Amorphous alumina (Al2O3) was produced by a sol-gel technique in order to make available its optical constants for possible astrophysical applications. Gradual annealing showed that the X-ray amorphousness of alumina ended somewhere between 723 and 873 K. Above this transition point, the structure changes into disordered γ-Al2O3. At T > 1273 K, crystalline α-Al2O3 (corundum) is formed.Mie calculations show that amorphous alumina exhibits a wide Al-O vibrational band, peaking at 11.5-11.8 μm and having a steep ``blue'' and an extended ``red'' wing. It may be an important contributor to the continuous opacity between the silicate bands in oxygen-rich circumstellar envelopes, whereas it is ruled out for the explanation of the 13 μm band. An average 13 μm band profile was derived from 51 IRAS low-resolution spectra of bright Mira stars and semiregular variables. Its shape, which is satisfactorily represented by a Lorentz profile, can be reproduced by Mie calculations with the data of α-Al2O3, but not with those of γ-Al2O3. The calculations show that the 13 μm band profile of α-Al2O3 is sensitive to grain shape. If α-Al2O3 is the absorber, a second band should be present at 21 μm. A close correlation was found between the strengths of the 13 μm band and the 10 μm silicate band. It suggests that the 13 μm band carrier could also be somehow connected with silicate dust. Experimental arguments supporting this attribution are presented.