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Is every non-zero projection in a C*-algebra $A$ a supremum or infimum (at least majorized by / majorizes) a family of pairwise orthogonal non-zero projections in $A$?

PS. Are there any cheap ways to generate abelian sub C*-algebras of a given C*-algebra (of course, having in mind Akemann's example of a non-separable C*-algebra with each abelian sub-C*-algebra separable)?

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I don't know if this is exactly what you're looking for, but the answer to your first question, as I understand it, is no. There are unital C$^*$-algebras that contain no projections other than 0 and 1, e.g. $C(X)$ for any connected (compact Hausdorff) space $X$.

If you want projections, then you generally want to work in von Neumann algebras, where they are plentiful. In von Neumann algebras you have good results about suprema and infima of projections, and about splitting projections, etc.

Edit: One way to produce projections is to use the continuous functional calculus. Suppose you have a normal element $x$ whose spectrum is disconnected, say $$ \mathrm{sp}(x)= K_1 \bigsqcup K_2.$$ Let $p$ be the function which is $1$ on $K_1$ and $0$ on $K_2$; then $p(x)$ and $(1-p)(x)$ are mutually orthogonal projections.

Of course it depends highly on the nature of your C$^*$-algebra whether you can produce such elements with disconnected spectrum.

When you say that you want orthogonal projections to generate abelian subalgebras, what exactly do you have in mind for these? Algebras generated by mutually orthogonal projections aren't that interesting in and of themselves. If you have finitely many of them, say $N$, then the algebra you get is just isomorphic to functions on the discrete space with $N$ points.

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Thanks. I want projections (mutually orthogonal) to generate abelian sub C*-algebras. In fact, I am interested in the ways of producing them. –  Joachim Gryn Jul 24 '11 at 23:13
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