Ein operatorwertiger Hahn-Banach Satz

We generalize Arveson's extension theorem for completely positive mappings [1] to a Hahn-Banach principle for matricial sublinear functionals with values in an injective C¹-algebra or an ideal in B($\text{H}$). We characterize injective W¹-algebras by a matricial order condition. We illustrate the matricial Hahn-Banach principle by three applications: (1) Let $\text{A, B, b}$ be unital C¹-algebras, $\text{b}$ a subalgebra of $\text{A}$ and $\text{B, B}$ injective. If ?: $\text{A}$ → $\text{B}$ is a completely bounded self-adjoint $\text{b}$-bihomomorphism, then it can be expressed as the difference of two completely positive $\text{b}$-bihomomorphism. (2) Let $\text{M}$ be a W¹-algebra, containing 1_$\text{H}$, on a Hilbert space $\text{H}$. If $\text{M}$ is finite and hyperfinite, there exists an invariant expectation mapping P of B($\text{H}$) onto $\text{M}$′. P is an extension of the center trace. (3) Combes [7] proved, that a lower semicontinuous scalar weight on a C¹-algebra is the upper envelope of bounded positive functionals. We generalize this result to unbounded completely positive mappings with values in an injective W¹-algebra.     

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