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Almost periodic operators in 𝑉𝑁(𝐺)

1990; American Mathematical Society; Volume: 317; Issue: 1 Linguagem: Inglês

10.1090/s0002-9947-1990-0943301-9

1088-6850

Ching Chou,

Spectral Theory in Mathematical Physics

Let G G be a locally compact group, A ( G ) A(G) the Fourier algebra of G G , B ( G ) B(G) the Fourier-Stieltjes algebra of G G and VN ( G ) {\text {VN}}(G) the von Neumann algebra generated by the left regular representation λ \lambda of G G . Then A ( G ) A(G) is the predual of VN ( G ) {\text {VN}}(G) ; VN ( G ) {\text {VN}}(G) is a B ( G ) B(G) -module and A ( G ) A(G) is a closed ideal of B ( G ) B(G) . Let AP ( G ^ ) = { T ∈ VN ( G ) : u ↦ u ⋅ T {\text {AP}}(\hat G) = \{ T \in {\text {VN}}(G):u \mapsto u \cdot T is a compact operator from A ( G ) A(G) into VN ( G ) } {\text {VN}}(G)\} , the space of almost periodic operators in VN ( G ) {\text {VN}}(G) . Let C δ ∗ ( G ) C_\delta ^*(G) be the C ∗ {C^*} -algebra generated by { λ ( x ) : x ∈ G } \{ \lambda (x):x \in G\} . Then C δ ∗ ( G ) ⊂ AP ( G ^ ) C_\delta ^*(G) \subset {\text {AP}}(\hat G) . For a compact G G , let E E be the rank one operator on L 2 ( G ) {L^2}(G) that sends h ∈ L 2 ( G ) h \in {L^2}(G) to the constant function ∫ h ( x ) d x \int {h(x)dx} . We have the following results: (1) There exists a compact group G G such that E ∈ AP ( G ^ ) ∖ C δ ∗ ( G ) E \in \text {AP}(\hat G)\backslash C_\delta ^*(G) . (2) For a compact Lie group G G , E ∈ AP( G ^ ) ⇔ E ∈ C δ ∗ ( G ) ⇔ L ∞ ( G ) E \in {\text {AP(}}\hat G{\text {)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G) has a unique left invariant mean ⇔ G \Leftrightarrow G is semisimple. (3) If G G is an extension of a locally compact abelian group by an amenable discrete group then AP ( G ^ ) = C δ ∗ ( G ) {\text {AP}}(\hat G) = C_\delta ^*(G) . (4) Let G = F r G = {{\mathbf {F}}_r} , the free group with r r generators, 1 > r > ∞ 1 > r > \infty . If T ∈ VN ( G ) T \in {\text {VN}}(G) and u ↦ u ⋅ T u \mapsto u \cdot T is a compact operator from B ( G ) B(G) into VN ( G ) {\text {VN}}(G) then T ∈ C δ ∗ ( G ) T \in C_\delta ^*(G) .