Subalgebras of \(C(\Omega,M_n)\) and their modules (Q819223)

The notion of a subhomogeneous C*-algebra is well-known: a C*-algebra \(A\) is subhomogeneous of degree not exceeding \(n\) if every irreducible representation of \(A\) has dimension not exceeding \(n\). It is also well-known that a C*-algebra \(A\) is subhomogeneous of degree not exceeding \(n\) if and only if \(A\) is *-isomorphic to a C*-subalgebra of the C*-algebra \(C(\Omega, M_n)\) of all \(M_n\)-valued continuous functions defined on a compact Hausdorff space \(\Omega\) (here, \(M_n\) is the algebra of all complex \(n\) by \(n\) matrices). In the article under review, the author studies the operator space version of this notion. An operator space \(X\) is called \(n\)-minimal if there exists a completely isometric embedding of \(X\) into \(C(\Omega,M_n)\). The class of \(1\)-minimal operator spaces coincides with the class of normed spaces equipped with their minimal operator space structure. The author poses a number of natural questions, which he summarises in the following general framework: Suppose that we are given a subcategory \(\mathcal{C}\) of the category of operator spaces and completely contractive maps, and that \(X\) is an \(n\)-minimal operator space belonging to \(\mathcal{C}\). Does there exist a complete isometry from \(X\) into \(C(\Omega, M_n)\) which moreover belongs to \(\mathcal{C}\)? As an extension of this problem, the author considers \(n\)-minimal operator spaces belonging to \(\mathcal{C}\) which possess additional structure, e.g., that of an operator module over an operator algebra. He obtains answers to the questions arising in the above framework in a number of situations. In particular, he shows that if an operator algebra \(A\) is an \(n\)-minimal operator space, then there exists a completely isometric homomorphism of \(A\) into \(C(\Omega, M_n)\). Corresponding results are true for operator systems (and completely positive maps) and TRO's (and ternary morphisms). The author shows that if a Hilbert module \(X\) over a C*-algebra \(A\) is \(n\)-minimal, then the module action can be realised concretely, through completely isometric embeddings, as operator multiplication in some \(C(\Omega, M_n)\). Moreover, the \(n\)-minimality of a Morita equivalence bimodule between C*-algebras is equivalent to the \(n\)-minimality of the algebras. The author also proves a similar result for operator modules, giving an \(n\)-minimal version of the Christensen--Effros--Sinclair representation theorem. In the proofs, use is made of the injective envelopes of the operator spaces in question, more generally, the injective envelopes of their Paulsen system, and the fact (shown in the paper) that the injective envelope and the second dual of an \(n\)-minimal operator space are again \(n\)-minimal. The author characterises the \(n\)-minimal injective operator spaces as the finite direct sums of spaces of the form \(C(\Omega, M_{k,l})\), where \(\Omega\) is Stonean, \(k,l\leq n\), and \(M_{k,l}\) is the space of all complex \(k\) by \(n\) matrices.