answers
1k
views
subgroup of SU(N) with maximal manifold dimension
Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...
- 1,207
10
votes
3
answers
1k
views
Compact subgroups of the unitary group of operators in a hilbert space
Is there a characterization for the compact subgroups of the unitary operators in a Hilbert space, where the unitaries are furnished with the norm topology? What about other topologies?
- 101
10
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1
answer
811
views
What is the significance of the Jiang Su algebra in classification of C$^*$ -algebras?
Something I've been thinking about for a while that I'm not sure I understand is why $\mathcal{Z}$ stability, as opposed to say $\mathcal{O}_\infty$-stability or even $\mathcal{K}$-stability is so ...
- 183
10
votes
2
answers
926
views
Continuity of the product map
Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...
- 4,705
10
votes
1
answer
518
views
For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$?
Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$.
For what kind of $C^*$ algebras $A$ does the following hold:
$$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\...
- 356
10
votes
1
answer
459
views
K-theory of ultrapowers
It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^*$-algebras. For instance, if $A$ is a $C^*$-algebra does $K_0(A^U)$ depend on ...
- 103
10
votes
1
answer
786
views
Examples of groups without the n-positive approximation property
Let $G$ be a locally compact group and let $A(G)$ be the http://eom.springer.de/f/f120080.htm>Fourier algebra of $G$, which we view as the predual of the group von Neumann algebra $\mathcal M(G)$. ...
- 4,624
10
votes
3
answers
859
views
Takesaki theorem 2.6
I originally posted this question on MSE and didn't get a satisfactory answer, even after putting a bounty on it. Hence, I thought I should ask here:
Consider the following theorem in Takesaki's book &...
- 175
10
votes
2
answers
793
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 $\...
- 7,047
10
votes
1
answer
1k
views
Separating vectors for C$^*$-algebras
(I asked this on math.stackexchange, without response).
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the ...
- 18.7k
10
votes
1
answer
350
views
Tensor product of a von Neumann algebra and $L_\infty $
Let $R$ be the hyperfinite $II_1$-factor. We know that $R$ is isomorphic to $R\otimes R$. So, $L_\infty(0,1) \otimes R$ is a von Neumann subalgebra of $R$.
I am not sure whether it is sure for any ...
- 617
10
votes
1
answer
677
views
Correspondences as generalized morphism between $C^*$-algebras
While reading about Morita equivalence in the category of $C^*$-algebras I met also the following notion: a correspondence between two $C^*$-algebras $A,B$ is a pair $(X,\varphi)$ where $X$ is a (...
- 9,330
10
votes
1
answer
776
views
Saito-Wright definition of Rickart C*-algebras
A C*-algebra is Rickart if for each $x\in A$ there is a projection $p\in A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $S\subset A$ is defined
as $$R(S)=\{a\in A\mid xa=0\, \forall x\...
- 810
10
votes
1
answer
449
views
A small corner w.r.t. a masa in a von Neumann algebra
Let $A \cong L^\infty[0,1]$ be a non-atomic maximal abelian *-subalgebra
in $M \cong B(L^2[0,1])$ (or any von Neumann algebra $M$).
Is the following true?
For every $T \in M$ and $\epsilon>0$, ...
- 10.1k
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
10
votes
1
answer
283
views
Faithful extreme traces on group C*-algebras
Let $G$ be a discrete amenable, residually finite, ICC(i.e. each non-trivial conjugacy class is infinite) group. Let $C^*_r(G)$ be the reduced group $C^*$-algebra of $G$. Since $G$ is ICC the (...
- 2,689
10
votes
1
answer
372
views
Real rank zero of group $C^*$-algebras
The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...
- 399
10
votes
1
answer
542
views
Strengthening of Connes' embedding conjecture
If $A$ and $B$ are $C^*$ algebras I will write $A \overline{\otimes} B$ for the maximal tensor product and $A \underline{\otimes} B$ for the minimal tensor product. If $G$ is a countable discrete ...
- 1,769
10
votes
1
answer
783
views
When do tensor products of C*-algebras commute with colimits?
Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...
- 976
10
votes
1
answer
605
views
Given a C-star dynamical system and a subgroup of the acting group, is the corresponding map on crossed product algebras necessarily an injection
Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and $A\...
- 1,702
10
votes
2
answers
688
views
Stable rank one and corners of $C^\ast$-algebras
Thanks to a result of Herman and Vaserstein in [3], Rieffel's notion of stable rank [4] coincides with the Bass stable rank [1] for every $C^\ast$-algebra $A$: we denote it by $\mathrm{sr}(A)$ and we ...
- 660
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
10
votes
1
answer
428
views
Direct sums of operator spaces
I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a ...
- 18.7k
10
votes
1
answer
1k
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Can we characterize the spatial tensor product of von Neumann algebras categorically?
The tensor product of commutative algebras is exactly their coproduct
in the category of commutative algebras.
In other words, if A and B are two commutative algebras,
then the covariant functor that ...
- 37.8k
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
10
votes
1
answer
1k
views
When are certain group C*-algebras exact?
This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic.
Anyway. There has been a lot of attention given to showing that for certain ...
- 25.8k