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We study (von Neumann) regular $^*$-subalgebras of $B(H)$, which we callR$^*$-algebras. The class of R$^*$-algebras coincides with that of"E$^*$-algebras that are pre-C$^*$-algebras" in the sense of Z. Sz\H{u}cs andB. Tak\'acs. We give examples, properties and questions of R$^*$-algebras. Weobserve that the class of unital commutative R$^*$-algebras has a canonicalone-to-one correspondence with the class of Boolean algebras. This motivatesthe study of R$^*$-algebras as that of noncommutative Boolean algebras. Weexplain that seemingly unrelated topics of functional analysis, like AFC$^*$-algebras and incomplete inner product spaces, naturally arise in theinvestigation of R$^*$-algebras. We obtain a number of interesting results onR$^*$-algebras by applying various famous theorems in the literature.

On regular $^*$-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras