FoldUnfold Table of Contents The Minkowski (Gauge) Functional The Minkowski (Gauge) Functional Recall from the Absorbent Sets page that if $X$ is a linear space and $E \subseteq X$ then $E$ is said to be absorbent if for every $x \in X$ there exists a $\lambda > 0$ such that $\lambda x \in E$. Given an absorbent subset $U$ of a linear space $X$ we can define a special type … [Read more...]

## Absorbent Sets

FoldUnfold Table of Contents Absorbent Sets Absorbent Sets Definition: Let $X$ be a linear space and let $E \subseteq X$. Then $E$ is said to be Absorbent or an Absorbing Set if for every $x \in X$ there exists a $\lambda > 0$ such that $\lambda x \in E$. Proposition 1: Let $X$ be a seminormed linear space. Then the open unit ball $B(0, 1) = \{ x \in X : p(x) and the … [Read more...]

## Absolutely Convex Sets

FoldUnfold Table of Contents Absolutely Convex Sets Example 1 Absolutely Convex Sets Definition: Let $X$ be a linear space and let $E \subseteq X$. Then $E$ is said to be Absolutely Convex if whenever $x, y \in E$ and $a, b \in \mathbf{F}$ are such that $|a| + |b| \leq 1$ then $(ax + by) \in E$. Proposition 1: Let $X$ be a linear space and let $E \subseteq X$. If $E$ is … [Read more...]

## Amenability of Abelian Groups

FoldUnfold Table of Contents Amenability of Abelian Groups Amenability of Abelian Groups Theorem 1: Let $G$ be an abelian group. Then $G$ is an amenable group. Recall that a group is said to be abelian (or commutative) if for all $g, h \in G$ we have that $gh = hg$. Proof: Let $A = \ell^1(G)$ and let $\sigma : A \to \mathbb{C}$ be defined for all $a \in A$ … [Read more...]

## ℓ^1(G) is Amenable if and only if G is an Amenable Group 2

FoldUnfold Table of Contents ℓ^1(G) is Amenable if and only if G is an Amenable Group 2 ℓ^1(G) is Amenable if and only if G is an Amenable Group 2 Theorem 1: Let $G$ be a group. Then the group algebra $\ell^1(G)$ is amenable if and only if the group $G$ is an amenable group. $\Leftarrow$ Let $G$ be an amenable group. Then there exists a positive invariant mean $\mu$ on $G$, … [Read more...]

## ℓ^1(G) is Amenable if and only if G is an Amenable Group 1

FoldUnfold Table of Contents ℓ^1(G) is Amenable if and only if G is an Amenable Group 1 ℓ^1(G) is Amenable if and only if G is an Amenable Group 1 Theorem 1: Let $G$ be a group. Then the group algebra $\ell^1(G)$ is amenable if and only if the group $G$ is an amenable group. Recall that if $G$ is a group then $\ell^1(G)$ is the Banach algebra of all complex-valued functions … [Read more...]

## Amenable Groups and Amenable Banach Algebras

FoldUnfold Table of Contents Amenable Groups and Amenable Banach Algebras Amenable Groups Amenable Banach Algebras Amenable Groups and Amenable Banach Algebras Amenable Groups Definition: Let $G$ be a group. An Invariant Mean of $\ell^{\infty}(G)$ is a positive linear functional $\mu : \ell^(G) \to \mathbf{R}$ with the following properties:a) $\mu (1) = 1$.b) If for each $h … [Read more...]

## Inner Bounded X-Derivations, B1(A, X)

FoldUnfold Table of Contents Inner Bounded X-Derivations, B1(A, X) Inner Bounded X-Derivations, B1(A, X) Recall from the Bounded X-Derivations, Z^1(A, X) page that if $\mathfrak{A}$ is a Banach algebra and $X$ is a Banach $\mathfrak{A}$-bimodule then a bounded linear operator $D : \mathfrak{A} \to X$ is said to be a bounded $X$-derivation if for all $a, b \in \mathfrak{A}$ … [Read more...]

## Bounded X-Derivations, Z1(A, X)

FoldUnfold Table of Contents Bounded X-Derivations, Z1(A, X) Bounded X-Derivations, Z1(A, X) Definition: Let $\mathfrak{A}$ be a Banach algebra, let $X$ a Banach $\mathfrak{A}$-bimodule. A Bounded $X$-Derivation is a bounded linear map $D : \mathfrak{A} \to X$ such that $D(ab) = D(a)b + aD(b)$ for all $a, b \in \mathfrak{A}$. The Set of All Bounded $X$-Derivations is … [Read more...]

## For Semi-Simple Commutative Banach Algebras A and B – A⊗B is Semi-Simple

FoldUnfold Table of Contents For Semi-Simple Commutative Banach Algebras A and B - A⊗B is Semi-Simple For Semi-Simple Commutative Banach Algebras A and B - A⊗B is Semi-Simple Proposition 1: Let $\mathfrak{A}$ and $\mathfrak{B}$ be semi-simple commutative Banach algebras over $\mathbb{C}$. Then $\mathfrak{A} \otimes \mathfrak{B}$ is semi-simple. Proof: Let $f \in … [Read more...]