On Bundles Related to Groupoids of Matrix Subalgebras

In the present paper we study some geometric objects and constructions related to groupoids of matrix subalgebras in a fixed matrix algebra. 1. The definition of groupoids In this paper we will freely use the notation from [3]. First recall some definitions and constructions [3]. 1.1. Groupoids Gk, l. Let Mkl(C) be the complex matrix algebra. Unital ∗-subalgebras isomorphic Mk(C) in some unital ∗-algebra A (in fact we deal with the case A = Mkl(C)) will be called k-subalgebras. Define the following category Ck, l. Its objects Ob(Ck, l) are k-subalgebras in the fixed Mkl(C), i.e. actually points of the matrix grassmannian Grk, l. For two objects Mk, α, Mk, β ∈ Ob(Ck, l) the set of morphisms MorCk, l(Mk, α, Mk, β) is just the space Homalg(Mk, α, Mk, β) of all unital ∗-homomorphisms of matrix algebras (i.e. actually isometric isomorphisms). Remark 1. Note that we do not fix an extension of such a homomorphism to an automorphism of the whole algebra Mkl(C), so it is not the action groupoid corresponding to the action of PU(kl) on Ob(Ck, l). It is interesting to note that if Gk, l would be an action groupoid for some topological group H acting on Gk, l, then H ≃ Frk, l (see [3]). This result follows from the homotopy equivalence BGk, l ≃ BPU(k) (see [3]) and the fact that for action groupoid G := X ⋊ H corresponding to an action of H on X the classifying space BG is homotopy equivalent to X× H EH [1]. Put G 0 k, l := Ob(Ck, l), Gk, l := ⋃ α, β∈Ob(Ck, l) MorCk, l(Mk, α, Mk, β). Clearly, Gk, l is a topological groupoid (in fact, even a Lie groupoid). As a topological space it can be represented as follows. Applying fiberwisely the functor Homalg(. . . , Mkl(C)) (see [3]) to the tautological Mk(C)-bundle Ak, l → Grk, l [3] we obtain the space Hk, l(Ak, l) which is exactly Gk, l. Being a groupoid, Gk, l has canonical morphisms: source and target s, t : Gk, l ⇉ G 0 k, l, composition m : Gk, l × t G s k, l Gk, l → Gk, l, identity e : G 0 k, l → Gk, l and inversion i : Gk, l → Gk, l. Let us describe first two of them in terms of topological spaces Grk, l ∼ G 0 k, l and Hk, l(Ak, l) ∼ Gk, l . The source morphism s : Hk, l(Ak, l) → Grk, l is just the bundle projection (recall that