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	:<math>\ell^1(S)	\{arphi:S ightarrow \mathbb{C}: \arphi\ = \sum_{x \in S}arphi(x) < \infty\}</math>

''l''¹(''S'') has a multiplicative identity, viz, the function δ_''e'' which is zero except at ''e'', where it takes the value 1. It has the involution

• (x) = \overline{arphi(x^---)}

• -algebra with identity.

• -algebra of contractions generated by ''S'' is the C

  ---

  enveloping algebra of ''l''¹(''S''). We can describe it as follows: For every state ''f'' of ''l''¹(''S''), consider the Cyclic Representation π_''f'' associated to ''f''. Then





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