All Questions
Tagged with oa.operator-algebras fa.functional-analysis
778 questions
0
votes
1
answer
365
views
When $\lambda$-commutativity implies commutativity?
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert $F$.
Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to $\lambda$-commute if there ...
- 724
1
vote
1
answer
98
views
If $\text{Im}(S^*M)\subseteq \text{Im}(M)$, is $\text{Im}(SM)\subseteq \text{Im}(M)$?
Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ the algebra of all bounded linear operators defined on $F$.
Assume that
$M\in \mathcal{B}(F)^+$ (i.e. $\langle Mx\;, \;x\rangle\geq 0$ for all ...
- 724
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
3
votes
1
answer
187
views
Algebraic tensor product of C*-algebras extends via ideals? Application to restriction theorem?
Is the following assertion and the proof below correct,
or am I missing something very important?
Moreover, would the corollaries be correct then?
Besides, I would also appreciate a lot any comment, ...
- 598
0
votes
1
answer
328
views
Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
- 407
2
votes
0
answers
164
views
An operator valued Egoroff's theorem
The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
- 4,058
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 ...
7
votes
2
answers
485
views
The von Neumann algebra generated by a non-closable operator
Let $H$ be a separable Hilbert space and let $M$ be a densely defined operator $\mathcal{D}(M) \subset H \to H$. It is closable iff its adjoint $M^{\star}$ is densely defined, and then its closure $\...
8
votes
1
answer
302
views
Does every integer map generate a von Neumann algebra of type I?
Consider a map $m: \mathbb{N} \to \mathbb{N}$ (we call it an integer map). Let $E_r$ be the set $m^{-1}(\{r\})$.
Let $H$ be the Hilbert space $\ell^2(\mathbb{N})$ and consider the densely defined ...
2
votes
1
answer
341
views
Closed two-sided ideals in $C(X,M_n)$
As is known (see Kadison-Ringrose, 3.4.1) each closed ideal $I$ in the $C^*$-algebra $C(X)$ of continuous functions on a compact space $X$ has the form
$$
I=\{f\in C(X): \ \forall x\in S\quad f(x)=0 \}...
- 7,426
10
votes
0
answers
201
views
Masas in SAW*-algebras
I asked this question three years ago at MSe but it has no response; let me try here.
Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, ...
- 11.3k
1
vote
0
answers
179
views
Positive square roots of inverse operators on different Sobolev spaces
Let $D$ be a self-adjoint (in the $H^0$-inner product) first-order differential operator on a manifold $M$, where $H^i$ stands for the $i$-th Sobolev space on $M$. Then $D$ extends to a bounded ...
- 1,903
1
vote
0
answers
233
views
Bochner integrals with values in a Hilbert $A$-module
I'm wondering whether there exists a generalisation of Bochner integration with values in a Hilbert $A$-module $M$, where $A$ is a general $C^*$-algebra rather than $\mathbb{C}$ (and whether there are ...
- 1,903
2
votes
1
answer
172
views
Why $\mathcal{B}_1(F)$ is not a subalgebra of $\mathcal{B}(F)$?
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $M\in \mathcal{B}(F)^+$ (i.e. $M^*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$.
I ...
- 724
3
votes
1
answer
158
views
Showing the following inclusion between two subalgebras of $\mathcal{B}(F)$
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $M\in \mathcal{B}(F)^+$ (i.e. $M^*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$).
I ...
- 724
2
votes
1
answer
941
views
Original statement of Naimark's dilation theorem
Naimark's dilation theorem in papers and textbooks is usually stated as:
Let $E$ be a regular, positive, $B(\mathcal H)$-valued measure on $X$. Then there exists a Hilbert space $\mathcal K$, a ...
- 3,984
2
votes
1
answer
238
views
Why is index unchanged after applying functional calculus?
Suppose $D$ is the Dirac operator on a closed spin manifold $M$, with spinors $S$. One can take the functional calculus of $D$ with respect to the continuous function $f:\mathbb{R}\rightarrow\mathbb{R}...
- 1,903
2
votes
1
answer
411
views
Problem of convergence of the following sequence
Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$. Let $T\in \mathcal{L}(E)$
be bounded linear operators from $E$ to $E$ and $M\in \...
- 724
2
votes
0
answers
205
views
relative amenability of von Neumann algebra
Let $\cal{M}$ be a finite von Neumann algebra and $\cal{N}$ be a von Neumann subalgebra of $\cal{M}$.
The von Neumann algebra $\cal{M}$ is is amenable relative to
$\cal{N}$ if there exists a norm ...
- 565
7
votes
0
answers
222
views
Can C*/W*-algebras be realized as (involutive?) monoid/co-monoid objects?
I would like to know how close one can get to realizing the category of C*-algebras as a category of monoid objects. Related (almost, but not quite, duplicate) questions are:
"Recovering a monoidal ...
- 146
1
vote
1
answer
160
views
When will the $G$-invariant measure space be isomophic to the tracial state space of the crossed product $C^\ast$-algebra
Suppose a countable discrete amenable group $G$ acts continuously on a infinite Compact Hausdorff space $X$, i.e. $\alpha:G\curvearrowright X$. Suppose $\alpha$ is minimal. Write $M_G(X)$ for all $G$-...
- 181
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
1
answer
157
views
Norm of "tensoring" with the identity
Consider a Banach space $E$ and a discrete set $X$. For an operator $T$ on $\ell^2(X)$ I can consider and induced operator $T'$ on the Bochner-Lebesgue space $\ell^2(X;E)$ of $E$-valued square-...
- 165
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. ...
- 7,057
1
vote
4
answers
367
views
Classification of $C^*$ algebras whose all non scalar elements have disconnected spectrum
To what extent have all unital $C^*$ algebras $A$ with the following property been classified? Is there a simple $C^*$ algebra with this property? Does $C(K)$ satisfy this property, where $K$ is an ...
- 356
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)$$
for every $\...
- 165
2
votes
2
answers
260
views
Bounded operators leaving dense subspace invariant
Let $A$ be a C$^*$-algebra. A pre-Hilbert $A$-module $H$ is a right $A$ module with a $A$-valued inner product (which is linear in the second variable and conjugate linear in the first variable) such ...
- 481
10
votes
0
answers
325
views
Are ideals in separable C*-algebras complemented subspaces?
Let $A$ be a separable C*-algebra and $J\subseteq A$ a closed two-sided ideal. Does this make $J$ into a complemented subspace of $A$? In other words, does the quotient map $A\to A/J$ have a ...
- 6,406
2
votes
1
answer
447
views
Why do Douglas-Muhly-Pearcy consider the following operator a co-isometry?
I been reading Nagy and Foias' book "Harmonic analysis of operators on Hilbert space". They prove the existence of a isometric (also unitary) dilation.
However Douglas-Muhly-Pearcy in http://...
- 243
7
votes
0
answers
1k
views
Books on von Neumann algebras
I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
- 455
5
votes
2
answers
216
views
On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilbert $ C^{\ast} $-module by a certain linear subspace
Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let
$$
\mathcal{E}_{I}
\stackrel{\text{df}}{=}
\{ x \in \mathcal{E} \mid \...
- 1,396
3
votes
1
answer
229
views
Symmetric diagonalizable operators and self-adjointness
Given a densely defined symmetric operator $L$ on a Hilbert space $H$, which is also assumed to be diagonalizable, will there always exist a unique extension of $L$ to a self-adjoint operator?
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
1
vote
1
answer
332
views
Every norm-continuous group of $C^*$-algebra automorphisms weakly inner?
Please, help out of the mind trap. In this prominent paper Kadison and Ringrose prove among other things the following
Corollary 8. Each norm-continuous representation of a connected topological
...
- 1,959
3
votes
1
answer
460
views
Norm inequality for convolution operators on groups
Let $G$ be a discrete, finitely generated group. Let $f\in \mathbb{C} G$ be given.
Consider $g\in G\setminus \operatorname{supp} f$ and let $\delta_g$ denote the Dirac delta at $g$.
Is it true ...
- 99
3
votes
0
answers
269
views
Finite dimensional representation of tensor product
Let $A$ and $B$ be $C^*$ algebras, and let $\pi:A \odot B \to B(H)$ be a $*$-representation of the algebraic tensor product on a finite dimensional Hilbert space $H$. Let $x \in A \odot B$. Since $H$ ...
- 1,769
8
votes
2
answers
759
views
If the diagonal of a positive operator is compact, is the operator itself compact?
Let $H$ be a separable Hilbert space with a fixed orthonormal basis $\{e_n\}_n$. For a bounded operator $T$ on $H$, the diagonal of $T$ is the unique operator $D_T$ on $H$ which is diagonal with ...
- 2,263
5
votes
1
answer
267
views
Induced group action of the left regular representation strongly continuous
Let $G$ be a compact group and let $\lambda: G \rightarrow \mathcal{U}(L^2(G))$ be the left regular representation, i.e. $\lambda_sf(t)=f(s^{-1}t)$. Why is the induced group action $\overline{\lambda}...
- 51
1
vote
1
answer
190
views
Bounded operators on the Stinespring representation space
Let $A$ be a $C^*$-algebra and let $\phi:A\to B(H)$ be a completely positive map. The Stinespring representation theorem constructs a representation of $A$ on a Hilbert space $K$, which is constructed ...
- 99
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
3
votes
0
answers
57
views
Integration of Weyl operators multiplied by quasifree state over a symplectic space
I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...
3
votes
1
answer
157
views
Self adjoint operators in Kasparov-Modules
In Blackadars book in 17.4.2 it says that for each element $x \in KK(A,B)$ there is a Kasparov module $(E,\pi ,T)$ such that $T=T^*$. Now, the argument for that is that if $(E,\pi, T)$ is any Kasparov-...
- 31
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},$$
and the map $\...
- 99
3
votes
0
answers
129
views
Equivariant $K$-homology with $G$-compact support
Let $G$ be a discrete countable group and let $A$ be $\sigma$-unital $G$-$C^*$-Algebra. For a proper locally compact Hausdorff $G$-space $X$ the equivariant $K$-homology with $G$ compact support and ...
- 31
2
votes
0
answers
450
views
Separable $\sigma$-unital sub-$C^*$-Algebras
Let $A$ be a $\mathbb{Z}_2$-graded $C^*$-Algebra. Then we can take the direct limit
$$
colim_{A_\sigma} KK_*(\mathbb{C}, A_\sigma)
$$
over all $\sigma$-unital graded $C^*$-sub-algebras $A_\sigma \...
- 21
1
vote
2
answers
386
views
About the quotient norm in the Calkin algebra
Recall that the Calkin algebra, is the quotient $B(H)/B_0(H)$, where $H$ is a Hilbert space and $B(H)$ and $B_0(H)$ are the algebra of bounded and compact operators on $H$.
Let $H$ be separable and $...
- 2,118
2
votes
1
answer
291
views
Homotopy equivalence of Kasparov's $KK$-Theory
The homotopy relation of Kasparov-Cycles is definied in Blackadar's book in 17.2.2. It is an equivalence relation. However, I really don't see a good argument for transitivity and can't find any ...
- 29
0
votes
1
answer
155
views
Completely positive map defined by the trace
Let $A=C^*_r(G)$ be the reduced group $C^*$-algebra of a finitely generated group. Consider the map $M_2(A)\to M_2(A)$,
$$\left[\begin{array}{ll}a&b\\c&d \end{array}\right]\mapsto \left[\begin{...
8
votes
2
answers
812
views
Weak*-norm continuous operators on von Neumann algebras
Let $M$ be a von Neumann algebra with predual $M_*$, and let $T\colon M\to M$ be a bounded, linear map. Let us say that $T$ is (sequentially) weak*-norm continuous if for every net (sequence) $(a_j)_j$...
- 3,497
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