David P. Blecher, N. Christopher Phillips
We initiate an investigation into how much the existing theory of (nonselfadjoint) operator algebras on a Hilbert space generalizes to algebras acting on L^p spaces. In particular we investigate the applicability of the theory of real positivity, which has recently been useful in the study of L^2-operator algebras and Banach algebras, to algebras of bounded operators on Lp spaces.
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We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for $C^*$-algebras, and we give many other applications.
We continue our study of operator algebras with contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the first part of this paper we do a more systematic development of this positivity and its associated ordering, proving many foundational facts. In much of this it is not necessary that the algebra have an approximate identity. In the second part we study interesting examples of operator algebras with cais, wh…
We continue our study of operator algebras with contractive approximate identities (cais) by presenting a couple of interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series, for example about whether, roughly speaking, `weak compactness' of an operator algebra, or the lack of it, can be seen in the spectra of its elements.
Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the present paper we generalize some part of this, and some other facts, to larger classes of Banach algebras.
We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity, such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a `large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0.
It is shown that a locally compact group $G$ is amenable if and only if some certain closed ideals of the Figà-Talamanca-Herz algebra $A_{p}(G)$ admit bounded $Δ$-weak approximate identities. Also, similar results are obtained for the function algebras $\mathfrak{L}A(G)$ and $C_{0}^{w}(G)$.
We make several contributions to our recent program investigating structural properties of algebras of operators on a Hilbert space. For example, we make substantial contributions to the noncommutative peak interpolation program begun by Hay and the first author, Hay and Neal. Another sample result: an operator algebra has a contractive approximate identity iff the linear span of the elements with positive real part is dense. We also extend the theory of compact projections to the most general case. Despite the title, our algebras are often allowed to have no approximate identity.
It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for real locally C*-algebras (projective limits of projective families of real C*-algebras), and for locally JB-algebras (projective limits of projective families of JB-algebras).
Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity, and the third one concerns bijective commutativity preserving linear maps.
A left ideal of any $C^*$-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in $C^*$-algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a $C^*$-algebra. We study left ideals from the perspective of `Hamana theory' and using the multiplier algebras introduced by the author. More generally, we develop some general theory f…
The properties of Banach algebras affected by differential identities
本文研究Banach代数在非空开子集上满足特定代数恒等式时的交换性，并通过例子说明假设限制的合理性。
Let $A$ be a commutative Banach algebra with non-empty character space $Δ(A)$. In this paper, we change the concepts of convergence and boundedness in the classical notion of bounded approximate identity. This work give us a new kind of approximate identity between bounded approximate identity and bounded weak approximate identity. More precisely, a net $\{e_α\}$ in $A$ is a \emph{c-w approximate identity} if for each $a\in A$, the Gel'fand transform of $e_αa$ tends to the Gel'fand transform of $a$ in the compact-open topology and we say $\{e_α\}$ is \emph{weakly bounded} if the image of $\{e_…
On the spectral identities and fundamental properties of one-sided Drazin inverses in Banach algebras
本文研究了Banach代数中单边（广义）Drazin逆的基本性质，引入单边强pi-正则性并证明其等价于单边Drazin可逆性，进而利用Jacobson引理推导出单边Drazin可逆谱及Fredholm型算子的谱恒等式。
Let $A$ be a Banach algebra with a non-empty character space. We say that a bounded net $\{e_α\}$ in $A$ is a bounded $Δ$-weak approximate identity for $A$ if, for each $a\in A$ and compact subset $K$ of $Δ(A)$, $||\widehat{e_αa}-\widehat{a}||_{K}=\sup_{ϕ\in K}|ϕ(e_αa)-ϕ(a)|\rightarrow 0$. For each $1<p<\infty$, we prove that the Figa-Talamanca Herz algebra, $A_{p}(G)$ has a bounded $Δ$-weak approximate identity if and only if $G$ is an amenable group. Also we give a sufficient condition for amenability of group $G$.
In [3] S. J. Bhatt and H. V. Dedania exposed certain classes of Banach algebras in which every element is a topological divisor of zero. We identify a new (large) class of Banach algebras with the aforementioned property, namely, the class of non-unital Banach algebras which admits either an approximate identity or approximate units. This also leads to improvements of results by R. J. Loy and J. Wichmann, respectively. If we observe that every single example that appears in [3] belongs to the class identified in the current paper, and, moreover, that many of them are classical examples of Bana…
In this paper we give an example of a Banach algebra $A$ and a closed ideal $I$ of $A$ such that the multiplier algebra of $I$ is equal to $A$ but $I$ does not have any bounded approximate identity. In the case that $I$ has an approximate identity, we give a necessary condition on $I$ for which $A=\mathcal{M}(I)$, where $\mathcal{M}(I)$ denotes the multiplier algebra of $I$. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier-Stieltjes algebra.
Approximate identities for Ideals in $L(L^p))$
本文刻画了 $L(L^p)$ 中具有左近似单位元的非平凡闭理想，证明只有紧算子理想满足该性质，并讨论了 $L(L^1)$ 中右近似单位元的存在与缺失情形。
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