The "twisted convolution" associated with the Weyl form of the canonical commutation relations for n degrees of freedom is decribed using ordinary convolution on a nilpotent central extension of additive phase space by the onedimensional torus. Twisted convolution determines several O*-algebras of quantum mechanical observables amongst which we study especially the algebra £f 2 (&> o) consisting of the oδf 2 -functions on phase space and mapped isometrically onto the Hilbert-Schmidt-operators by the Schrόdinger representation. The two last sections of the paper deal with "phase space quantum mechanics" from the point of view of twisted convolution: the WIGNER-MOYAL formalism and the entire function formalism of BABGMANN and SEGAL.
A n.c.JB*-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky forC*-algebras. Different characterizations of associativity are given.
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