A characterization of operator algebras
An operator space is a subspace of the algebra B(H) of all bounded linear operators on a Hilbert space H. A matricial operator algebra is a subalgebra of B(H). The norms \(\| \|_ n\) on \(M_ n(B(H))\), which come from the identifications \(M_ n(B(H))=B(H^ n)\), induces norms on \(M_ n(X)\) for every subspace X of B(H). An \(L^{\infty}\)-matricially normed space is a vector space X with a norm \(\| \|_ n\) on \(M_ n(X)\) (n\(\in {\mathbb{N}})\) such that \(\| \alpha x\beta \|_ n\leq \| \alpha \| \| x\|_ n\| \beta \|\) \((\alpha,\beta \in M_ n({\mathbb{C}})\), \(x\in M_ n(X))\) and \(\| x\oplus y\|_{n+m}=\max \{\| x\|_ n,\| y\|_ m\}\) \((x\in M_ n(X)\), \(y\in M_ m(X))\). If there is a bilinear map m: \(X\times X\to X\) such that \(\| m_ n(x,y)\|_ n\leq \| x\|_ n\| y\|_ n\) \((x,y\in M_ n(X))\) it is called a \(L^{\infty}\)-normed pseudo algebra. If \(m(m(x,y),z)=m(x,m(y,z))\) (x,y,z\(\in X)\) we omit the word ``pseudo''. The main result of the papers states that a unital \(L^{\infty}\)-normed pseudo-algebra A is completely isometrically isomorphic to a unital metricial operator algebra B, in the sense that there exists an algebra isomorphism T: \(A\to B\) such that \(\| T\|_{cb}=\sup \| T_ n\| \leq 1\) and \(\| T^{-1}\|_{cb}=\sup \| T_ n^{- 1}\| \leq 1,\) where \(T_ n: M_ n(A)\to M_ n(B)\) is defined by \(T_ n[a_{ik}]=[Ta_{ik}].\) One interesting point of the result is that the multiplication on A need not be assumed to be associative. In particular, the quotient of an operator algebra by a closed two-sided ideal is again an operator algebra up to complete isometric isomorphism.
\(L^{\infty }\)-matricially normed space
scientific article; zbMATH DE number 4175761
scientific article