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For a non-reflexive Banach space      X X       and      w ∈  X ∗  ∗  w \in {X^{{\ast }{\ast }}}      , two families of subalgebras of      B ( X ) , Γ  w = { T ∈  B ( X ) | T ∗  ∗  w = k w for some k ∈  C }  B(X),\;{\Gamma _w} = \{ T \in B(X)|{T^{{\ast }{\ast }}}w = kw\;{\text {for some}}\;k \in \mathbb {C}{\text {\} }}      , and      Ω  w = { T ∈  B ( X ) | T ∗  ∗  w ∈  w ⊕  X ^  } {\Omega _w} = \{ T \in B(X)|{T^{{\ast }{\ast }}}w \in w \oplus \hat X\}       for      w ∈  X ∗  ∗  ∖  X ^  w \in {X^{{\ast }{\ast }}}\backslash \hat X       with      Ω  w = B ( X ) {\Omega _w} = B(X)       for      w ∈  X ^  w \in \hat X      , were defined originally by Wilansky. We consider      X = c 0 X = {c_0}       and      X = l 1 X = {l_1}       and investigate relationships between the subalgebras for different      w ∈  X ∗  ∗  w \in {X^{{\ast }{\ast }}}      . We prove in the case of      c 0 {c_0}       that, for      w ∈  X ∗  ∗  ∖  X ^  w \in {X^{{\ast }{\ast }}}\backslash \hat X      , all      Γ  w {\Gamma _w}      ’s are isomorphic and all      Ω  w {\Omega _w}       ’s are isomorphic. For      X = l 1 X = {l_1}      , where it is known that not all      Γ  w {\Gamma _w}      ’s are isomorphic and not all      Ω  w {\Omega _w}       ’s are isomorphic, we show, surprisingly, that subalgebras associated with a Dirac measure on      β  N ∖  N \beta \mathbb {N}\backslash \mathbb {N}      , regarded as a functional on      l 1 ∗  l_1^{\ast }      , are isomorphic to those associated with some Banach limit (i.e., a translation invariant extended limit). We also obtain a representation for the operators in the subalgebras      { ∩  Γ  f | f is a Banach limit } \{ \cap {\Gamma _f}|f\;{\text {is a Banach limit}}\}       and      { ∩  Ω  f | f is a Banach limit } \{ \cap {\Omega _f}|f\;{\text {is a Banach limit}}\}       of      B ( l 1 ) B({l_1})      .

On some subalgebras of 𝐵(𝑐₀) and 𝐵(𝑙₁)