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Topological and algebraic properties of spaces of Lorch analytic mappings
Author(s) -
Mauro Guilherme V. S.,
Moraes Luiza A.,
Pereira Alex F.
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400108
Subject(s) - mathematics , topology (electrical circuits) , simple (philosophy) , topological space , space (punctuation) , connection (principal bundle) , topological algebra , spectrum (functional analysis) , algebraic number , banach algebra , algebraic topology , pure mathematics , banach space , algebra over a field , mathematical analysis , combinatorics , computer science , geometry , philosophy , physics , epistemology , quantum mechanics , homotopy , operating system
For a complex Banach algebra E , letH L ( U , E )be the space of the mappings from a connected subset U of E into E that are analytic in the sense of Lorch. We consider the spaceH L ( U , E )endowed with a convenient topology τ d which coincides with the topology τ b when U = E or U = B r ( z 0 ) = { z ∈ E ; ∥ z − z 0 ∥ < r }( z 0 ∈ E , r > 0 )and we study topological properties of ( H L ( U , E ) , τ d ) in connection with topological properties of the underlying space E . A description of the spectrum of ( H L ( B r ( z 0 ) , E ) , τ b ) is given and as a consequence, it is showed that the algebraH L ( B r ( z 0 ) , E )is semi‐simple if and only if E is semi‐simple.