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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2answers
446 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 ...
8
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2answers
374 views
$C^*$-algebra which is also a Hilbert space?
Does there exist a nontrivial (i.e. other than $\mathbb{C}$) example of a $C^*$-algebra which is also a Hilbert space (in the same norm, of course)?
For $\mathbb{C}^n$ with $n > 1$ the answer is ...
3
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2answers
422 views
Is a von Neumann algebra just a C*-algebra which is generated by its projections?
von Neumann algebras have the nice property that they are generated by their projections (the elements satisfying $e = e^{\ast} = e^2$) in the sense that they are the norm closure of the subspace ...
3
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1answer
102 views
Gelfand Topology and C*-algebras
Before we start here some notations to have no confusion:
Suppose $A$ is a commutative C*-algebra with unit. $\Sigma(A)$ is the Gelfand spectrum, given by all linear maps $\omega:A\rightarrow\Bbb{C}$ ...
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1answer
555 views
Why call this a spectral projection?
Regarding this question,
Why do spectral projections give norm approximations?
I have figured out what is meant by spectral projection, and have thus found the answer as well. A spectral projection ...
0
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1answer
104 views
Two questions from Dixmier's book on Von Neumann algebras
It seems something is going wrong with the preview I linked in some of my previous questions, so I will just type out the question. I am having trouble with Dixmier's proof of Corollary 5 on p. 46. ...
12
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7answers
678 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 ...
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 ...
4
votes
0answers
64 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 ...
4
votes
2answers
122 views
Duals via a Bilinear map
Let $E$ and $F$ be normed vector spaces. Then if $B$ is a bounded bilinear form on $E \times F$ then every $y \in F$ defines a bounded linear functional $f_y$ where $f_y(x)=B(x, y) \forall x \in E$. ...
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?
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 ...
6
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1answer
231 views
Matrices with entries in $C^*$-algebra
Let $\mathcal{A}$ be a $C^*$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution on ...
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 ...
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 ...
2
votes
1answer
77 views
Decomposition of representations
Let $A$ be a (possibly nonunital) Banach *-algebra, and $H$ be a Hilbert space. If $\pi: A \to B(H)$ is a *-homomorphism, i.e. a representation, then why must $\pi$ be equivalent to a direct sum of ...
2
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1answer
172 views
Ultraweak topology
In Stratila and Zsido, as well as some other sources, the ultraweak topology on $B(H)$ is taken to be the smallest topology for which every element in the closure of the span in $B(H)$ of the elements ...
1
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1answer
67 views
Showing that the WO closure of a *-algebra is a Von Neumann Algebra
I think it's best to defer to the source that I'm reading for a statement of exactly what I need to prove. Please refer to statement EP6 found on p.20 of this source.
The trouble is I don't follow ...
1
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1answer
137 views
Positive Linear Functionals on Von Neumann Algebras
Let $\omega$ be a positive linear functional on $M$ which is a Von Neumann Algebra. Suppose $\omega$ is completely additive (i.e. $\omega$ applied to a strongly convergent sum of mutually orthogonal ...
0
votes
1answer
165 views
Why do spectral projections give norm approximations?
First off, I'd like to ask:
If $H$ is a Hilbert space, and we have $A$ a bounded operator from $H$ to itself, $A$ being self adjoint (or normal), then if $A$ is compact there is a eigenspace ...
0
votes
2answers
136 views
In a C*-algebra, does $a \leq b$ imply $a^2 \leq b^2$?
While attempting to fill in the gaps in a proof of the Gelfand-Naimark-Segal representation theorem that I was given in a course in operator algebras, I found myself wondering whether, if ...
