By Dieudonne J.

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The set of equivalence classes of ∼ is denoted M(C(X)). 11 is an example of a direct limit; we often indicate this by writing M(C(X)) = lim M(n, C(X)). −→ We will usually identify M(n, C(X)) with its image in M(C(X)). We can view an element of M(C(X)) as a countably inﬁnite matrix with entries in C(X) and all but ﬁnitely many entries equal to 0. 12 Let X be compact Hausdorﬀ. Then M(C(X)) is an algebra without unit under the operations of matrix addition and multiplication and scalar multiplication.

Idempotents E and F in M(n, C(X)) are algebraically equivalent if there exist elements A and B in M(n, C(X)) such that AB = E and BA = F. 48 Preliminaries (a) Show that A and B can be chosen so that A = EA = AF = EAF and B = FB = BE = FBE. (b) Prove that algebraic equivalence deﬁnes an equivalence relation ∼a on the idempotents in M(n, C(X)). (c) Prove that E ∼s F implies that E ∼a F. (d) If E, F in M(n, C(X)) are algebraically equivalent, prove that diag(E, 0n ) is similar to diag(F, 0n ) in M(2n, C(X)).

Therefore ψ(a1 )+ψ(b2 )+ψ(c) = ψ(a2 )+ψ(b1 )+ψ(c), and so ψ(a1 −a2 ) = ψ(a1 )−ψ(a2 ) = ψ(b1 )−ψ(b2 ) = ψ(b1 − b2 ). By construction, the map ψ is a group homomorphism and is unique because it is determined by ψ. 7 Vect(X) vs. Idem(C(X)) 31 a group G(A) and a monoid homomorphism i : A −→ G(A) with the properties listed in the statement of the theorem. The map i is a monoid homomorphism, so there exists a group homomorphism i : G(A) −→ G(A) such that ij = i. Next, because j : A −→ G(A) is a monoid homomorphism, there exists a group homomorphism j : G(A) −→ G(A) such that ji = j.