\(C^\ast\)-convexity of norm unit balls (Q313530)

Let \(\mathbb{B}(\mathcal{H})\) be the \(C^*\)-algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\).NEWLINENEWLINEA subset \(\mathcal{K}\) of \(\mathbb{B}(\mathcal{H})\) is called \(C^*\)-convex if \(A_1,\dots ,A_k \in \mathcal{K}\) and \(C_1,\dots ,C_k \in \mathbb{B}(\mathcal{H})\) with \(\sum_{i=1}^k C_i^*C_i=I\) implies that \(\sum_{i=1}^k C_i^*A_iC_i \in \mathcal{K}\).NEWLINENEWLINEThe present paper is devoted to the characterization of norms on \(\mathbb{B}(\mathcal{H})\) whose unit balls are \(C^*\)-convex.NEWLINENEWLINEA norm \(\| \cdot \|\) on \(\mathbb{B}(\mathcal{H})\) is said to be an \(M\)-norm if NEWLINE\[NEWLINE\left\|\sum_{i=1}^kC_i^*X_iC_i\right\| \leq \max_{1 \leq i \leq k}\|X_i\| \;\;\;\;\left(X_i \in \mathbb{B}(\mathcal{H}), \;\;\;\sum_{i=1}^kC_i^*C_i=I \right).NEWLINE\]NEWLINENEWLINENEWLINENote that, given a norm \(\|\cdot \|\) on \(\mathbb{B}(\mathcal{H})\), the unit ball of \(\|\cdot \|\) is \(C^*\)-convex if and only if \(\|\cdot\|\) is an \(M\)-norm.NEWLINENEWLINEA norm \(\| \cdot \|\) on \(\mathbb{B}(\mathcal{H})\) is called an \(L\)-norm if NEWLINE\[NEWLINE\left\|\sum_{i=1}^k C_iXC_i^*\right\| \leq \|X\| \;\;\;\;\left(X \in \mathbb{B}(\mathcal{H}), \;\;\;\sum_{i=1}^kC_i^*C_i=I \right).NEWLINE\]NEWLINENEWLINENEWLINEThe author investigates connections between \(M\)-norms and \(L\)-norms and shows that the class of \(L\)-norms which are greater than an arbitrary norm \(\|\cdot\|\), and the class of \(M\)-norms which are less than an arbitrary norm \(\|\cdot\|\), both have minimal elements. In particular, it is proved that the trace norm \(\|\cdot\|_1\) is the minimal element in the class of \(L\)-norms greater than the operator norm \(\|\cdot\|_{\infty}\).NEWLINENEWLINEThere are also some constructions of \(M\)-norms and \(L\)-norms.