We present the local-band theory of the temperature dependence of up-and down-spin Green's functions in itinerant-electron ferromagnets. The results, calculated for a temperature-independent, local, exchange field strength, and under the assumption of the existence of short-range magnetic order, agree with recent measurements of angle-, energy-, and spin-resolved photoemission in nickel. States which contribute to the magnetization, not seen in the measurements, remain split in energy even at T c . Split states at T c are also expected in iron.PACS numbers: 75.10.Lp, 79.60.CnAngle-, energy-, and spin-resolved photoemission (ARPES) ideally is interpreted as the annihilation of a bulk electron of definite energy, momentum, and spin (and the creation of a high-energy electron which escapes and is measured). It measures the spectral density of the single-hole Green's function. The spin of this hole acts as a probe of the magnetic condition of the solid through which it propagates.The beautiful experiments of Hopster et al l using ARPES to study the magnetic properties of nickel have thus generated much interest, especially as it was claimed that no existing picture of hightemperature itenerant-electron magnetism is consistent with the results. It is our purpose here to show that the "local-band picture" is so consistent. It remains to be seen whether the "alloy analogy" picture is also consistent, as that theory has not been worked out for realistic bands. 2 The local-band picture 3,4 stresses the existence of short-range magnetic order which remains on a scale of about 20 A ( = 27r/a) above T c . This allows spin-split energy bands to be quite well defined locally. The alloy analogy 5,6 idea is to neglect short-range order and assume atomically random mean spins, enabling an exploitation of the coherent potential approximation.The question is to discuss the propagation of a hole starting with definite spin through two such different magnetic media. The results depend on the parameters of the hole state. There are two limiting cases. Namely, suppose the hole moves fast enough that the exchange field it sees changes direction in a time short compared to that which it takes its spin to align itself to the exchange field. The spin of such fast holes in effect sees the average magnetization, i.e., there is a motional narrowing. The spectral density will peak at different energies for spin up and spin down with the energy difference between peaks proportional to the magnetization. The spin of slow holes, on the other hand, follows the local magnetization and remains in approximate alignment or antialignment with it. There will be two peaks in the spectral density of each spin whose weights are coupled to the bulk magnetization, but which are split by the local magnetization.Thus the critical parameter is Ha v/A, where A is the spin splitting of the hole and v its velocity.(The minimum value of v is Ha/m, with m the hole mass.) For nickel, the probing holes in the actual experiment have small mass and relatively small A le...