Infinite dimensional Schur

Let A be an algebra over \mathbb{C} and let V be an irreducible representation of A with at most countable basis. Then any homomorphism of representations \phi:V\to V is a scalar operator.

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Infinite dimensional Schur

Your degree is a chance for expansion

Let X and Y be compact oriented smooth manifolds of the same dimension, with Y connected. Let f : X \to Y be a smooth function. Take y \in Y a regular value of f and, for x \in f^{-1}(y), define o(x) as +1 if f preserves orientation at x and as -1 if it reverses the orientation. We define the degree of f as the number d(f) = \sum_{x\in f^{-1}(y)}o(x).

Prove the following extension theorem: Assume Y=S^k is a k-dimensional sphere and X=\partial W is the boundary of a compact oriented (k+1)-manifold W. There exists a smooth extension F:W\to Y of f if and only if d(f)=0.

Your degree is a chance for expansion