Entropy of subshifts and Macaev norm (Q1876309)

Let \(H\) be a separable infinite-dimensional Hilbert space and \(\Sigma^0_\Phi\) be a symmetrically normed ideal with a symmetric norming function \(\Phi\). If \(\tau= (T_1,\dots, T_N)\) is an \(N\)-tuple of bounded linear operators, then the number \(k_\Phi(\tau)\) is defined by \(\liminf_{u\in{\mathbf F}({\mathbf H})^+_1}\,\max_{1\leq a\leq N}\,\| [u, T_a]\|\), where the inferior limit is taken with respect to the natural order on \({\mathbf F}({\mathbf H})^+_1\) and \([A, B]= AB- BA\). In the present paper, the exact value of Voiculescu's invariant \(k^-_\infty(\tau)\) is obtained, see \textit{D. Voiculescu} [J. Funct. Anal. 91, No. 1, 1--36 (1990; Zbl 0762.46051)].