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Let ${}_{\infty }\left(B_X\right)$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_X$ of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let ${}_u\left(B_X\right)$ be the subspace of ${}_{\infty }\left(B_X\right)$ of those functions which are uniformly continuous on $B_X$. A subset $B\subset B_X$ is a boundary for ${}_{\infty }\left(B_X\right)$ if $\parallel f\parallel =su{p}_{x\in B}\left|f\left(x\right)\right|$ for every $f{\in }_{\infty }\left(B_X\right)$. We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for ${}_{\infty }\left(B_X\right)$. On the other hand, for X = , the Schreier space,...