The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...
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19 views
Conditions under which $A$ is a W*-algebra for a positive map between C*-algebras $\phi : A \rightarrow B$
Let $A$ and $B$ be a C*-algebra.
Let $\phi : A \rightarrow B$ be a positive map.
Suppose that $B$ is a W*-algebra. Under what conditions on $\phi$ can we ensure that $A$ is also a W*-algebra?
-1
votes
2answers
29 views
Question about derivation in Jordan algebra
Let $(G,\circ)$ be a Jordan algebra, then $\sigma:G\to G$ given by $$c\mapsto a\circ(b\circ c)-b\circ(a\circ c),\quad\forall c\in G,$$ is a derivation, where $a$ and $b$ are two fixed elements of $G$.
...
7
votes
1answer
208 views
Property of partial traces
Consider the Kronecker product of $A \in M_m, B \in M_n$:
$A \otimes B = \left( \begin{matrix} a_{11}B&...&a_{1m}B\\ \vdots&\ddots\\a_{m1}B&...&a_{mm}B \end{matrix} \right)$
$A ...
4
votes
0answers
62 views
Comparison of positive elements and Hilbert C*-modules
I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras.
Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak ...
3
votes
0answers
44 views
Sub-unital maps between C*-algebras: is there any relevant result?
"In this section, we deal with positive linear maps $\phi : A \rightarrow M$ between two unital C∗-algebra $A$ and $M$ with units denoted by $I$. In fact, we may assume that $A$ is the unital ...
1
vote
1answer
50 views
K-theory, $K_{0}$ of algebra of compact operators
I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation:
Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
5
votes
1answer
51 views
characters of a $C^*$-algebra
I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone ...
4
votes
1answer
29 views
When is the image of a GNS representation WOT-dense?
Given a $C^*$-algebra $A$ and a state $\rho$ on $A$, let $\pi_\rho$ be the corresponding GNS representation on the Hilbert space $H_\rho$. I would like know when the image of $\pi_\rho$ is WOT-dense ...
4
votes
1answer
93 views
Quotients of C*-algebras
It is known that every unital separable C*-algebra is a quotient of the full group C*-algebra $C^*(F_I)$, where $F_I$ is the free group generated by some index set $I$.
Can we drop the ...
2
votes
0answers
61 views
Relation between noncommutative geometry and functional analysis
Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
2
votes
1answer
85 views
Quotients of the maximal tensor product
Let $A$ and $B$ be C*-algebras and let $\gamma$ be any C*-norm on the algebraic tensor product $A\odot B$. Why is $A\otimes_\gamma B$ a quotient of $A\otimes_{{\rm max}}B$, where $\otimes_{{\rm max}}$ ...
0
votes
0answers
50 views
Bounded Operator with Closed Range
I've read Martin Argerami's answer to this question. On the first line he claims that the range of $T$ is closed. Can somebody explain me why that's the case? For me it is not necessarily closed.
4
votes
0answers
63 views
C* algebra of bounded Borel functions
Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
1
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0answers
50 views
From positive definite function to Følner sequence -— a question on amenability and nuclearity
We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
3
votes
1answer
42 views
Block Matrices of Operators
I'm trying to prove the following: Consider the vector space of matrices of size $n\times n$ whose entries in $\mathcal B(H)$. Denote this vector space by $M_{n,n}(\mathcal{B(H)})$. We can define ...
5
votes
1answer
41 views
States on a C*algebra
A state $\omega$ on a unital $C^*$ algebra $A$ is called definite at $a\in A$ self-adjoint if $\omega(a^2)=\omega(a)^2$.
I proved that if we have such a definite state at $a$, then for all $b\in A$ ...
2
votes
1answer
58 views
On the Spectral Theorem
Let $H$ be a Hilbert space, $T\in B(H)$ be normal and $E$ its spectral measure.
a- Let $\delta >0$ , and let $M_{\delta}$ = $\left\{\lambda\in \sigma(T): |\lambda|\geq \delta\right\}$. ...
4
votes
1answer
97 views
States and positive elements in $C^*$-algebras
Let $A$ be a unital $C^*$-algebra and $w$ be a state (i.e a positive linear functional such that $\|w\|=w(1_A)=1$. I'm trying to prove the following:a) if $a$ is selfadjoint and $w(a^2)=w(a)^2$ then ...
2
votes
1answer
55 views
A question concering nuclearity
B. Blackadar in his book Operator algebras: Theory of C${}^\ast$-Algebras and Von Neumann Algebras defines a C*-algebra $A$ to be nuclear if for every C*-algebra $B$ the algebraic tensor product ...
2
votes
1answer
57 views
Algebra (Not *)-Isomorphisms of von Neumann algebras
Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
2
votes
1answer
102 views
Point spectrum in Hilbert spaces
Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
3
votes
2answers
96 views
Unitary operator in von Neumann algebra
Let $R\subseteq B(H)$ be a von Neumann algebra, and $U\in R$ be unitary. Prove that there is a self adjoint operator $A\in R$ such that $||A||\leq \pi$, and $U=\exp(iA)$ . Any idea how to start! Thank ...
3
votes
1answer
50 views
What is $\sigma_{\mathfrak{M}_{p}}(p\cdot a\cdot p)$ when $p=\chi_{S}(a)$ for closed $S\subseteq \sigma(a)$?
Let $\mathfrak{M}$ be a von-Neumann algebra and $a\in\mathfrak{M}$ a self-adjoint element. Let $S\subseteq\sigma(a)$ be a non-empty, closed subset. Let $p=\chi_{S}(a)$, which is a non-zero, ...
1
vote
1answer
41 views
differential equation of the square root of a matrix
If the differential equation governing the time dependent matrix $M(t)$ is
$\frac{dM(t)}{dt}=A.M(t).B$
or
$\frac{dM(t)}{dt}=A.M(t)+M(t).A$
where $A$ and $B$ are constant matrices, what is the ...
1
vote
0answers
55 views
Proving properties of exponential map on a Banach algebra
$$\exp(a) := \sum\frac {a^k}{k!}$$
Can you help me prove that:
$\exp$ is well defined (i.e. converges for all $a$ in $A$)
$\exp$ is continuous
$\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
1
vote
0answers
29 views
Are decomposable maps completely bounded?
By the word decomposable I mean a positive map $\phi:\mathcal{B(H)}\rightarrow \mathcal{B(K)}$; $\mathcal{H,K~}$ are some Hilbert spaces and $\phi=\psi_1+T\circ \psi_2$ where $T$ is the transpose ...
0
votes
0answers
28 views
A map that is $(n-1)$-positive but not $n$-positive
Let $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ be a linear map. $\phi$ is called $k$-positive if the map $\phi^{(k)} : M_{kn}(\mathbb{C}) \to M_{km}(\mathbb{C})$, defined by evaluating $\phi$ ...
5
votes
0answers
53 views
how to prove this element is strictly positive?
Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to prove: if $(e_n)$ is an approximate identity ...
1
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1answer
63 views
Is there any operator which its spectrum corresponding to a compact set?
we know that for each operator $T$ the spectrum $\sigma(T)$ is compact. Is the converse true I mean if we have a compact set $K\neq\emptyset$, is there any operator $T$ such that $\sigma(T)=K$?
I am ...
1
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1answer
37 views
Are self-adjoints elements norming?
Let $\mathsf A$ be a C*-algebra and let $\mathsf{A}^*$ be its dual space. Is it true that for $f\in \mathsf A^*$ we have
$$\|f\|=\sup\{|f(x)|\colon\; \|x\|=1\mbox{ and }x^* = x\}?$$
2
votes
1answer
56 views
ideals in $C^*$ algebra
Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$. I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact ...
1
vote
1answer
46 views
strictly positive in unital $C*$ algebra
Let $A$ be a unital $C*$ algebra. $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Prove that $a\in A_+$ is strictly positive iff $a$ is invertable. I proved one direction : if $a$ is ...
1
vote
0answers
25 views
Finding Strictly Positive Elements [duplicate]
I need to find the set of strictly positive elements in the $C^*$-algebra $C_0(\Omega)$ where $\Omega$ is a locally compact Hausdorff space.
Clearly, the set will be contained in $ \{ f \in ...
3
votes
2answers
58 views
Positive elements in $C^*$-algebras
I'm trying to prove the following, and I'm not sure if the proof is correct?
If $A,B$ are $C^*$-algerbas, and $f$ is a $*$-homomorphism from $A$ onto $B$ then $f(A_+)=B_+$.Proof: let $a\in A_+$ then ...
3
votes
2answers
43 views
example of positive but not completely positive operator
I was looking for some example of a positive operator which is not completely positive on a banach algebra. if I consider my banach algebra to be $\text{M}_n(\mathbb{C})$ of matrices over complex ...
1
vote
2answers
37 views
Structure of $L_1(G)$
I came across this while going through some basic examples of $C^*$ algebras.
If I consider $G$ as the set of cube roots of unity, what will be $L_1(G)$? I mean what will be the structure of elements ...
7
votes
1answer
237 views
Cube root in $ C^{*}$-algebra.
Let $A$ be a $C^*\text{-algbera}$ and $x\in A$. I'm trying to show thata)for $0<\alpha<\frac{1}{2}$, there exists $u\in A$ with $x=u(x^*x)^{\alpha}$ and $u^*u=(x^*x)^{1-2\alpha}$. b) there ...
0
votes
0answers
34 views
2 positive decomposable maps
A positive map $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ is said to be $k$-positive if the natural extension ...
2
votes
1answer
47 views
Determining whether an operator is Hermitian
The operator $F$ is defined by $F\psi(x)=\psi(x+a)+\psi(x-a) $, where $a$ is a nonzero constant. Determine whether or not $F$ is a Hermitian operator.
If the condition for $F$ to be Hermitian is ...
6
votes
2answers
83 views
Recovering a group from its C*-algebras and group algebra
Let $G$ and $H$ be locally compact groups. Does anyone know the answers to these questions?
Is it true that:
if $C^*(G)$ and $C^*(H)$ are $*$-isomorphic, then $G\cong H$?
if $C_r^*(G)$ and ...
0
votes
1answer
32 views
Help proving operator inequality
Given $P \geq 0$, I need to show that $2Tr(P^{5/2}) \leq Tr(P^3) + Tr(P^2)$. It's trivial to show that the RHS is the trace of a positive operator, but I'm at a loss on how to actually prove this ...
2
votes
2answers
44 views
Multiplicative linear functional on algebra of limit of polynomials
Let $A$ be the space of all functions which are limit of polynomials over the unit ball $D$.
Then $A$ is a commutative Banach algebra. Then how do I show that $A$ has no non zero multiplicative linear ...
1
vote
1answer
38 views
Linear functional on Banach algebra
Let $A$ be the space of all matrices of the form $\begin{pmatrix} a & b \\0 & a\end{pmatrix}$, $2\times2$ over complex field.
Then the spectrum of any element of $A$ comes out to be $\{a\}$. I ...
3
votes
0answers
112 views
Unitary operators - convergence problem
Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$.
Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A ...
0
votes
0answers
44 views
Operator monotone functions
By definition, I know that a function $f$ is operator monotone if $A - B \geq 0 \Rightarrow f(A) - f(B) \geq 0$. For instance, we have $A^2 \leq B^2 \Rightarrow A \leq B$ because the root function is ...
1
vote
1answer
47 views
Operator inequalities: $0 \leq A \leq B \Rightarrow Tr(A^p) \leq Tr(B^p)$?
It is trivial to show that $0 \leq A \leq B \Rightarrow Tr(A^2) \leq Tr(B^2)$, but does this generally hold for all $p >$ 2 as well?
0
votes
1answer
54 views
problem related to tensor product on Hilbert spaces
Let $K$ and $H$ be Hilbert spaces. Let $\{e_i:i\in I\}$ be an orthogonal basis of $H$. Define
$$
U_i:K\to K\overset{.}{\otimes} H: x\mapsto x\overset{.}{\otimes} e_i
$$
Assume ...
3
votes
1answer
85 views
Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.
If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then
$ a = u|a| $ for a unique unitary element $ u $ of $ A $.
If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $?
I ...
2
votes
1answer
50 views
Counterexample for a polar decomposition in von Neumann and $C^\ast$ algebras
For a von Neumann algebra, we have that partial isometry and positive operator of an operator in its polar decomposition belongs to the algebra, but in a $C^\ast$ algebra this may not be true.
Can ...
1
vote
1answer
48 views
State space is weak* compact
I'm trying to convince myself that the state space $S(A)$ of a unital $C^*$-algebra is weak* compact. I've proven that $S(A)$ is convex, and I feel that this should allow me to conclude weak* ...
