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A three-peak analysis of Ni(lll) photoelectron spectra, taken with high angular and energy resolution at 7.7 eV for 300 K^T ^ 1.25T C , is performed with the assumption of a temperature-independent exchange splitting and a many-body line shape. The intensity of the nonmagnetic peak between the two magnetic ones increases but its width decreases as T c is approached from below, while the intensity of the magnetic peaks decreases but remains finite. This strongly supports the notion that short-range magnetic order persists well above T c .PACS numbers: 79.60.Cn Angle-resolved photoelectron spectroscopy (ARPES) started to be useful for the investigation of the magnetic properties of Ni when a doublet structure produced by the exchange splitting A x of states at the top of the 3d band was identified in the (111) spectra at 300 K by three of the present authors. 1 Our results obtained for the photon energy co =10.2 eV were later confirmed by more refined experiments. 2 " 5 It also became clear that a similar structure with a larger splitting observed in the spectrum 1 at a; =16.8 eV originates from two different states in the d band and not from the exchange splitting. 3 The intensity variation of the structure observed in the ARPES spectra with o>, which is very similar for Cu 6 and Ni, 7 shows that a single exchange-split 3d state is only obtained for co < 13.5 eV. The exchange splitting derived from ARPES agrees well with the value needed as a fit parameter in calculations reproducing the observed oo dependence of the photoelectron spin polarization, 8 while selfconsistent calculations 9 give about twice that value. This discrepancy may be due to the inadequacy of the density-functional theory for strongly correlated systems as well as to final-state effects of the photoexcitation process. 10 In this paper we present new ARPES spectra containing the exchange doublet for temperatures in the range 300 K< T < 1.25T C , taken with a considerably higher resolution than that obtained in earlier experiments, 2 ' 3 together with a new lineshape analysis. This enables us to attack directly the basic problem concerning the magnetism of Ni, namely, to find a model which adequately describes the curious mixture of itinerant and local features displayed by this metal. For example, the nonintegral magneton number and the existence of A x are characteristic of a Stoner model, while the fact that kT c « A x contradicts it. On the other hand, the validity of the Curie-Weiss law for the susceptibility above T c is characteristic of a Heisenberg model. To make matters worse, the existence of spin waves above T c is alien to both models. 11 ' 12 The photoelectron spectrometer is basically the one described earlier, 13 except that we now use a high-intensity hollow-cathode H 2 discharge as the light source and a new electron mirror monochromator. 14 The angular and energy resolution of the spectrometer is 3° and 40 meV full width at half maximum for the spectra shown in Fig. 1. The geometrical configuration is shown as an inse...