A variant of Rudin's inequality for translation invariant projections on group algebras and its generalization to finite dimensional Banach spaces (Q1880189)

\textit{W. Rudin} showed in [Proc. Am. Math. Soc. 13, 429--432 (1962; Zbl 0105.09504)] that the orthogonal projection onto a translation invariant subspace \(E\) of \(L_1(G)\) for a compact abelian group \(G\) has minimal norm among all projections onto \(E\). The author extends this result to projections in general finite-dimensional normed spaces. To capture the notion of minimality in this setting, the author uses approximately minimal projections. If \(X=\mathbb R^n\) with some norm, an orthogonal projection \(P\) on \(X\) is approximately minimal if there exist constants \(D,\delta_0\) such that for all \(0<\delta<\delta_0\), any orthogonal projection on \(X\) with \(\| P-Q\| \leq \delta\) satisfies \(\| Q\| > \| P\| (1-D\delta^2)\). The main result is that \(P\) is approximately minimal if and only if \( 2 \| P\| = \min \| Q+Q^t\|\), where the minimum is taken over all projections \(Q\) with \(Q(X)=P(X)\). Moreover, this is also equivalent to the existence of an operator \(S\) on \(X\) with nuclear norm 1 such that trace\((SP)=\| P\| \) and such that \(S+S^t\) commutes with \(P\).