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I've run into a few problems in which reflexivity of a Banach space is given as a hypothesis. These problems are sometimes of the type where the banach space is specific/concrete, and sometimes it is just any Banach space. To me, reflexivity seems to be a hard condition to use. Is there an easy list of tricks to use this hypothesis? For example, does reflexivity imply any more tangible results via a standard theorem from functional analysis? An example of the type of answer I'm looking for is "Reflexivity often allows one to use the uniform boundedness principle" or "Reflexivity implies that the unit ball is weakly compact." But one of the reasons I'm asking these questions is because I feel I'm missing some other tricks that get used in functional analysis because I'm asked the question

"Show that every C* algebra that is reflexive as a Banach space is finite dimensional" and I feel that I simply don't know enough tricks/theorems to do this. (I could also use hints on this specific question, which might go some ways in revealing to me more tricks for reflexivity in general.)

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    @commenter: nice. Looks like$a$significant simplification of the proof suggested in Kadison's book.2012-10-07

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Suppose that $A$ is reflexive. By Sakai's theorem, $A$ is a von Neumann algebra, as it has a predual (being reflexive, $A$ is the dual of its dual). Being a von Neumann algebra, $A$ has abundance of projections, and in particular it has a maximal orthogonal set of projections $\{p_j\}_{j\in J}$. If $J$ is infinite, then there is a sequence $p_1,p_2,\ldots$ of pairwise orthogonal projections. This allows us to construct a copy of $\ell^\infty(\mathbb{N})$ inside $A$. As subspaces of reflexive Banach spaces are reflexive, we get a contradiction. This shows that every orthogonal family of projections in $A$ is finite, and this implies that $A$ is finite-dimensional.

The use of Sakai's theorem can be avoided by appealing to the universal representation; see exercise 10.5.17 in Kadison II, where more details for a proof can be traced.

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    Depends on what you call "elementary". I know about exercises in that book that are exactly the content of a paper. In any case, looking at the context of the question in the book, you can use the immediate previous question about masas to replace the first part of the proof (i.e. you only need to show that a masa contains a copy of $\ell^\infty(J)$, where the cardinality of $J$ agrees with the dimension of the masa).2012-10-02