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SUMMARY:Willian Franca (Federal University of Juiz de Fora\, Brazil)
DTSTART:20260330T150000Z
DTEND:20260330T160000Z
DTSTAMP:20260423T021152Z
UID:ENAAS/167
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ENAAS/167/">
 Applications of compact multipliers to algebrability of $(\\ell_{\\infty}\
 \setminus c_0)\\cup\\{0\\}$ and $(B(\\ell_2(\\mathbb{N}))\\setminus K(\\el
 l_2(\\mathbb{N}))\\cup \\{ 0\\}.$</a>\nby Willian Franca (Federal Universi
 ty of Juiz de Fora\, Brazil) as part of European Non-Associative Algebra S
 eminar\n\n\nAbstract\nIn present talk we deal with the class $\\mathcal{C}
 =\\mathcal{C}_1\\cup \\mathcal{C}_2$ where   $\\mathcal{C}_1$  (respective
 ly\,  $\\mathcal{C}_2$) is formed by all separable Uniform algebras  (resp
 ectively\, separable commutative  C$^*$-algebras) with no compact elements
 .  For a given algebra  $A$ in $\\mathcal{C}_1$ (respectively\, $A$ in $\\
 mathcal{C}_2$) we will show that $A$ is  isometrically isomorphic as algeb
 ra (respectively\, as C$^*$-algebra) to a subalgebra $M$ of $\\ell_{\\inft
 y}$ with $M\\subset (\\ell_{\\infty}\\setminus c_0)\\cup\\{0\\}.$ Under th
 e additional assumption that $A$ is non-unital we verify that  there exist
 s a copy of $M(A)$ (the multipliers algebra of $A$ which is non-separable)
  inside $(\\ell_{\\infty}\\setminus c_0)\\cup\\{0\\}$.\n\nFor an infinitel
 y generated abelian C$^*$-algebra $B\,$ we will study the least cardinalit
 y possible of a system of generators ($\\gn_{C^*}(B)$). In fact we will de
 duce that $\\gn_{C^*}(B)$ coincides with the smallest cardinal number  $n$
  such that an embedding of  $\\Delta(B)$ (= the spectrum of $B$)  in $\\ma
 thbb{R}^n$ exists - The finitely generated version of this result was  pro
 ved by Nagisa.\nIn addition\, we will introduce new concepts of algebrabil
 ity  in terms of $\\gn_{C^*}(B)$ ($(C^*)$-genalgebrability) and its natura
 l variations.\n\nFrom our methods we will infer that there is $^*$-isomorp
 hic copy of $\\ell_{\\infty}$ in $(\\ell_{\\infty}\\setminus c_0)\\cup\\{0
 \\}$. In particular\, $(\\ell_{\\infty}\\setminus c_0)\\cup\\{0\\}$ contai
 ns a copy of  every separable Banach space.\nMoreover\, all the positive a
 nswers of this work holds if we replace the set $(\\ell_{\\infty}\\setminu
 s c_0)\\cup\\{0\\}$ with $(B(\\ell_2(\\mathbb{N}))\\setminus K(\\ell_2(\\m
 athbb{N}))\\cup \\{ 0\\}.$\n\nThis is a joint work with Jorge J. Garcés (
 Universidad Politécnica de Madrid)\n
LOCATION:https://researchseminars.org/talk/ENAAS/167/
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