Tagged Questions

2
votes
1answer
101 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 ...
1
vote
1answer
111 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
1answer
137 views

Extreme points in the set of positive linear functionals of norm $\leq 1$

Let $A$ be a C* algebra, and $S$ the set of positive linear functionals on $A$ in the unit ball of $A^*$ (Which has the weak-* topology.) I am having difficulty seeing that all nonzero extreme points ...
2
votes
1answer
82 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 ...
3
votes
2answers
122 views

Why the weak * topology on the dual of a Banach space has the stronger meaning of locally compact

Let us say that for a Hausdorff topological space to be locally compact means that every point has a compact neighborhood. Why do locally compact have the property that if $x \in U$ and $U$ is open ...
1
vote
1answer
130 views

A specific example of the GNS construction

In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X$. ...
0
votes
1answer
86 views

Tensor product of Hilbert Algebras

A Hilbert algebra is an inner product space that is also a *-algebra where the various operations and structures interact according to some axioms. One of those axioms is that the linear operation ...
0
votes
1answer
109 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. ...
0
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0answers
84 views

weak closures of ideals [duplicate]

Possible Duplicate: Two questions from Dixmier's book on Von Neumann algebras On p. 46-47 in Dixmier's book on Von Neumann Algebras, which I just realized can be accessed through this ...
3
votes
0answers
108 views

Two questions about ultraweak and ultrastrong topology from Dixmier

You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below: Let $M$ be an ultraweakly ...
0
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0answers
90 views

polars in functional analysis in Dixmier

On page 39 of Dixmier's text on Von Neumann Algebras, he argues for Lemma 1, in which he tries to see that $\theta(L_1)=E_1$ using an argument about polars from functional analysis. I was hoping ...
4
votes
2answers
128 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 $f_y(x)=B(x, y) \forall x \in E$. ...
1
vote
0answers
165 views

Convergence of net sums of complex numbers, as well as operators

I have some questions concerning convergence of sums where the summands are complex number, although the real motivation of my question comes from Von Neumann algebras where sometimes the summands are ...
0
votes
0answers
111 views

Interchanging Strong Operator convergent sums

In the book on operator algebras by Stratila and Zsido, they discuss in Ch.2 the idea of taking a Hilbert space $H$ and an index set $I$ and associating to it the Hilbert space that is the direct sum ...
1
vote
2answers
109 views

Does the inequality $0\leq a\leq b$ in a C*-algebra imply $\|a\|\leq\|b\|$?

In relation to this question of mine: C* algebra inequalities I am wondering if it is true that if $0\leq a \leq b$ in a C* algebra, does one have $||a||\leq||b||$? If you need the C* algebra to be ...
1
vote
1answer
138 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 ...
2
votes
1answer
55 views

Classification of Type 1 factors

In the proof of this theorem, which says all of the type 1 factors (factors with minimal projections) are isomorphic to $B(\ell^2(I))$ for some $I$, I want to know a few things: The supposed ...
7
votes
1answer
146 views

Ideals in $C(X)$

Let $X$ be a Hausdorf Compact topological space. Please help me to show, for the purpose of understanding an example in some of my lecture notes, that the closed ideals in $C(X)$ are of the following ...
2
votes
1answer
103 views

Generation of Von Neumann Algebras

Suppose $M$ is a Von Neumann Algebra. (VNA) For me, these are subsets of some $B(H)$ that are $*$-algebras, containing the $1$ of $B(H)$, that are Weak Operator (WO) closed, or equivalently Strong ...
1
vote
1answer
615 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 ...