Let $\mathcal{H}$ be Hilbert space and $\mathfrak{B(}\mathcal{H}\mathcal{)}$ of all bounded linear operators on $\mathcal{H}$. Let $\mathcal{A}$ be a maximal commutative sub-algebra of $\mathfrak{B(}\mathcal{H)}$.


1. Is there an explicit formula a conditional expectation (a norm one projection) $\pi:\mathfrak{B(}\mathcal{H}\mathcal{)\longrightarrow A}$?
2. If Yes, I would be greatful if you could please give me it.
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A norm one projection
frege, maximal abelian subalgebras of von Neumann algebras are von Neumann algebras, hence are of the form $L\_\infty(\mu)$ for some measure $\mu$. See [this post](https://mathoverflow.net/questions/110461/direct-proof-of-injectivity-of-l-infty) of Bill Johnson to see a neat way to produce a projection. The problem is that in general masas of $\mathscr{B}(\mathcal{H})$ are elusive, so are the corresponding norm-one projections. 


In some particular cases, however, this is not difficult. For instance, when $\mathcal{H}=\ell\_2$ and you regard operators on $\mathcal{H}$ as matrices with respect to the canonical basis, you just chop off the diagonal to produce a projection.


