Transitivity for linear operators on a Banach space

Let G be the multiplicative group of invertible elements of E(X), the algebra of all bounded linear operators on a Banach space X. In 1945 Mackey showed that if $x_1,…,x_n$ and $y_1,…,y_n$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that $T(x_k) = y_k$, $k = 1,…,n$. We prove that some proper multiplicative subgroups of G have this property

On the non-existence of norms for some algebras of functions

Let C(Ω) be the algebra of all complex-valued continuous functions on a topological space Ω where C(Ω) contains unbounded functions. First it is shown that C(Ω) cannot have a Banach algebra norm. Then it is shown that, for certain Ω, C(Ω) cannot possess an (incomplete) normed algebra norm. In particular, this is so for $Ω = ℝ^n$ where ℝ is the reals

Centralizers for subsets of normed algebras

Let G be the set of invertible elements of a normed algebra A with an identity. For some but not all subsets H of G we have the following dichotomy. For x ∈ A either $cxc^{-1} = x$ for all c ∈ H or $sup {∥cxc^{-1}∥ : c ∈ H} = ∞$. In that case the set of x ∈ A for which the sup is finite is the centralizer of H

On Banach algebras with a Jordan involution

Let A be a Banach algebra. By a Jordan involution x→ x# on A we mean a conjugate-linear mapping of A onto A where x##= x for all x in A and (xy + yx)# = x#y# +y#x# for all $x, y$ in A. Of course any involution is automatically a Jordan involution. An easy example of a Jordan involution which is not an involution is given, for the algebra of all complex two-by-two matrices, by ≤ft(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right)#= ≤ft( \begin{array}{cc} \bar{a} & \bar{b }\\ \bar{c} & \bar{d} \\ \end{array}\right) In this note we provide one instance where a Jordan involution is compelled to be an involution. Say $x ∈ A$ is # -normal if x permutes with x# and # -self-adjoint if x = x#. Let y be #-normal. Then 2 (y#y)# = (y#y + yy#)# =2y#y so that y#y is #-self-adjoint. By [5, pp. 481-2]we know that (xn)# = (x#)n for all $x∈ A$ and all positive integers n. Also e# = e if A has an identity e

C*-seminorms

A necessary and sufficient condition is given for a*-algebra with identity to have a unique maximal C*-seminorm. This generalizes the result, due to Bonsall, that a Banach *-algebra with identity has such a*-seminorm