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Monogenic inverse semigroups and their \(C^*\)-algebras. (English) Zbl 0557.47027

Any inverse semigroup S may be viewed as an involutive semigroup of partial isometries on a Hilbert space \({\mathcal H}\). A preliminary classification of all monogenic (i.e. singly generated) inverse semigroups was given by L. M. Gluskin [Mat. Sb., Nov. Ser. 41(83), 23-36 (1957; Zbl 0208.030)]. Via the P. R. Halmos and L. J. Wallen theorem [J. Math. Mech. 19, 657-663 (1970; Zbl 0202.425)] which describes each \(T\in {\mathcal B}({\mathcal H})\) with \(T^ n\) a partial isometry for all \(n\in {\mathbb{N}}\), the complete list of monogenic inverse semigroups is readily described. The most interesting case is the free monogenic inverse semigroup \({\mathcal F}{\mathcal I}_ 1:\) two rather different generators for \({\mathcal F}{\mathcal I}_ 1\) are \(V\oplus V^*\) (where V is the usual unilateral shift) and \(\oplus^{\infty}_{n=2}J_ n\) (where \(J_ n\) is the finite shift with \(J^ n_ n=0).\)
There are two natural \(C^*\)-algebras associated with S, the enveloping \(C^*\)-algebra \(C^*(S)\) and the left regular representation \(C^*\)- algebra \(C^*_{\ell}(S)\). It is shown that \(C^*(S)=C^*_{\ell}(S)\) for each monogenic inverse semigroup and the \(C^*\)-structure is described in each case. For the case \(S={\mathcal F}{\mathcal I}_ 1\) it is shown that \(C^*(S)\) coincides with the \(C^*\)- algebra generated by the operator \(\oplus^{\infty}_{n=2}J_ n\) while \(C^*(S)\) strictly contains the \(C^*\)-algebra generated by the operator \(V\oplus V^*\). As a corollary it is noted that the star algebra \({\mathbb{C}}{\mathcal F}{\mathcal I}_ 1\) fails to have unique \(C^*\)-norm.

MSC:

47D03 Groups and semigroups of linear operators
46L05 General theory of \(C^*\)-algebras
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