Abstract. Hilbert triple systems were introduced into mathematics because of the role they play in a certain part of infinite-dimensional geometry. Recently the theory of involutive $H^\ast $-triples emerged as a triple analogue of $H^\ast $-algebras which are a classical topic in the Banach algebra theory. Our intention is to compare those theories. Structure theorems for associative Hilbert triple systems and associative $H^\ast $-triples, as well as the Wedderburn type theorems for general triple systems of both types, suggest that there might be an intimate connection between these two theories. We prove that this is in fact the case. More explicitly, we give two ways of constructing a simple involutive $H^\ast $-triple from a simple Hilbert triple and then prove that every simple $H^\ast $-triple can be obtained from a simple Hilbert triple via one of the above mentioned constructions. These results can be applied in order to give structure theorems for alternative Hilbert triple system and $JH^\ast $-triples.

AMS Subject Classification (1991): 17A40, 17A60, 17C65, 17D05, 46H25, 46K15, 46K70

Keyword(s): Hilbert triple system, H^\ast, involutive-triple, Hilbert module, isomorphism pair, polarized triple, alternative triple, Jordan triple

Received June 2, 1994 and in revised from November 15, 1994. (Registered under 5713/2009.)