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Let $(D, \succeq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A Banach limit $\ell$ on $D$ is a linear functional that satisfies $$\sup_{d \in D} \inf_{c \succeq d} f(c) \leq \ell(f) \leq \inf_{d \in D} \sup_{c \succeq d} f(c)$$ for all $f \in B$.

Banach limits exist by the Hahn-Banach theorem. In fact, if we allow our functions on $D$ to take values in an abstract Dedekind complete ordered vector space, then the existence of Banach limits is equivalent to the Hahn-Banach theorem.

Question. Are there Banach limits that satisfy $$\ell(fg) = \ell(f)\ell(g)$$ for all $f,g \in B$? If so, does this require something stronger than the Hahn-Banach theorem?

Let $(D, \succeq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A Banach limit $\ell$ on $D$ is a linear functional that satisfies $$\sup_{d \in D} \inf_{c \succeq d} f(c) \leq \ell(f) \leq \inf_{d \in D} \sup_{c \succeq d} f(c)$$ for all $f \in B$.

Banach limits exist by the Hahn-Banach theorem. In fact, if we say instead that our functions on $D$ take values in an abstract Dedekind complete ordered vector space, then the existence of Banach limits is equivalent to the Hahn-Banach theorem. But I am primarily interested in the real-valued case.

Question. Are there Banach limits that satisfy $$\ell(fg) = \ell(f)\ell(g)$$ for all $f,g \in B$? If so, does this require anything stronger than the Hahn-Banach theorem?

Let $(D, \preceq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A Banach limit $\ell$ on $D$ is a linear functional that satisfies $$\liminf(f) \leq \ell(f) \leq \limsup (f)$$ for all $f \in B$.

Banach limits exist by the Hahn-Banach theorem. In fact, if we allow our functions on $D$ to take values in an abstract Dedekind complete ordered vector space, then the existence of Banach limits is equivalent to the Hahn-Banach theorem.

Question. Are there Banach limits that satisfy $$\ell(fg) = \ell(f)\ell(g)$$ for all $f,g \in B$? If so, does this require something stronger than the Hahn-Banach theorem?

Let $(D, \succeq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A Banach limit $\ell$ on $D$ is a linear functional that satisfies $$\sup_{d \in D} \inf_{c \succeq d} f(c) \leq \ell(f) \leq \inf_{d \in D} \sup_{c \succeq d} f(c)$$ for all $f \in B$.

Banach limits exist by the Hahn-Banach theorem. In fact, if we allow our functions on $D$ to take values in an abstract Dedekind complete ordered vector space, then the existence of Banach limits is equivalent to the Hahn-Banach theorem.

Question. Are there Banach limits that satisfy $$\ell(fg) = \ell(f)\ell(g)$$ for all $f,g \in B$? If so, does this require something stronger than the Hahn-Banach theorem?