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Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.

Question: If there are $t\in\mathbb R$ and $\lambda\in\mathbb C$ with $|\lambda|=1$ such that $\cosh(t)(a^2+b^2)+\sinh(t)(\lambda ab+\overline\lambda ba)$ is a contraction, then it's possible to choose $\lambda=\pm1$ and still have a contraction (maybe for a different value of $t$).

There's obviously something in the background to do with hyperbolic metrics etc. etc. but I can't see how to leverage any general theory. So I've asked the question in a simple, clean way, but in a way which is maybe lacking context. The hope is that because a and b are self-adjoint, it should be enough to choose $\lambda$ real (i.e. self-adjoint) but I've no idea if this is true...

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Yes, somebody please edit the typo in the title... –  J. M. Nov 23 '10 at 10:45
    
@J.M. I "fixed" the title to what I assumed is the correct spelling. I am not entirely clear on what that has to do with the text of the question, so it may still be inappropriate. –  Willie Wong Nov 23 '10 at 11:01
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Well, it's a "norm minimisation" problem in disguise, right. I want to know which $t$ and $\lambda$ give the smallest norm to my cosh and sinh formula; my hope is that this occurs for real $\lambda$. –  Matthew Daws Nov 23 '10 at 11:46

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