Mathematics – Functional Analysis
Scientific paper
2005-08-04
Contemporary Math. 414 (2006) 147-162
Mathematics
Functional Analysis
16 pages. Three very minor mathematical typos corrected. Intended for the proceedings of GPOTS05
Scientific paper
For a Banach space $V$ we define its Lipschitz extension constant, $\cL\cE(V)$, to be the infimum of the constants $c$ such that for every metric space $(Z,\rho)$, every $X \subset Z$, and every $f: X \to V$, there is an extension, $g$, of $f$ to $Z$ such that $L(g) \le cL(f)$, where $L$ denotes the Lipschitz constant. The basic theorem is that when $V$ is finite-dimensional we have $\cL\cE(V) = \cP\cC(V)$ where $\cP\cC(V)$ is the well-known projection constant of $V$. We obtain some direct consequences of this theorem, especially when $V = M_n(\bC)$. We then apply techniques for calculating projection constants, involving averaging projections, to calculate $\cL\cE((M_n(\bC))^{sa})$. We also discuss what happens if we also require that $\|g\|_{\infty} = \|f\|_{\infty}$.
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