2
votes
0answers
14 views

Topological modules with enough continuous linear functionals.

Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be ...
2
votes
1answer
65 views

Help with proving: If $X$ is a Hilbert $A$-$B$-module, then $ \| _A \langle x,x \rangle \| = \| \langle x,x \rangle _B \| $ for all $x\in X $.

Sorry, I posted a related question last week on here, but I'm still having trouble and this is a little different, I hope it's OK. Thank you! ( proof that this is an isometric map (on a $C^*$-module) ...
0
votes
1answer
60 views

proof that this is an isometric map (on a $C^*$-module)

Are my steps right? I'm not sure about the statement in bold below. Let $A$ be a $C^*$-algebra. Let $X$ be an $A$-module. Let $x\in X$, let $a= \langle x,x \rangle $ Define $\lambda _a (z) = az$, ...
2
votes
1answer
60 views

Trying to prove: Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle $ is norm dense in $E$.

Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle$ is norm dense in $E$. I am having trouble proving this. I believe $\langle E,E\rangle $ is a $C^*$-algebra. If I can show this, then the ...
2
votes
1answer
136 views

For each $x$ in a Hilbert $A$-module $X$, there exists a unique $y\in X$ such that $x=y\langle y,y \rangle $.

How do I prove this lemma? Lemma: Suppose that $X$ is a Hilbert $A$-module. For each $x\in X$ there exists a unique $y\in X$ such that $x=y \langle y,y \rangle $. I don't know if this is needed but ...
3
votes
1answer
105 views

Proof of inequality$\Vert f_1S_1^*+f_2 S_2^*\Vert^2\geq\Vert f_1\Vert^2+\Vert f_2\Vert^2$ for functionals

I was trying to prove Remark 3.7 in this article, but failed. Here some necessary definitions. Let $\mathcal{A}$ be a unital $C^*$-algebra such that there exist $S_1,S_2\in\mathcal{A}$ with property ...
2
votes
1answer
114 views

Checking Ruan axioms

I need to check only one axiom for matrix bimodule made from certain bimodule. Here some preparatory definitions. Matricial approach. (See here for details) For arbitrary linear space $V$ denote by ...
3
votes
2answers
86 views

Dimension of an operator module

Let $H$ be an infinite dimensional Hilbert space. Consider unital $C^*$-sub-algebra of $\mathcal{A}\subset\mathcal{B}(H)$ such that there exist a family of isometries $\{S_n:n\in N\}$ with pairwise ...