All Questions
Tagged with fa.functional-analysis c-star-algebras
316 questions
4
votes
1
answer
196
views
When a normal functional is restricted to a vn Neumann sub-algebra
I have already asked this question and no comment(s) received up to now.
I am so curious to get feedback concerning the problem.
Let $M$ be a vn Neumann subalgebra in $B(H)$. Let $f$ and $g$ be ...
- 4,058
8
votes
1
answer
332
views
The double dual of the unitization of a $C^*$-algebra
I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the ...
- 267
7
votes
3
answers
676
views
Noncommutative torus as a von Neumann algebra
Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...
- 9,330
2
votes
2
answers
214
views
Commutative C*-rings
Let us consider the unital commutative $C^*$-algebra $C[0,1]$. We say $A\subseteq C[0,1]$ forms a C*-subring if it satisfies the following conditions:
1- $A$ is an involutive unital subring (closed ...
- 4,058
2
votes
0
answers
107
views
Finding non-zero elements with $x^*x\leq\frac{1}{n}$
For a given unital Banach *-algebra $A$, let us put $A_+=\{\sum_1^n x^*_ix_i : x_i\in A, n\in \mathbb{N}\}$. We write $x\geq0$ if $x\in A_+$.
What types of (non semi-simple) unital Banach *-algebras ...
- 4,058
0
votes
0
answers
47
views
An algebraic property that makes a per-C*-algebra complete
Let $A$ be a normed *-algebra with $\|x^*x\|=\|x\|^2$. Suppose that for every subset S of A, the left annihilator
${\displaystyle \mathrm {Ann} _{L}(S)=\{a\in A\mid \forall s\in S,as=0\}\,}$
is ...
- 4,058
3
votes
0
answers
130
views
Question about a paper on approximate identities
I am currently reading this paper on approximate identities of ternary Banach algebras. Assume that $(A, [.,.,.])$ is a ternary Banach algebra. A bounded net $(e_{\alpha}, f_{\alpha})$ is said to be ...
- 1,115
-2
votes
1
answer
138
views
Weak center is same as center for $C^{\ast}$-Algebra? [closed]
Let $A$ be a $C^{\ast}$-algebra. We say $A$ is weakly commutative if $ab^*c=cb^*a$ for all $a,b,c \in A$ and define weak center of $A$ as $$Z_w(A)= \{ v \in A : av^*c=cv^*a \;\forall a,c \in A \}.$$
...
- 1,115
4
votes
1
answer
200
views
Alternative proof of existence of absolute value of a functional on a C*-algebra
The usual proof of the existence of an absolute value of a functional on a C*-algebra $A$ uses the polar decomposition of normal functionals on $A^{**}$, which relies on the compactness of the unit ...
- 3,816
6
votes
0
answers
107
views
Real-world example of a Banach *-algebra with a nonzero *-radical
Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
- 3,816
1
vote
0
answers
183
views
G-abelian systems
Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...
2
votes
0
answers
116
views
General construction of enveloping C*-algebra, left/right-regular representation, etc
In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...
- 3,816
1
vote
1
answer
266
views
Proof of universality of Toeplitz algebra
It is well-known that the Toeplitz algebra $\mathcal{T}$ (that I view as concrete subalgebra of $\mathbb{B}(\ell^2(\mathbb{N})$) is the universal algebra generated by an isometry, that is, for any $C^*...
- 9,853
0
votes
1
answer
158
views
Showing a product on a character space is continuous
Quoting from Timmermann's An invitation to quantum groups and duality:
Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact
quantum group. Then there exists a compact group $G$ and ...
- 1,027
4
votes
0
answers
119
views
Relationship between canonical commutation relations and projective representations?
$\DeclareMathOperator\CCR{CCR}\DeclareMathOperator\Im{Im}\DeclareMathOperator\PU{PU}$Let $V$ be a real vector space equipped with an antisymmetric bilinear form $\omega$. Recall that there is a $C^\...
- 63.9k
0
votes
1
answer
495
views
Separability of an algebra is equivalent to separability of its spectrum
Let $A$ be a commutative C*-algebra.
I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.
Notes ...
- 143
4
votes
2
answers
254
views
$K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $*$-homomorphism which is injective on a dense $*$-sub-algebra of $A$. Can such a map have non-trivial kernel,...
- 969
9
votes
0
answers
176
views
Is the switch automorphism inner for continuous-trace $C^*$-algebras?
If $R$ is a commutative ring, and $A$ is an Azumaya algebra over $R$, then the switch (or flip, or exchange, etc.) automorphism of $A\otimes_R A$, given by $a\otimes b\mapsto b\otimes a$, is inner: it ...
- 272
1
vote
1
answer
499
views
When do completely positive maps have a closed image?
Let $\mathcal{A}, \mathcal{B}$ be C*-algebras. A map $\phi \colon \mathcal{A} \rightarrow \mathcal{B}$ is completely positive (cp) if it's linear, * preserving and all of its' coordinatewise ...
- 733
2
votes
0
answers
120
views
Need help in proving an inclusion between some subspaces of operators
The following question was first posted on Math.Stackexchange.com but unfortunately I didn’t get any answer. This might be obvious for many researchers but I can’t see how this is so, thus I am asking ...
- 1,115
6
votes
1
answer
347
views
Morita-invertible C*-algebras
I am familiar with the Morita theory of rings, and the hermitian Morita theory of rings with involution, and I am trying to understand some parallels and differences with the Morita theory of C*-...
- 272
0
votes
1
answer
159
views
Bisector Projection
Let $p,q$ be two projections of a $C^*$ algebra. A projection $l$ is called a bisector projection to $(p,q)$ if $$|pl-l|=|ql-l|$$ The motivation comes from the geometric intuition of "...
- 356
3
votes
0
answers
83
views
Reference request for representation theory of TRO
Let $H$ and $K$ be Hilbert spaces. Recall that a Ternary ring of operator(TRO) $V$ is a closed subspace of $B(H,K)$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$. I have recently started reading ...
- 1,115
1
vote
0
answers
109
views
Algebra structure on Haagerup tensor product of operator spaces
Let $A$ and $B$ be operator spaces. Is there any algebra structure on Haagerup tensor product of operator spaces such that the Haagerup tensor product becomes Banach Algebra?
Any references or ideas?
- 1,115
23
votes
3
answers
1k
views
Which $\ast$-algebras are $C^\ast$-algebras?
It's well-known that the norm on a $C^\ast$-algebra is uniquely determined by the underlying $\ast$-algebra by the spectral radius formula. Therefore there should be a way to axiomatize $C^\ast$-...
- 63.9k
0
votes
0
answers
127
views
On examples of action of C-star simple group on von Neumann algebra
Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
- 677
0
votes
1
answer
204
views
A certain class of representations
Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?
(The word "finite-dimensional" was ...
- 2,118
7
votes
0
answers
85
views
Analogue of Friedrichs extension for Hilbert $C^*$-modules
Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
- 1,903
4
votes
1
answer
110
views
Graded adjointable operators on a graded Hilbert space
Given a graded Hilbert space $\mathbf{H} = \bigoplus_{k \in \mathbb{N}} \mathbf{H}_k$, one might extend the notion of adjoint to a "graded adjoint" defined as follows: $L \in B(\mathbf{H})$ is said to ...
- 969
2
votes
1
answer
116
views
Extending $C^*$-norms from $*$-subalgebras
Let $A$ be a unital $*$-algebra, and $B$ a unital $*$-subalgebra of $A$. In addition, assume that there exists a $B$-$B$-sub-bimodule $C \subset A$, such that
$$
A \simeq B \oplus C,
$$
where $\...
- 969
2
votes
1
answer
180
views
Does the square root of a finite propagation operator have finite propagation?
Let $X$ be a non-compact manifold and let $C_0(X)$ act on $L^2(X)$ by pointwise multiplication.
We say $T\in\mathcal{B}(L^2(X))$ has finite propagation if there exists an $r>0$ such that: for all ...
- 1,903
6
votes
1
answer
119
views
Tensoring adjointable maps on Hilbert modules
Given a right Hilbert $A$-module $E$, and a right Hilbert $B$-module $F$, together with non-degenerate $*$-homomorphism $\phi:A \to \mathcal{L}_B(F)$, we can form the tensor product
$$
E \otimes_{\phi}...
- 969
2
votes
1
answer
124
views
Twisted canonical commutation relations
I am dealing with universal C*-algebra generated by $x,y$ with the following relations: $xy = qyx$, $x^{*}y = qyx^{*}$, $y^{*}x = qxy^{*}$, $x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$, $y^{*}y = q^2yy^{*} - (...
- 161
1
vote
0
answers
155
views
A criterion for abelian $C^*$ algebra [closed]
Let $A$ be a unital $C^*$ algebra such that for any two positive elements $x$, $y$ in $A$, whenever $x\leq y$ we have that $x^2\leq y^2$. Prove that $A$ is abelian.
- 1,879
5
votes
1
answer
499
views
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
2
votes
2
answers
399
views
Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different
In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
- 741
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
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
4
votes
1
answer
213
views
Complete positivity with infinite dimensional auxillary spaces
The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^*$ algebras of) the bounded operators on ...
- 143
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
8
votes
1
answer
172
views
Distance between subalgebras and positive elements in matrices
I repost here from stackexchange, as I was not given an answer there. (https://math.stackexchange.com/questions/2956530/distance-between-subalgebras-and-commutants-in-matrix-algebras)
This is a ...
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
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
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
7
votes
1
answer
199
views
If $\ $ $yx_n\to 0 $ for all $y$ in a C$^*$-algebra, Is it true that $x_n$ is weakly convergent to $0$?
$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $.
If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly
convergent to $0$ ?
For unitals this is trivial. ...
- 327
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
4
votes
1
answer
129
views
Semi-continuous fields of C*-algebras having dimension one on a dense set
Given a Hausdorff, locally compact space $X$, let us consider a semi-continuous field $\{A_x\}_{x\in X}$ of C*-algebras over $X$, such that $A_x$ is one-dimensional for every $x$ in a dense subset $D$ ...
- 2,263
2
votes
0
answers
97
views
Is $\ell_2(A)$ a Hilbert C$^*$-module with Opial property?
If $A=Mat_{n\times n}(\mathbb{C}) $, Is $\ell_2(A)$ a Hilbert $A$-module with Opial property?
Opial property: If ($w-\lim x_n=0 $) then $
(\liminf \lVert x_n\rVert<\liminf \lVert x_n-y \...
- 327