All Questions
Tagged with oa.operator-algebras fa.functional-analysis
778 questions
5
votes
1
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499
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Variations on Kaplansky Density
Let $A$ be a $C^*$-algebra and $\pi:A\rightarrow B(H)$ a faithful $*$-representation, so $M=\pi(A)''$ is a von Neumann algebra and $A\rightarrow M$ is an inclusion. von Neumann's Bicommutant Theorem ...
- 18.7k
3
votes
0
answers
227
views
Is there a noncommutative version of von Neumann's ergodic theorem? [closed]
The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem.
E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) ...
- 139
4
votes
1
answer
128
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Invariance of Finite Dimensional Woronowicz $\mathrm{C}^*$-ideals under the Antipode
Let $(A,\Delta)=:F(G)$ be a finite dimensional $\mathrm{C}^*$-Hopf algebra, and so the algebra of functions on a quantum group $G$.
Let $J$ be a closed ideal in $F(G)$ and $\pi:F(G)\rightarrow F(G)/J$...
- 1,027
6
votes
1
answer
200
views
Coarse index of Dirac operator on $\mathbb{R}$
Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index
$$\text{Ind}(...
- 1,903
1
vote
0
answers
253
views
When is the weak topology generated by a family of functions Baire?
Suppose we are given a locally compact space $X$ with $C_b(X)$ denoting the continuous bounded complex or real functions on $X$.
Now, if $A\subset C_b(X)$ is given, I am trying to figure out when the ...
- 173
0
votes
1
answer
174
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Does $\{\left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right||\varphi\in\{\psi\}^{\perp}\}$ split $\mathfrak{S}_1$?
Let $\mathfrak{S}_1$ be the space of trace-class self-adjoint operators on $L^2(\mathbb{R}^n)$, and $\psi\in L^2(\mathbb{R}^n)$ such that $\int |\psi|^2 = 1$. Is there a projection from $\mathfrak{S}...
user141436
1
vote
1
answer
128
views
Trace class operators in the unit ball of a finite dimensional subvector space of $B(H)$
Let $F\subset B(H)$ be a finite dimensional subvector space of the space of all bounded operators on a Hilbert space.
Question: Is there an upper bound for $$\{|tr(T)| \text{where} \quad T\in ...
- 356
5
votes
2
answers
642
views
Non-standard tensor products of inner product spaces
For two inner product spaces $(\mathcal{V}, (\cdot,\cdot)_V)$ and $(\mathcal{W}, (\cdot,\cdot)_W)$, we can put an inner product on their tensor product in the obvious way:
$$
(1) ~~~~ \langle v \...
- 643
1
vote
0
answers
105
views
The tower of path algebras associated to a tower of finite dimensional $C^*$-algebras is isomorphic to the original tower
Let $A_0\subseteq A_1\subseteq...$ be an infinite tower of unital inclusions of finite dimensional $C^*$-algebras and $B_0\subseteq B_1\subseteq ...$ be its associated infinite tower of path algebras. ...
- 235
4
votes
0
answers
99
views
Commuting flows problem for non-Lipschitz vector fields
Let $X$ be a continuous vector field on a (say compact) manifold $M$, if $X$ has ODE uniqueness then we can define its associated flow $\mathcal F_X:\mathbb R\times M\to M$ uniquely given by $\mathcal ...
- 938
0
votes
0
answers
54
views
On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
- 677
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
2
votes
1
answer
83
views
On existence of fixed point operator
Let $M$ be an infinite dimensional non-type $I$ factor, given $\xi$ in $\mathcal{H}$, does there exist a not identify operator $x$ in $M$ such that $x\xi=\xi$, I have tried with taking projection $P_{\...
- 677
-1
votes
2
answers
640
views
Invariance of spectrum under conjugation
Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
- 677
2
votes
0
answers
151
views
A Banach or $C^*$ algebraic analogy of a classical fact in real analysis
Let $A$ be a commutative unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$.
Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$.
Assume that for every $\phi\in \...
- 356
2
votes
0
answers
116
views
Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 993
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
3
votes
0
answers
297
views
Tensor product of compact operators on Banach modules
Let $A$ and $B$ be Banach algebras. Consider a right Banach $A$-module, $E$, and a right Banach $B$-module, $F$, as well as a Banach algebra morphism $\pi\colon A\to\mathcal L_B(F)$ into the bounded $...
- 320
3
votes
1
answer
206
views
Ultraproduct of non-commuative $L^p$-spaces
Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...
- 1,879
3
votes
2
answers
265
views
Ultraweak topology in abelian von Neumann algebras
Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff ...
- 159
1
vote
0
answers
277
views
Adjoint for a non-densely defined unbounded operator on a Hilbert space
Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...
- 993
8
votes
1
answer
277
views
Noncommutative Fredholm operators
Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely ...
- 99
6
votes
0
answers
181
views
Blocksum induces a unital H-space structure on the space of Fredholm operators
Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...
- 386
1
vote
1
answer
387
views
The compact open topology and the operator norm
Let $T:[-1,1]\rightarrow B(H)$ be a continuous family of bounded operators where $B(H)$ is endowed with the compact open topology for an infinite dimesional Hilbert space $H$. Is it true that if $T_0=...
- 99
4
votes
1
answer
476
views
Bicommutant theorem for commutative operator algebras
Let $\mathcal{B}(H)$ denote the space of bounded linear operators on a complex Hilbert space $H$. The von Neumann bicommutant theorem says:
Theorem. Suppose that $\mathcal{A}$ is a $C^*$-subalgebra ...
- 12.5k
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
340
views
Why is the definition of von Neumann trace independent of the choice of the Hilbert space?
A Hilbert module defined in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ ...
- 2,118
3
votes
0
answers
97
views
Is the set of points in the irreducible decompositions of this C$^{*}$ -algebra's representations closed?
Suppose $X$ and $Y$ are compact Hausdorff spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily ...
- 267
3
votes
1
answer
253
views
What is the story behind this Hilbert space in the definition of Hilbert Modules
Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert ...
- 2,118
7
votes
1
answer
491
views
Projections in the tensor product of von Neumann algebras
This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...
- 619
0
votes
1
answer
418
views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
2
votes
3
answers
663
views
center of a $C^*$-algebra
Does there exist a $C^*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?
I know the fact that the center of $\mathcal{K}(H)$ is $0$, but $\mathcal{K}(H)$ has no ...
- 451
2
votes
1
answer
107
views
tensor stability of block-positive matrices
Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation)
$\langle \psi |...
- 483
6
votes
1
answer
150
views
Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces
Let $A$ (resp. $B$) be a unital $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) ...
- 1,586
2
votes
0
answers
100
views
k-rank numerical range of an operator
Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...
- 231
4
votes
1
answer
443
views
Taylor spectrum of commuting operators
Taylor spectrum of commuting operators
Fom the following paper (M. Ch—o, H. Motoyoshi, B. Na¡cevska Nastovska: On the joint spectra of commuting tuples of operators and a conjugation) we have
Let ...
- 1,154
4
votes
1
answer
153
views
Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra?
$A$ is an infinite dimensional C$^*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an infinite dimensional C$^*$-algebra ...
- 327
8
votes
1
answer
547
views
Maps which are both completely positive and positive
Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
- 43.2k
3
votes
1
answer
388
views
Fundamental group and group measure space construction
Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
0
votes
1
answer
87
views
Convergence of self-adjoint elements in $\sigma$-weak topology
In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?
- 159
3
votes
0
answers
202
views
Anzai flow in noncommutative geometry
Consider Anzai flows (cf. Anzai: Ergodic Skew Product Transformations on the Torus, Osaka Math. J. 3 (1951), 83-99) on the two dimensional torus $T^2$. I would like to know if there exists some ...
5
votes
1
answer
306
views
Cartan subalgebra and group measure space construction
Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a Cartan subalgebra if moreover $\mathcal{N}...
6
votes
0
answers
237
views
A characterisation of certain $C^*$-algebras
I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
- 633
2
votes
1
answer
407
views
Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space
Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous ...
- 1,396
1
vote
1
answer
172
views
Examples of isomorphic W* algebra with non-homeomorphic weak topology
Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
- 1,586
3
votes
2
answers
207
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Commutative direct summands of C*-algebras
I have a question about commutative direct summands of $C$*-algebras.
Let $A$ be a $C$*-algebra (with unit) and suppose that its bidual $A^{**}$ has a commutative direct summand, that is, $A^{**}=B\...
- 633
1
vote
1
answer
193
views
When is $\inf_{n\geq0}x^n\neq0$?
Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $x$ be a positive element of $M$ with $\|x\|=1$. So, $(x^n)_{n\in\mathbb N}$ is a decreasing sequence of positive elements and $y:=\...
- 2,118
4
votes
0
answers
104
views
informative examples for understanding spectral triples
I am at the beginning of my thesis work and I am trying to understand spectral triples. I can recall the definition but I have no informative examples with which to make sense of it. What are some ...
- 111
2
votes
2
answers
723
views
Behaviour of Direct limit with quotient and double dual
I am trying to understand direct limit in category of $C^*$ algebras.
Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras?
Any references or ideas?
P....
- 1,115