Abstract:$C^*$-subalgebra of the algebra of all bounded operators on the Hilbert space $l^2$ generated by the multiplier algebra and a family of partial isometries satisfying some relations is covered in the paper. The ideals of the algebra under study, as well as the ideals of the quotient algebra, are considered over compact operators. It is demonstrated that the quotient algebra can be represented as a direct sum of two principal ideals and has nontrivial center.
Keywords:$C^*$-algebra, partial isometry, principal ideal, central projection, Calkin algebra.
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