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2 votes
1 answer
297 views

Predual theorem proof in Takesaki's volume I

Consider the following fragment from Takesaki's book "Theory of operator algebra I" (Section III.3 ,p133-134). Why is the boxed line true? I can see that $\epsilon: \widetilde{A}\to A$ is ...
1 vote
1 answer
188 views

Uniqueness of the predual of a W*-algebra

Consider the following fragments in Takesaki's "Theory of operator algebras" (volume I): Question: So, we have an abstract Banach space $F$ with $A \cong F^*$. In Lemma 3.6, one considers ...
2 votes
1 answer
111 views

About $\sigma$ strong$^*$-functionals and seminorms

I'm reading the book "Theory of operator algebras" by Takesaki. In this book, the $\sigma$-strong$^*$ topology on the space $B(H)$ (bounded operators on the Hilbert space $H$) is defined (...
0 votes
2 answers
123 views

$(ST)_{[13]}= S_{[13]}T_{[13]}$ for $S,T \in B(\mathcal{H}\otimes \mathcal{H}).$

Let $T\in B(\mathcal{H} \otimes \mathcal{H})$ where $\mathcal{H}$ is a Hilbert space. We can define operators $$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$ and if $\Sigma: \mathcal{H} \otimes \...
4 votes
1 answer
510 views

Strict topology and $*$-strong toppology on $B(H)$ coincide

In the paper Woronowicz - $C^*$-algebras generated by unbounded elements, I read that the $*$-strong operator topology on $B(H)$ and the strict topology on $B(H)$ coincide. I believe this means the ...
6 votes
1 answer
316 views

Compatibility of inductive and projective limits with dualization in functional analysis

Assume $(A_i)_{i \in I}$ is a family of locally convex topological vector spaces which are all moreover assumed to be Banach spaces. We assume moreover that $(A_i)_{i \in I}$ has additional structure ...
1 vote
0 answers
89 views

Do we have $M\hat{\otimes}_A N\cong M\otimes_A N$ if $M$ is a finitely generated projective $A$-module over a nuclear Frechet algebra $A$?

Let $A$ be a nuclear Frechet algebra with unit. Let $M$ be a right Frechet $A$-module and $N$ be a left Frechet $A$-module. Both $M$ and $N$ are assumed to be non-degenerate. We can define the ...