### 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.

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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.

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 $\Omega ={\mathbb{R}}^{n}$ where ℝ is the reals.

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 ${x}_{1},\dots ,{x}_{n}$ and ${y}_{1},\dots ,{y}_{n}$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that $T\left({x}_{k}\right)={y}_{k}$, $k=1,\dots ,n$. We prove that some proper multiplicative subgroups of G have this property.

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 $cx{c}^{-1}=x$ for all c ∈ H or $sup\parallel cx{c}^{-1}\parallel :c\in H=\infty $. In that case the set of x ∈ A for which the sup is finite is the centralizer of H.

Let T be a Fredholm operator on a Banach space. Say T is rootless if there is no bounded linear operator S and no positive integer m ≥ 2 such that ${S}^{m}=T$. Criteria and examples of rootlessness are given. This leads to a study of ascent and descent whether finite or infinite for T with examples having infinite ascent and descent.

The set of commutators in a Banach *-algebra A, with continuous involution, is examined. Applications are made to the case where A = B(ℓ₂), the algebra of all bounded linear operators on ℓ₂.

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