0
votes
1answer
20 views

Examples of hyperstonean space

From the abelian von Neumann algebra, I see the hyperstonean space as its spectrum´╝łanalogy with the C*-algebra). Now I want to see some examples of hyperstonean space. 1) Can you give me a ...
1
vote
1answer
48 views

Full (or universal) group C*-algebra of discrete group $\Gamma$

There is a quotation below (C*-Algebras and Finite-Dimensional Approximations): $ \qquad$We denote the $group~ring$ of $\Gamma$ by $\mathbb{C}[\Gamma]$. By definition, it is the set of formal sums ...
0
votes
2answers
43 views

A question on discrete group

There is a quotation below: For a discrete group $\Gamma$ , $f\in l^{\infty}(\Gamma)$ and $s, t\in \Gamma$. We let $s.f \in l^{\infty}(\Gamma)$ be the function $s.f(t)=f(s^{-1}t)$. My question is ...
1
vote
2answers
40 views

A question about discrete group

There is a quotation below: For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}$ for all $s, ...
2
votes
0answers
37 views

Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
2
votes
1answer
43 views

A question about compact Hausdorff space

Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. ...
1
vote
1answer
100 views

A question about convex set

I need to prove the closed set $C\subseteq \mathbb{R}_{+}$ is a convex. And let $x$, $y$ be arbitrary given in $C$, I have proved that $1/2(x+y)\in C$. Then does this means $C$ is convex ?
4
votes
1answer
197 views

The set of all continuous functions on a locally compact Hausdorff space.

I am reading a book about C*-algebra. There is a example that i could not understand. Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the set of all continuous functions vanishing at ...
3
votes
1answer
132 views

On the use of nets when defining operator topologies

Let's consider the strong operator topology and the weak operator topology on bounded operators of a infinite-dimensional Hilbert space $H$. When they define these operator topologies, some authors ...
7
votes
1answer
142 views

When is a $*$-homomorphism between multiplier algebras strictly continuous?

The strict topology on the multiplier algebra $M(A)$ of a C*-algebra $A$ is that generated by the seminorms $$ x\mapsto \| ax \|\qquad x\mapsto\| xa \| \qquad (x\in M(A), a\in A) $$ Whereas a ...
2
votes
1answer
78 views

A reference request for sums of $C^*$-algebras

Does anyone know where I can find a reference for the following well-known fact: Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s. Then ...
3
votes
1answer
75 views

What's the application of C*-algebra in topology?

C*-algebras are thought be be non-commutative topological spaces because of Gelfand's theorem that any commutative C*-algebra are isomorphic to C(X) for some locally compact Hausdorff space X. I've ...
1
vote
0answers
76 views

Definition by commutation property on structures : continuity and where?

(This is very vague, so sorry if there are approximations) I remember that one can define continuity as a commutation property of a function with the limit operation. Structurally, i think it maps a ...
2
votes
0answers
96 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
2
votes
1answer
185 views

How to make a topological subspace Hausdorff

If $X$ is a compact topological space then $C(X)$ is a C* algebra. I'm not going to attempt to discuss the locally compact case with $C_0(X)$ because usually the definition of that space requires the ...
3
votes
3answers
127 views

What if every function in $C(X)$ has finite spectrum?

Suppose that $X$ is a compact Hausdorf space, and that every continuous function on $X$ has finite range. How do I conclude that $X$ is a finite set, hence with discrete topology? So far, I have ...
3
votes
2answers
152 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 ...
16
votes
4answers
650 views

Are commutative C*-algebras really dual to locally compact Hausdorff spaces?

Several online sources (e.g. Wikipedia, the nLab) assert that the Gelfand representation defines a contravariant equivalence from the category of (non-unital) commutative $C^{\ast}$-algebras to the ...
3
votes
1answer
179 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 ...
3
votes
1answer
323 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 ...