In this paper, we introduce three topologies in the perfect matrix algebras ∑(λ). The neighborhood systems. V,(θ,N,M,ε), V_k(θ,N,M,ε), V_ω(θ,N,M,ε) of the point θ ∈ ∑(λ) are respectively called strong, k-and weak topologies, where ε > 0 and N, M are respectively the bounded, weak compact and finite sets of λ, λ. Obviously, ∑(λ) are locally convex algebras relative to the above three topologies.Our principal results are as follows:Theorem 1. The following propositions are equivalent:1°In ∑(λ), multiplication is eontinuous relative to the strong topology;°In λ there exists the bounded set N_o, whieh absorbs any bounded set N of λ, i.e. N a N_o for some a > 0;3°λ is a Banaeh space relative to the strong topology;4°∑(λ) is a Banach algebra relative to the strong topology;5°∑(λ) is a m-convex algebra relative to the strong topotogy, i.e. in ∑(λ) there exists a base for the strong neighborhood system {V_s} of the point θ∈∑(λ) satisfying V_s.V_s V_s.Corollary 1. If ∑() is not a Banach algebra relative to the strong topology, then it is also not a B_o-algebra.Corollary 2. If λ is perfect and convergence-free, then ∑(λ) is multiplicatively discontinuous and it is not a B_o-algebra relative to the strong topology.Corollary 3. If "gestufen" space λ is generated by the enumerable, non-negative and increasing "stufen" system {B~((n))}, then ∑(λ) is multiplieatively discontinuous and it is not a B_o-algebra relative to the strong topology.Theorem 2. The following propositions are equivalent:1°In ∑(λ), multiplication is continuous relative to the k-topology;2°λ and λ are sequentially separable, and in λ there exists the bounded set N', which absorbs any bounded set N of λ;3°In ∑(λ), multiplication is continuous relative to the strong topology which is equivalent to the k-topology.Theorem 3. In ∑(λ), multiplication is impossible to be eontinous relative to the weak topology.
