All Questions
49 questions
4
votes
0
answers
135
views
Automorphism-invariant positive linear functionals on $C*$-algebras
Let $A$ be a $C^*$-algebra. Does there exist a non-trivial positive linear functional $\nu\in A^*$ which is $\mathrm{Aut}(A)$-invariant? That is, $\nu\circ\alpha=\nu$ for all $\alpha\in\mathrm{Aut}(A)$...
- 1,959
1
vote
0
answers
108
views
Infinite tensor product of Hilbert spaces [duplicate]
Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
- 11
6
votes
0
answers
98
views
Conditions for completely positive maps to act homomorphically across multiple subalgebras
For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
- 61
9
votes
0
answers
240
views
What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.
What is known ...
- 146
2
votes
1
answer
166
views
Cocompact lattices in $\mathrm{Sp}(n, 1)$
This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
- 131
3
votes
1
answer
185
views
Is the weighted shift strong frequently hypercyclic?
One sided Shift
Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum_{i=1}^\infty |x_i-y_i|/2^i$ . Define the shift map $\...
- 757
9
votes
2
answers
516
views
Why operator systems?
A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
- 1,037
0
votes
1
answer
163
views
Regarding socle of a C* algebra
I wanted to know if the socle of a complex C*-algebra is essential?
Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
- 149
1
vote
0
answers
119
views
Invariant on C*-algebras-number of closed unbounded derivation it admitted
In working of the unbounded derivation of C*-algebras. I observed the following: For topological manifold $M$, the number of closed, linear independent, unbounded derivation it admitted on $C(M)$ is ...
- 523
1
vote
0
answers
346
views
Duality of maps on bounded vs trace-class operators (Schrödinger-Heisenberg dual)
$\newcommand\calH{\mathcal H}
\newcommand\calK{\mathcal K}
\newcommand\tr{\operatorname{Tr}}$I am looking for a (citable) reference for the following fact:
Bounded linear maps $g:T(\calH)\to T(\calK)$...
- 896
1
vote
0
answers
132
views
Can we construct non-closable unbounded derivation in abelian C* algebras?
Can we construct an unbounded derivation on abelian C* algebra which is not closable?
One of possible construction may be found in the paper by Bratteli and Robinson(Unbounded derivations of C*-...
- 523
9
votes
1
answer
669
views
Reference for "Every compact quasinilpotent operator is the limit of nilpotent ones"
It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
- 191
3
votes
1
answer
246
views
Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras
In "Weierstrass-Stone, the Theorem" by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following:
Let $\mathcal{A}$ be a subalegebra of $C(X, \mathbb{R})$ and $...
- 119
0
votes
0
answers
132
views
Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]
Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)?
Wikipedia says this is true.
However, it seems to me that ...
- 253
0
votes
1
answer
190
views
Commutator of translation invariant operators on $L^2(\mathbb{R})$
I have a question concerning the commutator of translation invariant operators on $L^2(\mathbb{R})$.
Recall that $S:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is translation invariant if $Su_t=u_tS$ for all $...
6
votes
1
answer
252
views
Arens regularity of Banach algebras
I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $*$-algebras". There he have discussed the Arens regularity ...
- 243
5
votes
1
answer
766
views
When are homomorphisms between Banach algebras contractions?
When are homomorphisms between Banach algebras contractions?
I recall from my student days that there are results which show that a positive answer to the above question holds under very general ...
- 2,472
10
votes
1
answer
429
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
5
votes
1
answer
301
views
Unbounded Component of the Fredholm Domain
Let $X$ be a Banach space and $T \in \mathcal L(X)$.
The authors Engel and Nagel introduce in their book "One-Parameter Semigroups for Linear Evolution Equations" on p. 248 the concept of the ...
- 301
7
votes
1
answer
394
views
Inverse limit in the category of $C^{\ast}$-algebras or operator spaces
Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces?
I tried to search but could not find a proper reference. Any reference or comments about ...
- 1,115
2
votes
1
answer
189
views
Need a reference of a fact given in B. Blackadar's Operator Algebras
I am reading Blackadar's book on Operator algebras. In $\Pi 9.6.5$ Blackader says that
Maximal Tensor products commute with arbitrary limits.
In the same book the proof of this fact is not given....
- 1,115
7
votes
1
answer
220
views
Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber?
The question is a special case of a previous question.
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection ...
- 9,019
5
votes
2
answers
285
views
Is $C^{\infty}(M)$ a projective Frechet $C^{\infty}(N)$-module for a smooth map $M\to N$ between compact smooth manifolds?
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\...
- 9,019
4
votes
0
answers
120
views
Reductive Operator Problem
In the 1972 paper ''An equivalent Formulation of the Invariant Subspace Conjecture'' Dyer, Pedersen, and Porcelli announce the following result:
The Invariant Subspace Problem has a positive ...
- 355
9
votes
0
answers
230
views
Using Property (T) to approximate invertible matrices
In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:
Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
4
votes
1
answer
201
views
closure of a separating set of pure states
Let $A$ be a unital C*-algebra, and let $\mathcal R$ be a separating family of irreducible representations of $A$. Each vector state of a representation in $\mathcal R$ is a pure state, and the span ...
- 974
4
votes
0
answers
263
views
Approximately inner conditional expectations of $II_{1}$ factors
In many contexts it is helpful to think of conditional expectations as averages of unitary conjugates, a standpoint vindicated by many standard techniques in the theory of finite von Neumann algebras. ...
- 7,067
5
votes
1
answer
203
views
Multiplier norm vs cb norm
Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra:
$$m_f(\alpha)(g)=f(g)\alpha(g)$$
for every $\...
- 165
10
votes
0
answers
325
views
Are ideals in separable C*-algebras complemented subspaces?
Let $A$ be a separable C*-algebra and $J\subseteq A$ a closed two-sided ideal. Does this make $J$ into a complemented subspace of $A$? In other words, does the quotient map $A\to A/J$ have a ...
- 6,406
3
votes
1
answer
261
views
CBAP for the full group $C^*$-algebra
Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by ...
- 99
1
vote
1
answer
441
views
Extensions of completely positive maps
It is known that for a completely bounded map $\psi:A\to B(H)$ there exist completely positive maps $\phi_1,\phi_2:A\to B(H)$ such that
$$\Vert \phi_i\Vert_{cb}=\Vert \psi\Vert_{cb},$$
and the map $\...
- 99
0
votes
3
answers
291
views
Smallest norms on crossed product $C^*$-algebras
Let $A$ be a commutative $C^*$-algebra with a discrete group $G$ acting on it. The reduced crossed product is the completion of the algebraic crossed product $C_c(G,A)$ in the reduced norm $\Vert \...
5
votes
1
answer
242
views
Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group
Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I ...
- 743
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
3
votes
1
answer
6k
views
About eigen-functions of the Gaussian kernel
If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
- 2,246
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
6
votes
1
answer
577
views
Who gave the generalized Stone-Weierstrass Theorem?
Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a closed self-adjoint subalgebra of $C(X)$ which contains the constants. Then $\mathcal{A}$ is the collection of continuous functions on $X$ ...
user14319
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
2
votes
0
answers
412
views
Two Definitions of Non-commutative $L^p$ space
Throughout, let $(\mathcal{M},\tau)$ be a von Neumann algebra $\mathcal{M}$, acting on a Hilbert space $H$, with normal semifinite faithful trace $\tau$.
In the survey article by Pisier and Xu, the ...
- 75
8
votes
2
answers
929
views
Literature on "real" $C^*$-algebras
I am trying to get a better understanding of "real" $C^*$-algebras. I encountered them in the paper
D. Voiculescu, Dual algebraic structures, J. Operator Theory 17(1987), 85-98,
which cites
G.G. ...
- 1,482
1
vote
1
answer
2k
views
PhD in operator algebras and non-commutative geometry [closed]
I do not know whether it is a good place to ask this question or not.
I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...
- 11
0
votes
0
answers
218
views
Series of linear maps: on a paper by Evans and Hanche-Olsen
I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...
- 421
9
votes
3
answers
2k
views
Generalizations and relative applications of Fekete's subadditive lemma
Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
- 10.5k
33
votes
3
answers
3k
views
Reference request for translating from Top to C*-alg
Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality — namely, that the categories of ...
- 18.7k
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
20
votes
2
answers
1k
views
P-adic C* algebras
I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-...
- 373
2
votes
1
answer
199
views
Uniqueness of free complements
Let $A,B$ be subfactors of a II$_1$ factor $M$ with $A*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a free complement for $A$ ...
- 1,411
4
votes
1
answer
875
views
equality in noncommutative Hölder inequality
Let $1\leq p,q,r\leq \infty$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Let $S_p$ denote the Schatten space. For any $x\in S_p$ and any $y\in S_q$ we have
$$
||xy||_{S_r} \leq ||x||_{S_p}||y||_{...
- 1,222
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 ...
- 213