Applications of compact multipliers to algebrability of $(\ell_{\infty}\setminus c_0)\cup\{0\}$ and $(B(\ell_2(\mathbb{N}))\setminus K(\ell_2(\mathbb{N}))\cup \{ 0\}.$

TL;DR

This paper explores the structure and algebrability of certain classes of algebras and operator spaces, demonstrating embeddings of complex algebraic structures into specific sets related to bounded and compact operators.

Contribution

It establishes isometric isomorphisms of separable uniform and C*-algebras into subalgebras of b, introduces new notions of algebrability, and shows embeddings of b and operator spaces into these sets.

Findings

Existence of b copies in the specified sets.

Embedding of non-separable multiplier algebras.

Introduction of *-algebrability concepts.

Abstract

In present work we deal with the class $\mathcal{C}=\mathcal{C}_1\cup \mathcal{C}_2$ where $\mathcal{C}_1$ (respectively, $\mathcal{C}_2$) is formed by all separable Uniform algebras (respectively, separable commutative C$^*$-algebras) with no compact elements. For a given algebra $A$ in $\mathcal{C}_1$ (respectively, $A$ in $\mathcal{C}_2$) we show that $A$ is isometrically isomorphic as algebra (respectively, as C$^*$-algebra) to a subalgebra $M$ of $\ell_{\infty}$ with $M\subset (\ell_{\infty}\setminus c_0)\cup\{0\}.$ Under the additional assumption that $A$ is non-unital we verify that there exists a copy of $M(A)$ (the multipliers algebra of $A$ which is non-separable) inside $(\ell_{\infty}\setminus c_0)\cup\{0\}$. For an infinitely generated abelian C$^*$-algebra $B,$ we study the least cardinality possible of a system of generators ($gen_{C^*}(B)$). In fact we deduce that $gen_{C^*}(B)$ coincides with the smallest cardinal number $n$ such that an embedding of $\Delta(B)$ (= the spectrum of $B$) in $\mathbb{R}^n$ exists - The finitely generated version of this result was proved by Nagisa. In addition, we introduce new concepts of algebrability in terms of $gen_{C^*}(B)$ ($(C^*)$-genalgebrability) and its natural variations. From our methods we infer that there is $^*$-isomorphic copy of $\ell_{\infty}$ in $(\ell_{\infty}\setminus c_0)\cup\{0\}$. In particular, $(\ell_{\infty}\setminus c_0)\cup\{0\}$ contains a copy of every separable Banach space. Moreover, all the positive answers of this work holds if we replace the set $(\ell_{\infty}\setminus c_0)\cup\{0\}$ with $(B(\ell_2(\mathbb{N}))\setminus K(\ell_2(\mathbb{N}))\cup \{ 0\}.$