Characterizations of Minimal Elements in a Non-commutative Lp-Space

For 1≤p<∞, let L_p(M,τ) be the non-commutative L_p-space associated with a von Neumann algebra M, where M admits a normal semifinite faithful trace τ. Using the trace τ, Banach duality formula and Gâteaux derivative, this paper characterizes an element a∈L_p(M,τ) such that (Formula presented.) where B_p is a closed linear subspace of L_p(M,τ) and ‖·‖_p is the norm on L_p(M,τ). Such an a is called B_p-minimal. In particular, minimal elements related to the finite-diagonal-block type closed linear subspaces (Formula presented.) (converging with respect to ‖·‖_p) are considered, where {ei}_i=1^∞ is a sequence of mutually orthogonal and τ-finite projections in a σ-finite von Neumann algebra M, and S is the set of elements in M with τ-finite supports.