论文标题
关于JB $^*$的乘数的严格拓扑 - 代数
On the strict topology of the multipliers of a JB$^*$-algebra
论文作者
论文摘要
我们在JB $^*$-代数的乘数代数上介绍了Jordan-Strict拓扑，这是一个概念，尽管在Jordan乘数进行了首次研究之后经过了四十年，但仍缺少这一概念。如果c $^*$ - 代数$ a $被视为JB $^*$ - 代数，则$ M（a）$的J-Strict拓扑恰恰是精心研究的C $^*$ - 严格的拓扑。我们证明，每个JB $^*$ - 代数$ \ Mathfrak {a} $在其乘数代数$ M（\ Mathfrak {a}）$中是J-strict tement，而后一个代数是J-STRICT完成的。我们表明，在JB $^*$之间，持续过滤的Jordan同构，三倍同态和正交性保留运算符 - 代数允许J-Trict连续扩展到乘数代数之间的相应类型的操作员。我们在jb $^*$ - 代数$ \ mathfrak {a} $的乘数代数上表征J-strict连续函数，我们确定$ m（\ mathfrak {a}）$的双重二键相对于J-STRICT对J-STRICT对J-STRICT topology Is IsmorphiC to Ismorphic to Ismorphic to Ismorphic to Ismorphic in Ismorphic to $ ismorphic to $ asmorphic to $ asmorphic。我们还提出了乘数代数的J-Strict拓扑的第一个应用，表明在额外的假设中，$ \ Mathfrak {a} $和$ \ Mathfrak {b} $是$σ$ - Unital Jb $^*$ - 代数 - 代数 - 每个过渡性的Jordan $^*$ - hommormorphism（相应地相差）从$ \ mathfrak {a} $保存运算符）到$ \ mathfrak {b} $承认延伸到过滤的J-STRICT连续jordan $^*$ - 同型 - 同型（分别是三倍同构或连续的矫正运算符），从$ m（\ mathfrak）上$ m（\ mathfrak {b}）$。
We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB$^*$-algebras admit J-strict continuous extensions to the corresponding type of operators between the multipliers algebras. We characterize J-strict continuous functionals on the multipliers algebra of a JB$^*$-algebra $\mathfrak{A}$, and we establish that the dual of $M(\mathfrak{A})$ with respect to the J-strict topology is isometrically isomorphic to $\mathfrak{A}^*$. We also present a first applications of the J-strict topology of the multipliers algebra, by showing that under the extra hypothesis that $\mathfrak{A}$ and $\mathfrak{B}$ are $σ$-unital JB$^*$-algebras, every surjective Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $\mathfrak{A}$ onto $\mathfrak{B}$ admits an extension to a surjective J-strict continuous Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $M(\mathfrak{A})$ onto $M(\mathfrak{B})$.
