完备矩阵代数Ⅰ——乘法连续性问题

<正> 完备空间与完备矩阵环是Kothe.G.于1934年引入的,近年来由于在分析中最有用的一类线性拓扑空间——核空间的种种研究的影响,又给Kothe理论带来了许多新的进展,出现了大量文章.然而,至今为止还未见到有把完备矩阵环当作拓扑代数来加以探讨的工作,为此,我们准备着手对这样一类显然有其重要性的拓扑代数进行详细的讨论.

PERFECT MATRIX ALGEBRAS I——The problem of multiplieative eontinuity——

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.