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Gleason parts for algebras of holomorphic functions in infinite dimensions
Revista Matemática Complutense ( IF 0.8 ) Pub Date : 2019-09-23 , DOI: 10.1007/s13163-019-00324-z
Richard M. Aron , Verónica Dimant , Silvia Lassalle , Manuel Maestre

For a complex Banach space X with open unit ball \(B_X,\) consider the Banach algebras \(\mathcal {H}^\infty (B_X)\) of bounded scalar-valued holomorphic functions and the subalgebra \(\mathcal {A}_u(B_X)\) of uniformly continuous functions on \(B_X.\) Denoting either algebra by \(\mathcal {A},\) we study the Gleason parts of the set of scalar-valued homomorphisms \(\mathcal {M}(\mathcal {A})\) on \(\mathcal {A}.\) Following remarks on the general situation, we focus on the case \(X = c_0,\) giving a complete characterization of the Gleason parts of \(\mathcal {M}(\mathcal {A}_u(B_{c_0}))\) and, among other things, showing that every fiber in \(\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))\) over a point in \(B_{\ell _\infty }\) contains \(2^c\) discs lying in different Gleason parts.

中文翻译:

无限维全纯函数代数的Gleason零件

对于具有开放单位球\(B_X,\)的复杂Banach空间X,请考虑带界标量值全纯函数的Banach代数\(\ mathcal {H} ^ \ infty(B_X)\)和子代数\(\ mathcal { \(B_X。\)上的连续连续函数的A} _u(B_X)\)\(\ mathcal {A},\)表示任一代数,我们研究了标量值同态\(\ mathcal \(\ mathcal {A}。\)上的{M}(\ mathcal {A})\)在对一般情况进行说明之后,我们集中讨论\(X = c_0,\)的情况,从而完整描述了格里森的零件\(\ mathcal {M}(\ mathcal {A} _U(B_ {C_0}))\)并且除其他外,显示\(\ mathcal {M}(\ mathcal {H} ^ \ infty(B_ {c_0}))\)中的每根光纤在\(B _ {\ ell _ \ infty}中的一点上\)包含位于不同格里森零件中的\(2 ^ c \)光盘。
更新日期:2019-09-23
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