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Injective envelopes of $C^*$-algebras as maximal rigid multiplier covers

Tomasz Kania

Abstract

The injective envelope of a $C^*$-algebra, introduced by Hamana in his foundational 1979 papers, has become a central tool in noncommutative analysis. For a compact Hausdorff $X$, injectivity on the commutative side amounts to extremal disconnectedness: $I(C(X))\cong C(K)$ with $K$ extremally disconnected. Gleason's classical construction \cite{Gleason} of $K$ is intricate; Błaszczyk \cite{Blaszczyk} later gave a strikingly concise route: first \emph{maximise} the regular topology under an irreducibility constraint, then \emph{compactify} to obtain $G(X)$. Indeed, in Błaszczyk argument, maximality is the driver and extremal disconnectedness is the consequence.\smallskip Our aim is to transpose this to the noncommutative setting. The multiplier algebra $M(E)$ is the natural analogue of the Čech--Stone compactification: $M(C_0(Y))\cong C(βY)$. We introduce \emph{$A$-multiplier covers} $(E,ι)$ and a rigidity notion paralleling Hamana's. The punchline is that a~\emph{maximal rigid} cover forces $M(E)$ to be a rigid essential extension of $A$, hence identifies canonically with $I(A)$.

Injective envelopes of $C^*$-algebras as maximal rigid multiplier covers

Abstract

The injective envelope of a -algebra, introduced by Hamana in his foundational 1979 papers, has become a central tool in noncommutative analysis. For a compact Hausdorff , injectivity on the commutative side amounts to extremal disconnectedness: with extremally disconnected. Gleason's classical construction \cite{Gleason} of is intricate; Błaszczyk \cite{Blaszczyk} later gave a strikingly concise route: first \emph{maximise} the regular topology under an irreducibility constraint, then \emph{compactify} to obtain . Indeed, in Błaszczyk argument, maximality is the driver and extremal disconnectedness is the consequence.\smallskip Our aim is to transpose this to the noncommutative setting. The multiplier algebra is the natural analogue of the Čech--Stone compactification: . We introduce \emph{-multiplier covers} and a rigidity notion paralleling Hamana's. The punchline is that a~\emph{maximal rigid} cover forces to be a rigid essential extension of , hence identifies canonically with .

Paper Structure

This paper contains 9 sections, 12 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hamana's injective envelope
  4. Multiplier algebras and quotienting ideals
  5. Multiplier algebras and quotienting ideals
  6. Multiplier covers and maximality
  7. Main theorem: $M(E_{\max})$ realises $I(A)$
  8. The commutative case revisited
  9. Examples and remarks

Key Result

Theorem 2.1

For every operator system $A$ there exists an injective operator system $I(A)$ and a unital complete order embedding $j:A\hookrightarrow I(A)$ such that: Moreover, $I(A)$ is unique up to a unique complete order isomorphism over $A$. If $A$ is a $C^\ast$-algebra, then $I(A)$ is a $C^\ast$-algebra and $j$ is a $*$-monomorphism. Equivalently, $I(A)$ is the unique rigid essential extension of $A$: wh

Theorems & Definitions (29)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 19 more