On compact completely bounded maps of \(C^*\)-algebras
Let \(A\) and \(B\) be \(C^*\)-algebras. For a linear map \(\phi\) of \(A\) into \(B\), it is possible to define associated maps \[ \phi\otimes\text{id}_ n :: A\otimes M_ n \to B\otimes M_ n, \] and \(\phi\) is said to be completely positive if each \(\phi\otimes\text{id}_ n\) is positive, and completely bounded if \(\sup_{n} \|\phi\otimes\text{id}_ n\|<\infty.\) This quantity is denoted \(\| \phi \|_{cb}\) when it exists. The main theorem says that the following four statements are equivalent. (1) Either A is finite dimensional or B satisfies that all its irreducible representations are on Hilbert spaces at most some positive integer. (2) Every bounded linear map of A into B is completely bounded. (3) Every compact linear map of A into B is completely bounded. (4) Every compact linear map of A into B is, in the norm \(\|.\|_{cb}\), a limit of maps with finite rank. The proof is based on [\textit{R. R. Smith}, J. Lond. Math. Soc., II. Ser. 27, 157-166 (1983; Zbl 0527.46052)] and [\textit{J. Tomiyama} and the author, J. Oper. Theory 10, 141-152 (1983; Zbl 0517.46043)]. Further related results are included. As an application of the above result of Smith, it is shown that if, on the algebraic tensor product \(A\odot B\), the greatest cross norm is equivalent to the projective \(C^*\)-cross norm, then A or B is finite dimensional.
scientific article; zbMATH DE number 3846241
scientific article