The unit ball of an injective operator space has an extreme point

TL;DR

This paper proves that the unit ball of an injective operator space has an extreme point by introducing $AW^*$-TROs and demonstrating their ideal decompositions, thus answering a previously open question.

Contribution

It establishes that the unit ball of an injective operator space has an extreme point and introduces the concept of $AW^*$-TROs with ideal decompositions.

Findings

The unit ball of an $AW^*$-TRO has an extreme point.

Injective operator spaces can be decomposed into ideals within an $AW^*$-algebra.

$AW^*$-TROs admit an algebrization with a quasi-identity.

Abstract

We define an $AW^*$-TRO as an off-diagonal corner of an $AW^*$-algebra, and show that the unit ball of an $AW^*$-TRO has an extreme point. In particular, the unit ball of an injective operator space has an extreme point, which answers a question raised in the author's previous work [Journal of Operator Theory, 76(2) (2016), 219-248] affirmatively. We also show that an $AW^*$-TRO (respectively, an injective operator space) has an ideal decomposition, that is, it can be decomposed into the direct sum of a left ideal, a right ideal, and a two-sided ideal in an $AW^*$-algebra (respectively, an injective $C^*$-algebra). In particular, we observe that $AW^*$-TRO, hence an injective operator space, has an algebrization which admits a quasi-identity.

Equations

$$X=X_T\stackrel{\infty }{\oplus }{X}_L\stackrel{\infty }{\oplus }{X}_R$$

$$\iota (X)=\iota (X_T)\stackrel{\infty }{\oplus }\iota (X_L)\stackrel{\infty }{\oplus }\iota (X_R).$$

Full text

Mathematics subject classification 2010. Primary 47L07, 46M10; Secondary 47L25, 46L07, 46L45, 47L20, 46H10, 47L30

Key words and phrases. Extreme point, injective operator space, ternary ring of operators (TRO), ideal decomposition, quasi-identity, algebrization, $AW^{*}$-algebra, monotone complete $C^{*}$-algebra

Recall that an operator space $X$ is called a triple system or a ternary ring of operators (TRO for short) if there exists a complete isometry $\iota$ from $X$ into a $C^{*}$-algebra such that $\iota(x)\iota(y)^{*}\iota(z)\in\iota(X)$ for all $x,y,z\in X$. A theorem of Ruan and Hamana (independently) states that an operator space $X$ is injective if and only if it is an off-diagonal corner of an injective $C^{*}$-algebra, i.e., there exist an injective $C^{*}$-algebra ${\mathcal{A}}$ and projections p,q\in\mbox{{\mathcal{A}}} (meaning $p=p^{2}=p^{*}$ and $q=q^{2}=q^{*}$) such that $X$ is completely isometric to p\mbox{{\mathcal{A}}}q (Theorem 4.5 in [14] and Theorem 3.2 (i) in [2]). In particular, an injective operator space is a TRO. Noting that an injective $C^{*}$-algebra is monotone complete and hence an $AW^{*}$-algebra, the Ruan-Hamana theorem motivates the following definition. (The reader is referred to [15] for a modern account of and recent progress in monotone complete $C^{*}$-algebras and $AW^{*}$-algebras.)

1. (1)

Our definition of an $AW^{*}$-TRO is weaker than the one given in [12] (Definition 6.2.1) where an $AW^{*}$-TRO is defined as a TRO whose linking $C^{*}$-algebra is an $AW^{*}$-algebra. This condition is so strong that even some injective operator spaces fail to be $AW^{*}$-TROs in this sense. For instance, a countably-infinite-dimensional column Hilbert space is an injective operator space ([13]) and hence a TRO, however, its linking $C^{*}$-algebra is not unital, and so is not an $AW^{*}$-algebra. In our belief, disqualifying an injective operator space, which is an off-diagonal corner of an $AW^{*}$-algebra, from being an $AW^{*}$-TRO is not befitting to its name, so in this paper we use the term “$AW^{*}$-TRO” in the sense of Definition 1 above, and hence an injective operator space is an $AW^{*}$-TRO. Also this definition is consistent with that of a $W^{*}$-TRO which is an off-diagonal corner of a $W^{*}$-algebra but its linking $C^{*}$algebra need not be a $W^{*}$-algebra. 2. (2)

With this modified definition of $AW^{*}$-algebras and \mathscr{L}_{T}:=\begin{bmatrix}p\mbox{{\mathcal{A}}}p&p\mbox{{\mathcal{A}}}q\\ q\mbox{{\mathcal{A}}}p&q\mbox{{\mathcal{A}}}q\end{bmatrix}\subseteq\mbox{{\mathbb{M}}}_{2}(\mbox{{\mathcal{A}}}), where ${\mathcal{A}}$, $p$, and $q$ are as in Definition 1, all Theorems, Corollaries, and Lemmas in Sections 6.2 and 6.3 of [12] are valid except for Statement 6.2.2 and Corollary 6.2.6 there.

From the proof above we obtain “ideal decompositions” for $AW^{*}$-TROs and injective operator spaces similar to the ones done for TROs with predual in [7]. The technique we use here is to embed an off-diagonal corner into the diagonal corners which is a modification of the technique developed in [1] and is employed in [7].

[TABLE]

so that there is a complete isometry $\iota$ from $X$ into an $AW^{*}$-algebra (respectively, an injective $C^{*}$-algebra) in which $\iota(X_{T})$, $\iota(X_{L})$, and $\iota(X_{R})$ are a two-sided, left, and right ideal, respectively, and

[TABLE]

1. (1)

In the proof above it is also possible to define \iota:X\to\mbox{{\mathcal{B}}} by $\iota(x):=x_{L}e^{*}\oplus e^{*}(x_{R}+x_{T})$ for $x\in X$. 2. (2)

A TRO $X$ with predual can be considered as an off-diagonal corner of a von Neumann algebra ${\mathcal{A}}$ (see the beginning of the proof of the Theorem in [7]), thus the above argument gives an alternate and simpler proof of the Theorem in [7] noting that ${\mathcal{B}}$ in the above proof is a von Neumann algebra in this case. The simplicity of this alternate proof is attributed to the use of the comparison theorem for projections.

The following corollary is straightforward from the corollary above. The reader is referred to [5], [6], or [9] for quasi-multipliers and algebrizatioins of operator spaces, and Definition 4.2 (i) in [8] for quasi-identities.

Acknowledgment

The author wishes to express his gratitude to Professor Kazuyuki Saitô for finding and fixing a gap in the initial draft of the proof of Theorem 3.