##
**Extremally rich JB\(^{*}\)-triples.**
*(English)*
Zbl 1385.46008

In order to measure how close a point \(x\) of a Banach space \(X\) is to being an extreme point of the unit ball \(X_1\), R. M. Aron and R. H. Lohman [Pac. J. Math. 127, No. 2, 209–231 (1987; Zbl 0662.46020)] defined the \(\lambda\)-function by
\[
\lambda(x)=\sup\{\lambda' \in (0,1]: \exists y\in X_1,\exists e\in\partial_e(X_1), \quad x=\lambda' e+(1-\lambda')y\},
\]
where \(\partial_e(X_1)\) denotes the set of extreme points of \(X_1\). \(X\) is said to have the \(\lambda\)-property if, for each \(x\in X\), the \(\sup\) above is over a non-empty set (in which case, of course, \(\lambda(x)>0\)), and is said to have the uniform \(\lambda\)-property if \(\inf\lambda(X_1)>0\). L. G. Brown and G. K. Pedersen [Math. Scand. 81, No. 1, 69–85 (1997; Zbl 0898.46051)] investigated the \(\lambda\)-function for unital \(C^*\)-algebras (which here and below are denoted by \(A\)). They defined and used the set \(A_q^{-1}=A^{-1}\partial_e(A_1) A^{-1}\) of quasi-invertible elements of \(A\) (where \(A^{-1}\) denotes the group of invertible elements of \(A\)) and call \(A\) extremally rich if this set is dense in \(A\). \(A\) is extremally rich iff its unit ball \(A_1\) is the convex hull of its extreme points. (For an arbitrary Banach space, it is known that it has the \(\lambda\)-property iff its unit ball is the sequential closure of its extreme points.) Also, \(A\) is extremally rich iff it has the \(\lambda\)-property, which happens iff it has the uniform \(\lambda\)-property.

It is natural to consider the above in the setting of JB\(^*\)-triples (which form a wide class of Banach spaces containing, for example, \(C^*\)-algebras and JB\(^*\)-algebras, and which share many Banach space properties with the mentioned examples). The authors of the paper under review have done so in several previous papers. For example, an element \(x\in E\) (here and below, \(E\) denotes a JB\(^*\)-triple) has been defined to be BP quasi-invertible (with ‘BP’ for ‘Brown-Pedersen’) if there exists \(y\in E\) such that \(B(x,y)=0\). Here, \(B(x,y)\) is the Bergmann operator associated to \(x,y\in E\), on \(E\). (To have a vague idea why the Bergmann operator intervenes naturally, recall Kadison’s well-known result that a norm one element \(v\in A\) is extreme in \(A_1\) iff \(v\) is a partial isometry such that \((1-vv^*)A(1-v^*v)=\{0\}\) – but the latter is the same as \(B(v,v)=0\) if one considers \(A\) as a JB\(^*\)-triple.) The set of BP quasi-invertible elements is denoted by \(E_q^{-1}\). It contains \(\partial_e(E_1)\) and coincides with the set of quasi-invertible elements defined by Brown and Pedersen [loc. cit.] when \(E\) is a unital \(C^*\)-algebra.

In the paper under review, \(E\) is called extremally rich if \(E_q^{-1}\) is dense in \(E\). This, too, fits the definition of Brown and Pedersen in the sense that a \(C^*\)-algebra is extremally rich as a \(C^*\)-algebra iff it is extremally rich as a JB\(^*\)-triple. JBW\(^*\)-triples are known to be extremally rich but there are other extremally rich JB\(^*\)-triples (such as \(C^*\)-algebras of topological stable rank \(1\)).

Among the main open questions, there is the one of knowing whether \(E\) is extremally rich iff if it has the (uniform) \(\lambda\)-property (as is the case for \(C^*\)-algebras, see above). It is known that JBW\(^*\)-triples satisfy the uniform \(\lambda\)-property. As a step towards an answer, the authors show that, if \(E\) is extremally rich, then for \(a\in E\setminus E_q^{-1}\), \(\|a\|<1\), one has \(\lambda(a)=1/2\). This is a consequence of the exact computation of the distance of an element \(x\in E\) to \(\partial_e(E_1)\) when \(E\) is extremally rich. This, in turn, is a partial (yet satisfactory) answer to another open problem, the one of generalizing the computations of Brown and Pedersen for \(C^*\)-algebras.

Extremally rich JB\(^*\)-triples are shown to be stable under quotients and \(\ell_\infty\)-sums; some characterizations in terms of \(E_q^{-1}\) of extremal richness are given; the relationship between \(E_q^{-1}\) and other notions like Jordan invertibility or von Neumann regularity is commented upon.

Another open question concerns the so-called quadratic conorm \(\gamma^q\) defined on \(E\) and a characterization of its points of continuity. The authors show that, if \(E\) is extremally rich, then \(\gamma^q\) is continuous at \(a\in E\) iff either \(\gamma^q(a)=0\) (which is equivalent to \(a\) being not von Neumann regular) or \(a\in E\setminus E_q^{-1}\). This result is new even for \(C^*\)-algebras.

It is natural to consider the above in the setting of JB\(^*\)-triples (which form a wide class of Banach spaces containing, for example, \(C^*\)-algebras and JB\(^*\)-algebras, and which share many Banach space properties with the mentioned examples). The authors of the paper under review have done so in several previous papers. For example, an element \(x\in E\) (here and below, \(E\) denotes a JB\(^*\)-triple) has been defined to be BP quasi-invertible (with ‘BP’ for ‘Brown-Pedersen’) if there exists \(y\in E\) such that \(B(x,y)=0\). Here, \(B(x,y)\) is the Bergmann operator associated to \(x,y\in E\), on \(E\). (To have a vague idea why the Bergmann operator intervenes naturally, recall Kadison’s well-known result that a norm one element \(v\in A\) is extreme in \(A_1\) iff \(v\) is a partial isometry such that \((1-vv^*)A(1-v^*v)=\{0\}\) – but the latter is the same as \(B(v,v)=0\) if one considers \(A\) as a JB\(^*\)-triple.) The set of BP quasi-invertible elements is denoted by \(E_q^{-1}\). It contains \(\partial_e(E_1)\) and coincides with the set of quasi-invertible elements defined by Brown and Pedersen [loc. cit.] when \(E\) is a unital \(C^*\)-algebra.

In the paper under review, \(E\) is called extremally rich if \(E_q^{-1}\) is dense in \(E\). This, too, fits the definition of Brown and Pedersen in the sense that a \(C^*\)-algebra is extremally rich as a \(C^*\)-algebra iff it is extremally rich as a JB\(^*\)-triple. JBW\(^*\)-triples are known to be extremally rich but there are other extremally rich JB\(^*\)-triples (such as \(C^*\)-algebras of topological stable rank \(1\)).

Among the main open questions, there is the one of knowing whether \(E\) is extremally rich iff if it has the (uniform) \(\lambda\)-property (as is the case for \(C^*\)-algebras, see above). It is known that JBW\(^*\)-triples satisfy the uniform \(\lambda\)-property. As a step towards an answer, the authors show that, if \(E\) is extremally rich, then for \(a\in E\setminus E_q^{-1}\), \(\|a\|<1\), one has \(\lambda(a)=1/2\). This is a consequence of the exact computation of the distance of an element \(x\in E\) to \(\partial_e(E_1)\) when \(E\) is extremally rich. This, in turn, is a partial (yet satisfactory) answer to another open problem, the one of generalizing the computations of Brown and Pedersen for \(C^*\)-algebras.

Extremally rich JB\(^*\)-triples are shown to be stable under quotients and \(\ell_\infty\)-sums; some characterizations in terms of \(E_q^{-1}\) of extremal richness are given; the relationship between \(E_q^{-1}\) and other notions like Jordan invertibility or von Neumann regularity is commented upon.

Another open question concerns the so-called quadratic conorm \(\gamma^q\) defined on \(E\) and a characterization of its points of continuity. The authors show that, if \(E\) is extremally rich, then \(\gamma^q\) is continuous at \(a\in E\) iff either \(\gamma^q(a)=0\) (which is equivalent to \(a\) being not von Neumann regular) or \(a\in E\setminus E_q^{-1}\). This result is new even for \(C^*\)-algebras.

Reviewer: Hermann Pfitzner (Orléans)

### MSC:

46B20 | Geometry and structure of normed linear spaces |

46L70 | Nonassociative selfadjoint operator algebras |