The author deals with the so-called local multiplier algebra \(M_{\text{loc}}(A)\) of a \(C^*\)-algebra \(A\). Recall that a closed two-sided ideal \(I\) in \(A\) is essential if its annihilator is equal to zero. The intersection of two essential ideals is essential, so the set of essential ideals of \(A\) forms a directed set. If \(I\) and \(J\) are essential ideals with \(I\supseteq J\), then there is an injective homomorphism from \(M(I)\), the multiplier algebra of \(I\), into \(M(J)\). Thus the set of multiplier algebras of essential ideals is a direct set. By definition, \(M_{\text{loc}}(A)\) is the closure of the algebraic inductive limit of this system; so \(M_{\text{loc}}(A)\) is a \(C^*\)-algebra. For an arbitrary \(C^*\)-algebra \(A\) let \(\text{Prim}(A)\) denote the space of primitive ideals, \(\text{MinPrimal}(A)\) the space of minimal closed primal ideals, and \(\text{MinSpec}(A)\) the space of minimal (not necessarily closed) primal ideals of \(A\). These spaces are equipped with hull-kernel topologies. Let Glimm \((A)\) denote the set of Glimm ideals equipped with the strongest topology for which the natural surjection \(\text{Prim}(A)\to\text{Glimm}(A)\) is continuous. Recall that \(C^*\)-algebras with extremally disconnected primitive space (i.e., disjoint open subsets have disjoint closures) are said to be boundedly centrally closed. In this paper the following basic results are proved: Theorem 1. A \(C^*\)-algebra \(A\) is boundedly centrally closed if and only if \(A\) is quasi-standard and \(\text{Glimm}(A)\) is extremally disconnected. -- Theorem 2. Let \(A\) be a boundedly centrally closed \(C^*\)-algebra. Then \(\text{Glimm}(M(A))\) is homeomorphic to \(\text{MinSpec}(M(A))\), where \(M(A)\) is the multiplier algebra of \(M\). -- Theorem 3. If \(A\) is a \(C^*\)-algebra, then \(M_{\text{loc}}(A)\) is standard. -- Corollary. If \(A\) is a \(C^*\)-algebra, then each minimal primal ideal of \(M_{\text{loc}}(A)\) is prime. -- Theorem 4. Let \(A\) be a \(C^*\)-algebra with \(\text{MinPrimal}(A)\) separable and set \(B=M_{\text{loc}}(A)\). The mapping \(P\mapsto P\cap A\) \((P\in\text{MinSpec}(B))\) is a homeomorphism from \(\text{MinSpec}(B)\) onto \(\text{MinSpec}(A)\).