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F / 2 Y . G/invariant. f /. s/Á/ D . s 1 /f /Á 2 Y . fV /. s/Á/ 2 Y , since Y is closed. 2. This is an immediate consequence of 1. 7 (Schur’s Lemma). Let . ; H/ be a unitary representation of a locally compact group G. The following conditions are equivalent: 1. is irreducible. 2. Every nonzero vector in H is cyclic. 3. g/; g 2 G, is a scalar multiple of the identity. Proof. 1 ” 2. Suppose that is irreducible. Let 2 H be nonzero. g/ ; g 2 G, is G-invariant and hence dense in H, since is irreducible.
GC / / \ g: Evidently g D g for every . 39. 3) Œg; g for every Proof. gC / . ad/. gC / , so ŒgC ; gC : We have the following bracket relation. 40. Let . ; V / be a finite-dimensional g-module. ad/ we have that T . g0 /V all 2 T . /. 1 Lie Groups and Lie Algebras 19 Proof. gC / . Vi /i of V and let j be the index for which v 2 Vj nVj C1 . T /. X /v/ . X /. X /. T /X X /v mod gj C1 : v/ v/ This shows that inductively on i , . T / . C /IdV /i . v/ mod gjC1 : Hence for i big enough, it follows that .
H/ of unitary operators on H such that for any x 2 H the mapping G 3 g 7! x/ 2 H is continuous relative to the topology of H. The Hilbert space H is called the representation space of , and we agree to denote it by H. / or H . According to the dimension of H. /, is said to be finite- or infinite-dimensional. 8. An invariant subspace of means a subspace L of H. L/ L for any g 2 G. When there is no closed invariant subspace other than f0g and H. /, is said to be irreducible. 9. Let ; be two unitary representations of G.
C-star-algebras. Hilbert Spaces by Unknown Author