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 ...
3
votes
1answer
103 views
Clarification on Dirac notation
I am new to the Dirac notation, so would appreciate some clarification.
Suppose $\Psi=\psi_1+\psi_2$ where $\Psi$ is normalized and $H$ is a linear operator such that $H\psi_1=E_1\psi_1$ and ...
1
vote
1answer
43 views
Where have I gone wrong? — operators wavefunctions, etc
Aim: Show that $-{\hbar^2\over 2m} {d^2\over dx^2}\psi(x)-{\hbar^2\over m}\mathrm{sech}^2(x)\psi(x)=E\psi(x)$
is equivalent to
$\hat{O}^\dagger\hat{O}\psi(x)=({2mE\over \hbar^2}+1)\psi(x)$
where ...
2
votes
1answer
64 views
Conjugates in integrals
Is it always true that $\int f(x)^*M^\dagger Mf(x) dx=\int (Mf(x))^*(Mf(x)) dx$? where $M$ is an operator. If so, is there a simple proof that this is so? THanks.
3
votes
1answer
138 views
Dense *-subalgebras of C*-algebras and their intersections with sub-C*-algebras
Consider the following question:
Let $A$ be a normed space containing a closed subset $B\subseteq A$ and a dense subset $D\subseteq A$. Is $B \cap D$ necessarily a dense subset of $B$?
My conclusion ...
11
votes
1answer
322 views
Renorming $\mathcal{B}(\mathcal{H})$?
Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
1
vote
1answer
152 views
Scalar operators and commutators
Given a scalar operator $S$ and vector operators $V_1, V_2$, show that the commutator $[S,V_1\times V_2]= [S,V_1]\times V_2+V_1\times [S,V_2]$.
I don't quite understand what a scalar operator is. But ...
2
votes
0answers
48 views
Question considering masas
Suppose $M_1, M_2$ are type II_1 factors, A is a masa in $M_1\otimes M_2$, can we find a *-automorphism $\phi$ on $M_1\otimes M_2$ such that there are two masas $A_1\subset M_1,A_2\subset M_2$ and ...
5
votes
1answer
125 views
Is the centre of a C*-algebra a sub-C*-algebra?
I believe that the answer is affirmative and I would be grateful to any comments on my attempt (see below) of proving this.
Let $A$ be a C*-algebra and denote by $Z(A)$ the centre of $A$.
First of ...
4
votes
0answers
296 views
Double dual of the space $C[0,1]$
The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
2
votes
1answer
266 views
Topologies and Continuity in Operator Theory
I am studying Operator Theory right now, but I have not had much exposure to topology. I am trying to pick it up along the way, and I am wondering about a probably simple point:
What is the ...
1
vote
2answers
316 views
Basic Spectral Theory Problem: Finding the Point/Continuous Spectrum of an Operator
I have the following problem:
Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for ...
2
votes
3answers
213 views
Eigenvalues of operators
I have a linear operator $T$ which acts on the vector space of square $N\times N$ matrices in this way:
$T(A)=0.5(A-A^\mathrm{t})$
($A^\mathrm{t}$: the transpose of a matrix $A$).
I need to prove ...
0
votes
0answers
147 views
A form of the Baker-Hausdorff equation
I wonder how many different ways are there of writing the Baker-Hausdorff equation! This is a form which I recently encountered and haven't been able to figure out how it comes,
$e^ae^Xe^b = ...
11
votes
2answers
447 views
Is there an algebraic homomorphism between two Banach algebras which is not continuous?
According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces.
Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...
12
votes
7answers
682 views
Reference for spectral sequences
What are good expositions of spectral sequences, which include a thorough introduction to the topic as well as the most important examples of applications - maybe with an emphasis an topological ...
