An $H_\infty$-spectrum (Bruner-May-McLure.Steinberger 86) is a ring spectrum (i.e. a monoid object in the stable homotopy category) equipped with extra structure modelling extended power operations.
R. Bruner, Peter May, James McClure, M. Steinberger, $H_\infty$-Ring Spectra and their Applications, Lecture Notes in Mathematics 1176, Springer 1986 (pdf)
Justin Noel, H-infinity is not E-infinity, Proceedings of the 4th Arolla Conference (arXiv:0910.3566)
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