On the finite dimensionality of every irreducible unitary representation of a compact group

Autor:	Leopoldo Nachbin
Rok vydání:	1961
Předmět:	Pure mathematics Unitary representation Induced representation Applied Mathematics General Mathematics Restricted representation Unitary group Trivial representation Peter–Weyl theorem Compact quantum group Representation theory of finite groups Mathematics
Zdroj:	Proceedings of the American Mathematical Society. 12:11-12
ISSN:	1088-6826 0002-9939
DOI:	10.1090/s0002-9939-1961-0123197-5
Popis:	We shall prove that every irreducible unitary representation of a compact group is finite dimensional. Our argument is a variation of known proofs and it hardly could be based on an idea different from those already current. It makes no use of the Peter-Weyl theorem and of compact or Hilbert-Schmidt operators and seems simpler than the proofs in [1; 2; 3; 4]. Its crucial point is that the prospectively finite dimension of the representation Hilbert space is expressible by a known integral formula. Let 3e 5 0 be a Hilbert space and x-> U. be a group homomorphism of a compact group G into the group U(aO) of all unitary operators in 30, such that the scalar product (t| Ux,q) is a continuous function of x G for all (, CJe. Suppose that this representation is irreducible, namely that there is no closed vector subspace of 30 invariant under all U., except the trivial ones 0 and 30. Then there results that 30 is finite dimensional. In fact, let (, 71, (', 1'Cj3. Denoting complex conjugation by a star, since
Databáze:	OpenAIRE
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