By Michael Spivak

Publication via Michael Spivak, Spivak, Michael

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We are now ready to derive a matrix version of a Wiener-Levy type theorem. We note that each f ∈ A(G, Mn ) vanishes at inﬁnity and therefore the closure of its range in Mn contains 0 if G is not compact. 22. Let f ∈ A(G, Mn ) and let U be an open subset of Mn containing the closure f (G). Then for any holomorphic map F : U −→ Mn satisfying F(0) = 0 if G is non-compact, there exists a function ϕ ∈ A(G, Mn ) such that ϕ (γ ) = F( f (γ )) (γ ∈ G). We will denote ϕ by F( f ). Proof. We need to show that the function F ◦ f : G −→ Mn belongs to A(G, Mn ).

Proof. Let σ = f · λ for some f ∈ L1 (G, Mn ). Let 1 (1 Then we have 1 f )(x) = IMn ⊗ f (x) f ∈ L1 (G, Mn2 ). f : G −→ Mn2 be the function (x ∈ G). 3 Spectral Theory 57 Consider the C*-dynamical system (Mn2 , G, ι ) in which Mn2 acts on the Hilbert 2 space Cn and the action ι is trivial.

X ∈ G. Therefore ⎛ det σ T ⎜ .. fT ∗⎝ . ⎞ ⎟ T T T ⎠ = f ∗ σ ∗ Adj σ = 0 det σ T which gives fiTj ∗ det σ = 0 where fiTj = f ji ∈ L∞ (G). We can apply the equivalence (i) ⇔ (iii) to det σ and L∞ (G). This implies that f ji is constant for all i, j. Hence f is constant. 15. In the above lemma, (i) =⇒ (iii) can be extended to non-abelian groups in which case, (i) or (ii) implies that 0 is not an eigenvalue of σ (π ) for each π ∈ G\{ι }. However, (iii) =⇒ (i) fails for non-abelian groups. Let σ be an adapted probability measure on a non-amenable group G.