Homotopy theory of bundles whose fibers are matrix algebras (Q2487396)

Let \(A_k, \;k > 1\), be a locally trivial bundle over \(X\) whose fibre is the matrix algebra \(M_k({\mathbb C})\) of \((n\times n)\)-matrices over \({\mathbb C}\), where \(X\) is a finite complex. This bundle \(A_k\) is called a bundle of algebras if there exists a bundle mapping of \(A_k\) into a trivial bundle \(X\times M_{k\ell}({\mathbb C})\) for some \(\ell\) such that for any \(x\in X\) the fibre \((A_k)_x\) can be immersed into \(M_{k\ell}({\mathbb C})\) as a central simple subalgebra. In particular, when \((k, \ell)=1\), it is called a floating bundle of algebras. The present paper deals with a homotopy theory of floating bundles of algebras. In Section 1 the author defines a canonical bundle of algebras over \(\text{ Gr}_{k,k\ell}\), where \(\text{ Gr}_{k,k\ell}\) is a homogeneous space parametrizing subalgebras in \(M_{k\ell}({\mathbb C})\) which are isomorphic to \(M_k({\mathbb C})\), and shows that this bundle plays a role of the universal bundle for the \(A_k\)'s (Proposition 1.19). Section 2 considers an analogue of \(K\)-theory associated with floating bundles of algebras. For this the author introduces a relevant stable equivalence relation on the set of floating bundles of algebras over \(X\). As the stable equivalence relation of the usual vector bundles defines the \(K\)-functor, this relation together with operation induced by the tensor product of floating bundles of algebras defines a contravariant functor on the category of finite \(CW\)-complexes with values in the category of abelian groups. Then one sees that the classifying space for its equivalence classes is just the direct limit \(\varinjlim \text{ Gr}_{k,k\ell}\) with respect to \(k\) and \(\ell\) such that \((k, \ell)=1\) (Theorem 2.5). The author proves that \(\varinjlim \text{ Gr}_{k,k\ell}\simeq \text{ BSU}\) (Theorem 2.10), and finally shows that \(\varinjlim \text{ Gr}_{k,k\ell}\) is isomorphic to \(\text{ BSU}_\otimes\) as an \(H\)-space (Theorem 2.17). Here \(\text{ BSU}_\otimes\) denotes the \(\text{ BSU}\) with structure of \(H\)-space induced by the tensor product of virtual \(\text{ SU}\)-bundles of virtual dimension 1.