Questions tagged [c-star-algebras]
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$. Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).
2,742
questions
2
votes
1
answer
33
views
Does the ultra-weak topology coincide with the weak topology on the unit ball?
Just let me say first, I am no expert neither in $C^*$-algebras nor in $W^*$-algebras. But I came across the following question:
Let $A$ be a $C^*$-algebra. Then its bidual $A^{**}$ is also a $C^*$-...
- 4,976
1
vote
1
answer
14
views
Definition of transpose map in Murphy's C* algebras
Given a continuous map $\theta : \Omega \rightarrow \Omega'$, the author defines its transpose $\theta^t : C(\Omega) \rightarrow C(\Omega')$ by $f \mapsto f\circ \theta$. Then he states that it is a ...
- 301
2
votes
1
answer
40
views
A question about a projection in a W$^*$-algebra
Let $M$ be a W$^*$-algebra algebra over a Hilbert space $\mathcal H$. Let $N$ be a W$^*$-subalgebra of $M$ and $\phi:M \to N$ be a normal positive linear map from $M$ onto $N$. Let us consider the ...
- 1,879
1
vote
1
answer
31
views
Normal Hahn-Banach extension of a von Neumann algebra
Let $M\subseteq B(H)$ be a von Neumann algebra. Let $\omega \in M_*$ be a normal functional. In a paper I am reading$^{(\dagger)}$, it is said that it is possible to find a normal functional $\...
- 356
0
votes
0
answers
50
views
Any reference for the mathematics of Quantum Mechanics with infinite degrees of freedom?
I am looking for a book, or lecture notes or even courses available on YouTube where there is a good and detailed discussion on the mathematical aspects of Quantum Mechanics with infinite degrees of ...
- 89
0
votes
1
answer
33
views
Suficient condition for an unital $C^*$-algebra homomorphism to be surjective on postive elements.
I am trying to find sufficient conditions for an unital $C^*$-algebra homomorphism $\alpha:A \rightarrow B$ to be surjective on positive elements, that is, $\alpha(A_+)=\alpha(B_+)$. For example, if ...
1
vote
1
answer
45
views
Nuclear $C^*$-algebras have the WEP-property.
Exercise: Let $A$ be a nuclear $C^*$-algebra. Show that $A$ has Lance's WEP-property, i.e. show that there exists a ucp map $\Phi: B(H_u) \to A^{**}$ such that $\Phi(a)= a$ where $A\subseteq A^{**}\...
- 356
2
votes
1
answer
31
views
Closed ideals of the continuous functions on the locally compact space $X$ vanishing at infinity, $C_0(X)$, with Stone-Weierstrass theorem
I am reading the book "Operator Algebras and Quantum Statistical Mechanics" by O. Bratteli and D. W. Robinson, and Example 2.1.9 says the following:
Let $\mathfrak{A}=C_0(X)$, the ...
- 155
1
vote
2
answers
43
views
Approximating a self adjoint element with invertible elements
I would like to prove the following proposition:
Let $x\in \mathcal{A}$ be a self adjoint element of unital C* algebra $\mathcal{A}$. Then for any $\epsilon >0$, there is an invertible element $y\...
- 135
2
votes
1
answer
54
views
Projective tensor product of operator spaces
Consider the following fragment from Effros and Ruan's book "Operator spaces"
Why is a decomposition as in the red box possible? In fact, it is not even clear to me that any such ...
- 356
4
votes
0
answers
37
views
Smooth vectors for the torus action on the irrational rotation algebra
There exists a canonical action of the group $S^1\times S^1$ on the irrational rotation algebra $A_\theta$, which is the universal C*algebra generated by two unitaries $u$ and $v$ satisfying $uv=e^{i2\...
3
votes
1
answer
31
views
Abstract versus concrete operator spaces
Definition: An (abstract) operator space $X$ is a linear space together with a sequence $\{\|\cdot\|_n\}_{n=1}^\infty$ of complete norms such that
$\|x\oplus y\|_{m+n}= \max\{\|x\|_m, \|y\|_n\}$ for $...
- 356
1
vote
3
answers
166
views
A factor von Neumann algebra is a prime algebra.
Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ denotes the algebra of all bounded linear operators on $\mathcal{H}.$
Recall that a von Neumann algebra $\mathcal{U}\...
- 11
1
vote
0
answers
59
views
Series of operators
Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}...
- 89
1
vote
1
answer
57
views
$M\xi$ forms a unital Hilbert algebra for some cyclic-separating vector $\xi$
Let $M$ be a von Neumann algebra over a Hilbert space $\mathcal H$ and $\xi\in \mathcal H$ be a cyclic-separating vector for $M$, that is, $[M\xi]=[M'\xi]=\mathcal H$, where $M'$ is the commutant of $...
- 1,879
2
votes
1
answer
51
views
Simple proof that norm of operator is greater than norm of real part
I'm looking at Takesaki's Theory of Operator Algebras I, chapter I.4 exercise 3. The following statement is implicitly used:
Let $A$ be a unital $C^*$-algebra and $a, b\in A$ selfadjoint. Then $\Vert ...
- 2,046
2
votes
1
answer
27
views
The center of a von Neumann algebra may or may not have minimal projections
Let $\mathcal H$ be a Hilbert space with $\dim \mathcal H=\infty$. Let $M \subseteq B(\mathcal H)$ be a von Neumann algebra acting on $\mathcal H$. Let $Z$ denotes the Center of $M$. Now a projection $...
- 1,879
1
vote
1
answer
62
views
Does Hilbert $C^*$-modules have orthonormal basis?
Let $E$ be a (right) Hilbert $A$-module where $A$ is unital $C^*$-algebra. I call a subset $S$ of $E$ to orthonormal if $\langle x,x\rangle=1$ ($1$ denotes the unit in $A$) for all $x\in S$ and $\...
- 1,060
1
vote
0
answers
57
views
Noncommutative analog compact open topology
Recall that for any locally compact Hausdorff topological space $X$, there is so called compact open topology on $C(X)$ https://en.wikipedia.org/wiki/Compact-open_topology. Is there a noncommutative ...
1
vote
1
answer
47
views
If all unital Banach algebras contain maximal ideals, then why doesn't $M_n(\mathbb{C})$ have proper ideals?
In Murphy's textbook on C*algebra, he writes:
So all unital Banach algebra should contain maximal ideals, but on a StackExchange post, there is an explicit example $M_n(\mathbb{C})$ of an unital (non-...
- 4,185
2
votes
1
answer
39
views
$\sup \{upu^*: u \in \mathcal U( M)\}$ is a central projection of a von Neumann algebra $M$
Let $M$ be a von Neumann algebra acting on the Hilbert space $\mathcal H$. Let $0 \ne p \in M$ be a projection. Now consider a projection $q \in M$ defined by
$$q:=\sup \{upu^*: u \text{ is an unitary ...
- 91
5
votes
0
answers
67
views
Let $A$ be a unital $C^*$-algebra, $a\in A,\ x,y\in A_{sa}$. Does there exist a state $\phi$ on $A$ such that $\phi(xa^*ay)=\lVert a\rVert^2\phi(xy)$?
If $A$ is a commutative, unital $C^*$-algebra, then $A=C(X)$ for some compact, $T_2$ space $X$. Then $a=f,x=g$ and $y=h$ are continuous functions on $X$. Then there is $x_0\in X$ such that $\lVert f\...
- 3,053
1
vote
1
answer
41
views
Strictly positive element in a von Neumann algebra
If $A$ is a $C^*$-algebra and $a\in A^+$. $a$ is said to be strictly positive if $aA$ is dense in $A$.
Let $M$ be a von Neumann algebra. Does there exist another sufficient and necessary condition to ...
- 1,723
1
vote
0
answers
34
views
Smeared polynomials of creation and annihilation operators
Given a massive free scalar field. We can define the quantum *-algebra of observables as a subset $\mathcal{E}\subseteq C^{\infty}(\mathcal{F})$ given by
$$
\mathcal{E}:= \lbrace 1,a[f],\overline{a[g]}...
- 81
2
votes
1
answer
30
views
What are the most general structures that admit a singular value decomposition for each element?
I was looking at *-algebra and C*-algebra and as far as I understood, the latter guarantees each element to have a singular value decomposition, or at least, that every element $a$ is such that $u a v$...
- 165
2
votes
1
answer
49
views
Unitary "phase" in *-algebra
I'm looking into unital *-algebras and I'm trying to understand if the following is true:
If $A, B$ are elements of the algebra such that $AA^* = BB^*$ then there exists a unitary element $U$ such ...
- 165
0
votes
0
answers
27
views
GNS construction when no approximate identity
Let $A$ be a normed $*$-algebra, let $\alpha$ be a continuous positive form on $A$, which has finite variation, meaning there exists $C>0$ such that $|\alpha (a)|^2 \leq C \cdot \alpha (a^* a)$ for ...
- 1,083
1
vote
1
answer
64
views
Von Neumann algebra decomposition as integral of factors and mixed state decomposition as sum of irreducible states
My question is the following. It is known that any Von Neumann algebra can be uniquely decomposed as integral over algebra factors. It is also know that any mixed state can be uniquely expressed as ...
- 89
1
vote
1
answer
84
views
Doubt on Von Neumann algebra decomposition as integral of factors [closed]
I am trying to understand the Von Neumann decomposition, according to which every Von Neumann Algebra can be uniquely decomposed as integral (or direct sum) of factors. More specifically, I am trying ...
- 89
4
votes
1
answer
109
views
A question on disjointness preserving operator
In "Disjointness preserving operators on $C^*$ algebras" by Manfred Wolff, Arch. Math 62, 248-253, 1994 it is presented the concept of zero divisor preserving map as follows:
Let $A$...
- 816
0
votes
0
answers
26
views
An Ideal Correspondence For Twisted C*-Dynamical Sytems?
Back in the 90's, Nilsen proved the following result for normal $C^{*}$-dynamical systems:
Suppose that $A$ is a $C^{*}$-algebra, and $G$ is a locally compact group.
Let $\delta$ be a coaction of $G$ ...
- 1,017
1
vote
1
answer
50
views
The equation $ab+ba=0, a=a^*, b=b^*$ in a $C^*$ algebra
Does the equation $ab+ba=0$, $a=a^*, b=b^*$ in a $C^*$ algebra implies that ab=0?
What is the universal unital $C^*$ algebra subject to the above relations with boundedness condition $|a|=|b|=1$
- 816
2
votes
1
answer
46
views
Theory for finite-dimensional representations of C*-algebras
As the title suggests, I am interested if there is any well-developed theory of finite-dimensional representations of $C^*$-algebras (the algebra itself may not necessarily be finite-dimensional). For ...
- 447
0
votes
1
answer
58
views
Modules over C*-algebras and W*-algebras
I've been reading this paper by Rieffel and I'm a bit confused about the some of the claims, found in Section 1 (Modules over operator algebras). Let $B$ be a $C^\ast$-algebra and let $n(B)$ be its ...
- 63
3
votes
1
answer
49
views
Topology on a $II_1$-factor is induced by the $\|.\|_2$-norm
Let $(M,\tau)$ be a $II_1$-factor, i.e., $\tau$ is a faithful normal state on $M$ and $M$ has a trivial center.
The $\|\cdot\|_2$-norm induced by $\tau$, i.e., $\|a\|^2=\tau(a^*a)$ for $a\in M$ ...
- 171
4
votes
0
answers
86
views
Can the Stinespring Dilation Theorem extend to *unbounded* operators?
I have some (possibly basic) questions about $C^*$-algebras, the Stinespring Theorem (Theorem 3.6 in Takesaki's book), and unbounded operators. This is motivated by quantum mechanics, where unbounded ...
- 141
0
votes
1
answer
29
views
The image of a $C^*$-algebra is closed under an isometry
I need to show that $\mathcal{A}$ closed and $\|{\phi(A)}\|=\|{A}\|$ imply that $\phi(\mathcal{A})$ is closed in $C(\Delta(X))$ (I don't think this is necessary but $\mathcal{A}$ is a $C^*$-algebra, $\...
1
vote
1
answer
23
views
Structure space of the continuous functions on $X$ is homeomorphic to $X$
I have two questions regarding the proof of a proposition in Landsman's notes on $C^∗$-Algebras, Hilbert $C^∗$-modules, and Quantum Mechanics. (Note that $\Delta(\mathcal{A})$ is the structure space ...
0
votes
0
answers
12
views
Morita Equivalence and Imprimitivity 'sub-bimodules'
I'm currently trying to figure out the following problem, and I'm not sure if the answer is obviously true / false.
Suppose that $A,B$ are Mortia equivalent $C^{*}$-algebras with imprimitivity ...
- 1,017
0
votes
1
answer
24
views
Imai - Takai duality for reduced crossed products?
I'm currently looking for a citation to a statement of Imai-Takai duality, specifically for reduced $C^{*}$-algebra crossed products, and I can't seem to find one.
It's clear to me that such a result ...
- 1,017
1
vote
1
answer
80
views
Halperin Projection Convergence vs Tensor Product
Let $\mathcal{A}$ be a $\mathrm{C}^*$-algebra, and consider projections $p_1,\dots,p_N$ and $q_1,\dots,q_N$, all in $\mathcal{A}$. Using Halperin (I. Halperin, The product of projection operators. ...
- 8,208
2
votes
0
answers
183
views
An 'infinitely conditioned' state in a C*-algebra?
Edit: I have worked on the problem a bit, and thus have refocussed. The first part has now been asked at MO.
Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra. Let $f\in \mathcal{A}$ self-adjoint ...
- 8,208
2
votes
0
answers
35
views
Conditional expectation from $\mathcal{A}\rtimes_r\Gamma$ onto $\mathcal{A}\rtimes_r H$
I wish to show the following: Let $\Gamma$ be a discrete group, $H$ a subgroup of $\Gamma$ and $\mathcal{A}$ be a unital $\Gamma$-$C^*$-algebra. I want to prove that the map $E_H: C_c(\Gamma,\mathcal{...
- 171
3
votes
0
answers
98
views
Properties of reducible representations
I have the following doubt. Let's assume we have two mixed states $\rho_1 = \Sigma_i a_i \omega_i^{1}$ and $\rho_2 = \Sigma_i b_i \omega_i^{2}$ on the same algebra, where the states $\omega$ are all ...
- 89
2
votes
1
answer
75
views
$\|\tau\|=trace(|B|)$
Let $M_n(\mathbb{C})$ the matrix space, $\tau:M_n(\mathbb{C})\to\mathbb{C}$ defined by
$$
\tau(A)=trace(AB)
$$
We consider $M_n(\mathbb{C})$ with norm $\|A\|^2=\sup\{|\lambda|\in\mathbb{C}:\lambda$ is ...
- 562
0
votes
0
answers
40
views
calculate $K_0(C(Z_n))$ and $K_1(C(Z_n)$)
For each natural number $n$ let $Z_n$ be the $n$-clover obtained by forming the disjoint union of $n$ circles, choosing one point in each circle, and then identifying these $n$ points, so that they ...
- 11
0
votes
0
answers
45
views
When is $\{(g f_1, g f_2) \mid g \in C(X)\}$ dense in $L^2(\mu) \oplus L^2(\nu)$ where $\mu \perp \nu$ and $(f_1, f_2)\in L^2(\mu) \oplus L^2(\nu)$
This is from Conway's operator theory book, chapter 6, exercise 3.
If $X$ is compact and $\mu$ is a positive Borel measure on $X$, then $\pi_\mu$ : $C(X) \rightarrow \mathcal{B}\left(L^2(\mu)\right)$ ...
- 669
1
vote
1
answer
43
views
When is the Representation given by the Multiplication Operator Injective?
If $X$ is compact and $\mu$ is a positive Borel measure on $X$, then $\pi_\mu$ : $C(X) \rightarrow \mathcal{B}\left(L^2(\mu)\right)$ defined by $\pi_\mu(f)=M_f$ is a representation of $C(X)$
Question: ...
- 669
2
votes
1
answer
16
views
Inclusion of G-Hilbert spaces induce homotopy equivalence after stabilization?
In the paper
https://arxiv.org/pdf/math/0001094.pdf
proposition 6.1, we need to show that the functor $[-]_s$ which sends an equivariant $*$-homomorphism $f$ to its stable homotopy class $[id_{\mathbb{...
- 183
2
votes
1
answer
57
views
Extension of states on $C^*$-algebras (unique extension compared to a sub algebra)
The following question is related to this, which was asked earlier on the forum. I am still a bit confused by the situation and decided to ask a separate question about it.
Let $C\subset B\subset A$ ...
- 171