In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when / is holomorphic on a larger sector.We also examine how certain properties of this functional calculus, such as the existence of a bounded H°° functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive L p space, then T has a bounded H°° functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.1991 Mathematics subject classification (Amer. Math. Soc): 47A60.

The Fourier algebra A(G) of a locally compact group G is the space of matrix coefficients of the regular representation, and is the predual of the yon Neumann algebra VN(G) generated by the regular representation of G on L 2 (G). A multiplier m of A (G) is a bounded operator on A (G) given by pointwise multiplication by a function on G, also denoted m. We say m is a completely bounded multiplier ofA (G) if the transposed operator on VN(G) is completely bounded (definition below). It may be possible to find a net ofA (G)-functions, (m i : ie I) say, such that mi tends to 1 uniformly on compacta, and, for some L in IR+, ][millMo<=L(l[ IIMo being the completely bounded operator norm). We define A~ to be the infimum of all values of L, as we consider all possible nets of this type ; in particular A~ is set equal to + oo if there is no such net. In this paper, we calculate A~ for all non-compact real-rankone simple Lie groups with finite center: If G is locally isomorphic to SO(l,n) or S U(I, n) (where n > 2), then A~ = 1 ; if G is locally isomorphic to Sp (1, n) (with n > 2), then Aa = 2n -1, and ifG is locally isomorphic to the exceptional Lie group F4(_ 2o ), then AG=21. The second-named author [16] has shown that if G is simple and of real rank greater than one, then A~ = + oo ; he has also shown, that ifFis a lattice in G, then A G = Ar, and that the von Neumann algebras of lattices F and F' contained in the Lie groups G and G' cannot be isomorphic unless A~ = Aa,. Consequently, if F and F' are lattices in Sp(l,n) and Sp(l,n') respectively and n4:n', then the von Neumann algebras of the two lattices are not isomorphic.

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We prove a Hörmander-type spectral multiplier theorem for a sublaplacian on SU(2), with critical index determined by the Euclidean dimension of the group. This result is the analogue for SU(2) of the result for the Heisenberg group obtained by D. Müller and E.M. Stein and by W. Hebisch.

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