Banach synaptic algebras

TL;DR

This paper characterizes Banach synaptic algebras as those isomorphic to self-adjoint parts of Rickart C*-algebras and explores their relation to AW*-algebras and generalized Hermitian algebras.

Contribution

It provides a characterization of Banach synaptic algebras using representation theorems and establishes their connection to well-known operator algebra structures.

Findings

Banach synaptic algebras are isomorphic to self-adjoint parts of Rickart C*-algebras.

Conditions are given for a Banach synaptic algebra to be isomorphic to an AW*-algebra.

Relationships between synaptic algebras and generalized Hermitian algebras are analyzed.

Abstract

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Stormer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C*-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW*-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.

Equations

$$C(a):=\{b\in A:aCb\},C(M):=\bigcap _{b\in M}C(b),$$

$$CC(M):=C(C(M),\text{ and }CC(a):=C(C(a)).$$

$$a\odot b:=\frac{1}{2}(ab+ba)=\frac{1}{2}[(a+b{)}^2-a^2-b^2]=\frac{1}{4}[(a+b{)}^2-(a-b{)}^2].$$

$$cd=0\Rightarrow (c^{\ast }c)(d{d}^{\ast })=0\iff (c^{\ast }c{)}^{\text{o}}(d{d}^{\ast })=0\Rightarrow [(c^{\ast }c{)}^{\text{o}}d][d^{\ast }(c^{\ast }c{)}^{\text{o}}]=0\iff$$

$$[(c^{\ast }c{)}^{\text{o}}d][(c^{\ast }c{)}^{\text{o}}d{]}^{\ast }=0\iff (c^{\ast }c{)}^{\text{o}}d=0\iff d=[1-(c^{\ast }c{)}^{\text{o}}]d\iff d=c^{\prime }d.$$

$$s^{\ast }sc{c}^{\ast }=0\text{ for all }s\in S\iff c=p_{T}c.$$

Full text

1 Introduction

A synaptic algebra [5, 10, 11, 12, 13, 14, 15, 16, 17, 18, 29] (see Section 4 below) is a generalization of several structures based on operator algebras. The adjective ‘synaptic’ is derived from the Greek word ‘sunaptein,’ meaning to join together; indeed synaptic algebras unite the notions of an order-unit normed space [1, p. 69], a real unital special Jordan algebra [28], a convex effect algebra [22], and an orthomodular lattice [3, 25].

Important examples of synaptic algebras include the JW-algebras, the AJW-algebras, and the spin factors of D. Topping [34, 8], as well as the ordered special Jordan algebras of T. Sarymsakov et al. [32]. Also the generalized Hermitian algebras introduced and studied in [7, 9] are synaptic algebras. Moreover, the self-adjoint parts of von Neumann algebras [31], Rickart C*∗-algebras [23, §3], and AW∗*-algebras [26] are synaptic algebras. Whereas most of the preceeding examples are Banach (i.e., norm-complete) algebras, a synaptic algebra need not be norm-complete (e.g., [5, Example 1.2]).

We shall be using results of Erik Alfsen, Frederic Schultz, and Erling Størmer in [2] pertaining to so-called JB-algebras to prove that a synaptic algebra is isomorphic to a Rickart JC-algebra (i.e., the self-adjoint part of a Rickart C*∗*-algebra of bounded linear operators on a Hilbert space) if and only if it is a Banach algebra.

In Section 2 we review some basic definitions pertaining to $\ast$-algebras, we recall the definition and some of the relevant properties of a JB-algebra in Section 3, and in Section 4, we present the axioms SA1–SA8 for a synaptic algebra and remark on the extent to which these axioms hold for a JB-algebra. In Theorem 4.4, we prove that a Banach synaptic algebra is a special JB-algebra. In Section 5, we relate the carrier and Rickart properties and prove that a synaptic algebra is Banach if and only if it is isomorphic to the self-adjoint part of a Rickart C*∗-algebra (Theorem 5.3). In Section 6, we present some additional properties of a synaptic algebra, in Section 7 we review some facts about blocks and C-blocks in a synaptic algebra, and in Section 8, we characterize Banach synaptic algebras that are isomorphic to the self-adjoint part of an AW∗*-algebra. Finally, in Section 9, we investigate some of the relationships between synaptic algebras and so-called generalized Hermitian (GH) algebras.

2 Preliminaries

In this paper we use ‘iff’ as an abbreviation for ‘if and only if,’ the notation ‘$:=$’ means ‘equals by definition,’ ${\mathbb{R}}$ is the ordered field of real numbers, and ${\mathbb{N}}:=\{1,2,3,...\}$ is the well-ordered set of natural numbers.

A Jordan algebra $J$ [28] with Jordan product $\odot$ is unital iff there is a unit element $1\in J$ such that $a\odot 1=a$ for all $a\in J$. The Jordan algebra $J$ is special iff it can be embedded in an associative algebra $R$ in such a way that $a\odot b=\frac{1}{2}(ab+ba)$ for all $a,b\in J$, where $ab+ba$ is calculated in $R$ [28].

Let ${\mathcal{C}}$ be an associative algebra over the complex numbers. If there is a unit element $1\in{\mathcal{C}}$ such that $1c=c1=c$ for all $c\in{\mathcal{C}}$, then ${\mathcal{C}}$ is said to be unital. If there is an “adjoint mapping” $c\mapsto c^{\ast}$ on ${\mathcal{C}}$ such that, for all $c,d\in{\mathcal{C}}$ and for every complex number $\alpha$, (1) $(\alpha c)^{\ast}={\bar{\alpha}}c^{\ast}$, (2) $(c+d)^{\ast}=c^{\ast}+d^{\ast}$, (3) $(cd)^{\ast}=d^{\ast}c^{\ast}$, and (4) $(c^{\ast})^{\ast}=c$, then ${\mathcal{C}}$ is called a $\ast$-algebra. An isomorphism from one $\ast$-algebra to another is understood to be an algebra isomorphism that preserves the adjoint mapping. If the $\ast$-algebras are unital, then such an isomorphism automatically preserves $1$.

Let ${\mathcal{C}}$ be a $\ast$-algebra. Then an element $c\in{\mathcal{C}}$ is said to be self-adjoint iff $c=c^{\ast}$, and a self-adjoint idempotent $p=p^{\ast}=p^{2}\in{\mathcal{C}}$ is called a projection. We denote by ${\mathcal{C}}^{sa}$ the set of all self-adjoint elements in ${\mathcal{C}}$ and we refer to ${\mathcal{C}}^{sa}$ as the self-adjoint part of ${\mathcal{C}}$. Clearly, if $c\in{\mathcal{C}}$, then $c+c^{\ast}$, $\frac{1}{i}(c-c^{\ast})$, $cc^{\ast},c^{\ast}c\in{\mathcal{C}}^{sa}$, and if ${\mathcal{C}}$ is unital, then $1\in{\mathcal{C}}^{sa}$. Moreover, ${\mathcal{C}}^{sa}$ is a special real Jordan algebra under the Jordan product $c\odot d:=\frac{1}{2}(cd+dc)$ for all $c,d\in{\mathcal{C}}^{sa}$. Also, if ${\mathcal{C}}$ is unital, then the Jordan algebra ${\mathcal{C}}^{sa}$ is unital.

An (abstract) unital C*∗-algebra is defined to be a unital Banach ${\ast}$-algebra ${\mathcal{C}}$ such that the norm satisfies $\|cc^{\ast}\|=\|c\|^{2}$ for all $c\in{\mathcal{C}}$. In what follows, we shall consider only unital C∗ algebras; thus we shall omit the adjective “unital.” An isomorphism of one C∗*-algebra onto another is understood to be an isomorphism of ${\ast}$-algebras that is also an isometry.

2.1 Lemma**.**

Let ${\mathcal{C}}$ be a C∗-algebra, let $c\in{\mathcal{C}}$, and let $p=p^{2}$ be a projection in ${\mathcal{C}}$. Then: (i)* $cc^{\ast}=0\Leftrightarrow c=0$. (ii) $cc^{\ast}=pcc^{\ast}\Leftrightarrow c=pc$.*

Proof.

(i) $cc^{\ast}=0\Leftrightarrow 0=\|cc^{\ast}\|=\|c\|^{2}\Leftrightarrow c=0$. (ii) Suppose that $cc^{\ast}=pcc^{\ast}$. Then $(1-p)p=0$, so $(1-p)c[(1-p)c]^{\ast}=(1-p)cc^{\ast}(1-p)=(1-p)pcc^{\ast}(1-p)=0$, whence by (i), $(1-p)c=0$, i.e., $c=pc$. The converse is obvious. ∎

If ${\mathfrak{H}}$ is a complex Hilbert space then ${\mathcal{B}}({\mathfrak{H}})$ denotes the unital $\ast$-algebra of all bounded linear operators on ${\mathfrak{H}}$ under the formation of operator adjoints and with the uniform operator norm. A norm-closed $\ast$-subalgebra ${\mathcal{C}}$ of ${\mathcal{B}}({\mathfrak{H}})$ with $1\in{\mathcal{C}}$ can be shown to be a C*∗-algebra and as such, it is called a concrete C∗-algebra. By a classical result of I. Gelfand and M. Neumark [20], every C∗-algebra is isomorphic to a concrete C∗*-algebra.

If ${\mathcal{C}}$ is a $C^{\ast}$-algebra, then the self-adjoint part ${\mathcal{C}}^{sa}$ of ${\mathcal{C}}$ is not only a special real unital Jordan algebra, but it turns out to be a JB-algebra as per Section 3 below. By definition, a JC-algebra is the self-adjoint part ${\mathcal{C}}^{sa}$ of a concrete ${\mathcal{C}}^{\ast}$-algebra, i.e., it is a norm closed unital special Jordan algebra of self-adjoint operators on a complex Hilbert space [34]. Thus, a JC-algebra is also a JB-algebra.

Both JB-algebras and synaptic algebras are so-called order-unit normed spaces, according to the following definition.

2.2 Definition** ([1, pp. 67–69]).**

An order-unit space is a real partially ordered vector space $V$ with a positive cone $V^{+}:=\{v\in V:0\leq v\}$ and with a distinguished element $1\in V^{+}$, called the order unit such that:

1. (1)

For every $v\in V$, there exists $n\in{\mathbb{N}}$ such that $v\leq n1$.

An order-unit normed space with order unit $1$ is defined to be an order-unit space $V$ with order unit $1$ such that:

1. (2)

$V$ is archimedean, i.e., if $v,w\in V$ and $nv\leq w$ for all $n\in{\mathbb{N}}$, then $-v\in V^{+}$ (equivalently, $v\leq 0$). 2. (3)

The order-unit norm $\|\cdot\|$ on $V$ is defined by $\|v\|:=\inf\{\lambda\in{\mathbb{R}}:0<\lambda\text{\ and\ }-\lambda 1\leq v\leq\lambda 1\}$.

3 JB algebras

In this section we review the definition, some notation, and some facts pertaining to JB-algebras as per [2].

3.1 Definition** ([2, page 13]).**

A JB-algebra $B$ is both an order-unit space with order unit $1\in B$ and a (not necessarily special) unital Jordan algebra over ${\mathbb{R}}$. Moreover, $B$ is a Banach space under a norm $\|\cdot\|$ that satisfies the following conditions for all $b,c\in B$:

1. (1)

$\|b\odot c\|\leq\|b\|\|c\|$. 2. (2)

$\|b^{2}\|=\|b\|^{2}$. (Note: $b^{2}:=b\odot b$.) 3. (3)

$\|b^{2}\|\leq\|b^{2}+c^{2}\|$.

3.2 Theorem** ([2, Theorem 2.1]).**

Let $B$ be a JB-algebra. Then:**

1. (i)

$B$* is a (not necessarily special) real Jordan algebra with Jordan product $\odot$, and $b\odot 1=b$ for all $b\in B$.* 2. (ii)

$B$* is an order-unit normed space with order unit $1$ and the given norm $\|\cdot\|$ on $B$ is the order-unit norm.* 3. (iii)

The positive cone in $B$ satisfies $B^{+}:=\{b\in B:0\leq b\}=\{b^{2}:b\in B\}$. 4. (iv)

Under $\|\cdot\|$, $B$ is a norm-complete (i.e., a Banach) space. 5. (v)

For all $b\in B$, $-1\leq b\leq 1\Rightarrow 0\leq b^{2}\leq 1$.

Conversely, if $B$ is a norm-complete order-unit normed space with order unit $1$ and also a (not necessarily special) real unital Jordan algebra, and if condition (v) above holds, then $B$ is a JB-algebra under the order-unit norm.

An isomorphism of one JB-algebra onto another is understood to be (1) an order, (2) a linear, and (3) a Jordan isomorphism. In view of (1) and (2), a JB-algebra isomorphism is an isometry.

The self-adjoint part ${\mathcal{C}}^{sa}$ of a C*∗*-algebra ${\mathcal{C}}$ is organized into a special JB-algebra as follows: ${\mathcal{C}}^{sa}$ hosts the special Jordan product $c\odot d:=\frac{1}{2}(cd+dc)$ for $c,d\in{\mathcal{C}}^{sa}$ and the positive cone is given by $({\mathcal{C}}^{sa})^{+}=\{cc^{\ast}:c\in{\mathcal{C}}\}$.

3.3 Definition** ([2, Equation (2.24)]).**

If $B$ is a JB-algebra and $b\in B$, then the mapping $U_{b}\colon B\to B$ is defined by $U_{b}c:=2b\odot(b\odot c)-b^{2}\odot c$ for all $c\in B$.

Clearly $U_{b}$ is linear on $B$ and it turns out that $U_{b}$ is positive, hence order preserving on $B$ [2, Proposition 2.7]. If $B$ is a special Jordan algebra, then $U_{b}c=bcb$ for all $b,c\in B$.

The commutative unital Banach algebra in the next definition plays an important role in [2] where it is written as $C(b)$ rather than as $\Gamma(b)$ [2, page 14]. Here we have changed the notation to avoid confusion with the notion of “commutant” in a synaptic algebra.

3.4 Definition**.**

If $B$ is a JB-algebra and $b\in B$, then $\Gamma(b)$ is the commutative unital Banach algebra obtained by forming the norm closure of the associative Jordan subalgebra of $B$ consisting of all real polynomials in $b$.

A key result is [2, Proposition 2.3], which provides a functional representation theorem asserting that there is an order and algebraic isomorphism of $\Gamma(b)$ onto the partially ordered commutative real Banach algebra $C(X,{\mathbb{R}})$, under pointwise partial order and operations, of all continuous real-valued functions on a compact Hausdorff space $X$.

4 Synaptic algebras

To help fix ideas in the following definition, the reader may think of $R$ as the C*∗*-algebra ${\mathcal{B}}({\mathfrak{H}})$ of all bounded linear operators on a complex Hilbert space $\mathfrak{H}$ and of $A$ as the self-adjoint part ${\mathcal{B}}^{\rm sa}({\mathfrak{H}})$ of ${\mathcal{B}}({\mathfrak{H}})$.

4.1 Definition** ([5, Definition 1.1]).**

Let $R$ be a real or complex linear associative algebra with unity element $1$ and let $A$ be a real vector subspace of $R$. If $a,b\in A$ and $M\subseteq A$, we write $aCb$ iff $a$ and $b$ commute (i.e. $ab=ba$ as calculated in $R$) and we define

[TABLE]

[TABLE]

We call $C(M)$ the commutant of $M$ and $CC(M)$ the bicommutant of $M$. The vector space $A$ is a synaptic algebra with enveloping algebra $R$ iff the following conditions are satisfied:

1. SA1.

$A$ is an order-unit normed space with order unit $1$, positive cone $A^{+}=\{a\in A:0\leq a\}$, and $\|\cdot\|$ is the corresponding order-unit norm. 2. SA2.

If $a\in A$ then $a^{2}\in A^{+}$. 3. SA3.

If $a,b\in A^{+}$, then $aba\in A^{+}$. 4. SA4.

If $a\in A$ and $b\in A^{+}$, then $aba=0\Rightarrow ab=ba=0$. 5. SA5.

If $a\in A^{+}$, there exists $b\in A^{+}\cap CC(a)$ such that $b^{2}=a$. 6. SA6.

If $a\in A$, there exists $p\in A$ such that $p=p^{2}$ and, for all $b\in A$, $ab=0\Leftrightarrow pb=0$. 7. SA7.

If $1\leq a\in A$, there exists $b\in A$ such that $ab=ba=1$. 8. SA8.

If $a,b\in A$, $a_{1}\leq a_{2}\leq a_{3}\leq\cdots$ is an ascending sequence of pairwise commuting elements of $C(b)$ and $\lim_{n\rightarrow\infty}\|a-a_{n}\|=0$, then $a\in C(b)$.

4.2 Standing Assumptions**.**

In what follows, we assume that $A$ is a synaptic algebra with enveloping algebra $R$ and with unit $1$. We assume that $1\not=0$, i.e., $A\not=\{0\}$. The “unit interval” in $A$ is denoted by $E:=\{e\in A:0\leq e\leq 1\}$ and elements in $E$ are called “effects.” The idempotents in $A$ are called “projections” and the set of all projections in $A$ is denoted by $P:=\{p\in A:p=p^{2}\}$.

The set $E$ of effects is organized into a convex effect algebra [6, 22] under the partial binary operation provided by the restriction of the addition operation on $A$. Under the restriction of the partial order on $A$, the set $P:=\{p\in A:p=p^{2}\}$ of projections in $A$ is an orthomodular lattice (OML) [3, 25] with orthocomplementation $p\mapsto p^{\perp}:=1-p$ [5, §5].

Below we comment briefly on each of the axioms SA1–SA8 and we compare and contrast SA1–SA8 with features of a JB-algebra $B$.

\bullet\By SA1 and Theorem 3.2, both $A$ and $B$ form order-unit normed spaces with order units $1$ and with order unit norms $\|\cdot\|$.

\bullet\By SA2, $A$ forms a special real Jordan algebra under the Jordan product

[TABLE]

Clearly, $a\in A\Rightarrow a\odot 1=a$, so $A$ is a unital Jordan algebra. As in Definition 3.3, for each $a\in A$, we define the quadratic mapping $U_{a}\colon A\to A$ by $U_{a}b:=2a\odot b-a^{2}\odot b=aba$. (In [5], $aba$ is written as $J_{a}b$; here we use $U_{a}b$ for consistency with [2].) Obviously, $U_{a}$ is linear and by [5, Theorem 4.2] it is order preserving. Let $a,b,c\in A$. Then $aCb\Rightarrow ab=ba=a\odot b\in A$. As $a^{2}\in A$ and $aCa^{2}$, it follows that $a^{3}=a\odot a^{2}\in A$, and by induction, $a^{n}\in A$ for all $n\in{\mathbb{N}}$. Consequently, $A$ is closed under the formation of real polynomials in $a$. Thus, as in Definition 3.4, we define $\Gamma(a)$ to be the norm closure of the set of all real polynomials in $a$. It can be shown (see below) that $\Gamma(a)\subseteq CC(a)$; hence, if $b\in B$, then $\Gamma(b)$ can be regarded as an analogue in $B$ of the bicommutant in $A$.

\bullet\Axiom SA3 specifies that, for $a,b\in A^{+}$, $U_{a}b\in A^{+}$. But, as mentioned above, we actually have the stronger condition $U_{a}b\in A^{+}$ for all $a\in A$ and all $b\in A^{+}$. Likewise, as mentioned above, if $b\in B$ and $c\in B^{+}$, then $U_{b}c\in B^{+}$.

\bullet\Axiom SA4 can be written as $U_{a}b=0\Rightarrow ab=ba=a\odot b=0$ for all $a\in A$ and all $b\in A^{+}$. Analogously, by [2, Proposition 2.8], $B$ has the weaker property that if $b,c\in B^{+}$, then $U_{b}c=0\Rightarrow b\odot c=0$.

\bullet\By SA5, if $a\in A^{+}$, then there exists $b\in A^{+}\cap CC(a)$ such that $b^{2}=a$, and by [5, Theorem 2.2], $b$ is uniquely determined by $a$. Of course, we write $a^{1/2}:=b$ and refer to $a^{1/2}$ as the square root of $a$. As a consequence of SA2 and SA5, the positive cone in $A$ is given by $A^{+}:=\{a\in A:0\leq a\}=\{a^{2}:a\in A\}$. The absolute value and the positive and negative parts of an element $a\in A$ are defined by $|a|:=(a^{2})^{1/2}$, $a^{+}:=\frac{1}{2}(|a|+a)$, and $a^{-}:=\frac{1}{2}(|a|-a)$, respectively. Then $a=a^{+}-a^{-}$, $|a|=a^{+}+a^{-}$, and $a^{+}a^{-}=0$. Analogously, by [2, Equations (2.9) and (2.10], $B$ has the property that if $b\in B^{+}$, there exists $c\in\Gamma(b)$ such that $c^{2}=b$. (Curiously, there is no indication that the “square root” $c$ of $b$ can be chosen to be in $B^{+}$.)

\bullet\By SA6, if $a\in A$, there exists a projection $p\in P$ such that, for all $b\in A$, $ab=0\Leftrightarrow pb=0$, and by [5, Theorem 2.7], $p$ is uniquely determined by $a$. We define $a^{\text{\rm o}}:=p$ and refer to $a^{\text{\rm o}}$ as the carrier of $a$. By [5, Theorem 2.10 (vi)], $a^{\text{\rm o}}\in CC(a)$. Also, for $a,b\in A$, we have $ab=0\Leftrightarrow a^{\text{\rm o}}b=0\Leftrightarrow a^{\text{\rm o}}b^{\text{\rm o}}=0\Leftrightarrow b^{\text{\rm o}}a^{\text{\rm o}}=0\Leftrightarrow ba=0$. For a self-adjoint linear operator $T$ in ${\mathcal{B}}^{\rm sa}({\mathfrak{H}})$, $T^{\text{\rm o}}$ is the (orthogonal) projection onto the closure of the range of $T$. In general, the JB-algebra $B$ will fail to satisfy SA6.

\bullet\By SA7, if $a\in A$ with $1\leq a$, there exists $b\in A$ such that $ab=ba=1$. Of course, $b$ is called the inverse of $a$ and as usual is written as $a^{-1}:=b$. Clearly, $a^{-1}$ is uniquely determined by $a$, and $a^{-1}\in CC(a)$. As a consequence of [2, Equation (2.21) and Proposition 2.4], $1\leq b\in B$ implies that $b$ is invertible in $B$ and the inverse of $b$ belongs to $\Gamma(b)$.

\bullet\Axiom SA8, in the presence of the remaining axioms, turns out to be equivalent to the condition that, for every $a\in A$, the commutant $C(a)$ is norm-closed in $A$. (see [5, Theorem 8.11]). Therefore, for $M\subseteq A$, both $C(M)$ and $CC(M):=C(C(M))$ are norm-closed in $A$. In particular, the bicommutant $CC(a)$ is norm closed in $A$ and since every real polynomial in $a$ belongs to $CC(a)$, it follows (as was mentioned above) that $\Gamma(a)\subseteq CC(a)$. Furthermore, the positive cone $A^{+}$ is norm-closed in $A$ [5, Theorem 4.7 (iii)]. Unless $B$ is a special Jordan algebra, finding a reasonable analogue of axiom SA8 in $B$ may be problematic.

Thus, the synaptic algebra $A$ and the JB-algebra $B$ may differ in the following important respects: (1) Although $B$ is a Banach algebra, $A$ need not be Banach. (2) Although $A$ is a special Jordan algebra, $B$ need not be special. (3) $B$ might not satisfy the carrier axiom SA-6. (4) Unless $B$ is a special Jordan algebra, axiom SA-8 might not have a reasonable interpretation in $B$.

Of course an isomorphism (or for emphasis, a synaptic isomorphism) of one synaptic algebra onto another, we mean a bijection that is (1) an order, (2) a linear, and (3) a Jordan isomorphism. Clearly, a synaptic isomorphism is an isometry.

In the next lemma we collect some properties of the (order-unit) norm in the synaptic algebra $A$. Note that part (vi) of the lemma corresponds to part (v) of Theorem 3.2 for a JB-algebra.

4.3 Lemma**.**

Let $a,b,e\in A$. Then:**

1. (i)

$-\|a\|\leq a\leq\|a\|$. 2. (ii)

$-b\leq a\leq b\Rightarrow\|a\|\leq\|b\|$. 3. (iii)

$\|a\odot b\|\leq\|a\|\|b\|$. 4. (iv)

$\|a^{2}\|=\|a\|^{2}$. 5. (v)

$\|a^{2}\|\leq\|a^{2}+b^{2}\|$. 6. (vi)

$-1\leq a\leq 1\Leftrightarrow a^{2}\in E$. 7. (vii)

$e\in E\Leftrightarrow e^{2}\leq e$.

Proof.

(i) Part (i) follows from [1, Proposition II.1.2].

(ii) Suppose that $-b\leq a\leq b$. By (i), $b\leq\|b\|$, whence $-\|b\|\leq-b\leq a\leq b\leq\|b\|$, and therefore $\|a\|\leq\|b\|$.

(iii) See [5, Lemma 1.7 (iv)].

(iv) See [5, Lemma 1.7 (ii)].

(v) As $0\leq a^{2}\leq a^{2}+b^{2}$, we have $-(a^{2}+b^{2})\leq a^{2}\leq a^{2}+b^{2}$, whence (v) follows from (ii).

(vi) Part (vi) follows from [5, Lemma 1.7 (i)].

(vii) By [5, Lemma 2.5 (i)], $e\in E\Rightarrow e^{2}\leq e$. Conversely, suppose $e\in A$ with $e^{2}\leq e$. Clearly, $0\leq e$. Also, $0\leq(1-e)^{2}=1-2e+e^{2}$, whence $0\leq e-e^{2}\leq 1-e$, so $0\leq e\leq 1$, i.e., $e\in E$. ∎

4.4 Theorem**.**

If $A$ is a Banach synaptic algebra, then $A$ is a special JB-algebra.

Proof.

Suppose that $A$ is Banach. By parts (iii), (iv), and (v) of Lemma 4.3, $A$ satisfies conditions (1), (2), and (3) in Definition 3.1. ∎

5 Representation of a Banach synaptic algebra as the self-adjoint

part of a Rickart C*∗*-algebra

5.1 Definition**.**

Let ${\mathcal{C}}$ be a $C^{\ast}$-algebra and let ${\mathcal{C}}^{sa}$ be the self-adjoint part of ${\mathcal{C}}$. Note that every projection $p=p^{*}=p^{2}$ in ${\mathcal{C}}$ belongs to ${\mathcal{C}}^{sa}$.

1. (1)

${\mathcal{C}}$ is called a Rickart C*∗*-algebra [30] iff, for each $c\in{\mathcal{C}}$, there is a projection $c^{\prime}\in{\mathcal{C}}$ such that, for all $d\in{\mathcal{C}}$, $cd=0\Leftrightarrow d=c^{\prime}d$. 2. (2)

${\mathcal{C}}^{sa}$ has the Rickart property iff, for every $b\in{\mathcal{C}}^{sa}$, there exists a projection $p\in{\mathcal{C}}^{sa}$ such that for all $g\in{\mathcal{C}}^{sa}$, $bg=0\Leftrightarrow g=pg$. A JC-algebra with the Rickart property is called a Rickart JC-algebra. 3. (3)

${\mathcal{C}}^{sa}$ has the carrier property iff, for every $b\in{\mathcal{C}}^{sa}$, there exists a projection $b^{\text{\rm o}}\in{\mathcal{C}}^{sa}$ such that for all $g\in{\mathcal{C}}^{sa}$, $bg=0\Leftrightarrow b^{\text{\rm o}}g=0$.

5.2 Lemma**.**

Let ${\mathcal{C}}$ be a $C^{\ast}$-algebra. Then the following conditions are mutually equivalent: (i)* ${\mathcal{C}}$ is a Rickart C∗-algebra. (ii) ${\mathcal{C}}^{sa}$ has the Rickart property. (iii) ${\mathcal{C}}^{sa}$ has the carrier property.*

Proof.

(i) $\Rightarrow$ (ii). Assume (i), let $b,g\in{\mathcal{C}}^{sa}$ and put $p:=b^{\prime}$. Then $p$ is a projection, $p\in{\mathcal{C}}^{sa}$, and $bg=0\Leftrightarrow g=b^{\prime}g\Leftrightarrow g=pg$.

(ii) $\Rightarrow$ (iii). Assume (ii), let $b,g\in{\mathcal{C}}^{sa}$, and put $b^{\text{\rm o}}:=1-p$. Then $b^{\text{\rm o}}$ is a projection, $b^{\text{\rm o}}\in{\mathcal{C}}^{sa}$, and $bg=0\Leftrightarrow g=pg\Leftrightarrow(1-p)g=0\Leftrightarrow b^{\text{\rm o}}g=0$.

(iii) $\Rightarrow$ (i). Assume (iii), let $c,d\in{\mathcal{C}}$, and put $c^{\prime}=1-(c^{\ast}c)^{\text{\rm o}}$. It will suffice to prove that $cd=0\Leftrightarrow d=c^{\prime}d$. We have

[TABLE]

[TABLE]

To prove the converse, we begin by putting $b:=c^{\ast}c\in{\mathcal{C}}^{sa}$, so that $c^{\prime}=1-(c^{\ast}c)^{\text{\rm o}}=1-b^{\text{\rm o}}$. Then by Definition 5.1 (3) with $g:=c^{\prime}=1-b^{\text{\rm o}}\in{\mathcal{C}}^{sa}$, we have $b^{\text{\rm o}}c^{\prime}=b^{\text{\rm o}}(1-b^{\text{\rm o}})=b^{\text{\rm o}}-(b^{\text{\rm o}})^{2}=b^{\text{\rm o}}-b^{\text{\rm o}}=0$, whence $0=bc^{\prime}=b(1-b^{\text{\rm o}})=b-bb^{\text{\rm o}}$, and it follows that $b=bb^{\text{\rm o}}$, i.e., $c^{\ast}c=c^{\ast}c(c^{\ast}c)^{\text{\rm o}}$. Therefore, $c^{\ast}c\left(1-(c^{\ast}c)^{\text{\rm o}}\right)=0$, and we have $c^{\ast}cc^{\prime}=0$. Consequently, $0=c^{\prime}c^{\ast}cc^{\prime}=(cc^{\prime})^{\ast}(cc^{\prime})$, and it follows that $cc^{\prime}=0$, whence $d=c^{\prime}d\Rightarrow cd=cc^{\prime}d=0$. ∎

5.3 Theorem**.**

*The synaptic algebra $A$ is norm-complete *(i.e., Banach) iff it is isomorphic to the self-adjoint part ${\mathcal{C}}^{sa}$ of a Rickart C∗-algebra ${\mathcal{C}}$.

Proof.

Suppose that $A$ is a Banach synaptic algebra. Then, by Theorem 4.4, $A$ is a special JB-algebra; hence, by [2, Lemma 9.4], $A$ is isomorphic to a JC-algebra, which by definition is the self-adjoint part ${\mathcal{C}}^{sa}$ of a C*∗-algebra ${\mathcal{C}}$ of bounded linear operators on a complex Hilbert space. The synaptic algebra $A$ has the carrier property, whence ${\mathcal{C}}^{sa}$ has the carrier property, so by Lemma 5.2, ${\mathcal{C}}$ is a Rickart C∗-algebra. The converse is proved by a straightforward verification that axioms SA1–SA8 hold for the self-adjoint part of a Rickart C∗*-algebra. ∎

5.4 Corollary**.**

If $A$ is a Banach synaptic algebra, then $P$ is a $\sigma$-complete OML.

Proof.

By [4, Theorem 1.8.1], the lattice of projections in a Rickart C*∗*-algebra is $\sigma$-complete. ∎

6 Additional properties of a synaptic algebra

We continue to assume that $A$ is a synaptic algebra and that $P$ is the orthomodular lattice (OML) of projections in $A$. As both $A$ and $P$ are partially ordered sets (posets for short), we begin by reviewing some terminology.

Let ${\mathcal{P}}$ be a poset and let ${\mathcal{Q}}\subseteq{\mathcal{P}}$. A supremum (an infimum) of ${\mathcal{Q}}$ in ${\mathcal{P}}$ is a least upper bound (a greatest lower bound) for $Q$ in ${\mathcal{P}}$. Note that the supremum, if it exists (the infimum, if it exists) of the empty subset of ${\mathcal{P}}$ is the smallest element (the largest element) in ${\mathcal{P}}$. The subset ${\mathcal{Q}}\subseteq{\mathcal{P}}$ is upward directed (downward directed) iff, for every pair $\{a,b\}\subseteq{\mathcal{Q}}$, there exists $c\in{\mathcal{Q}}$ with $a,b\leq c$ (with $c\leq a,b$).

The poset ${\mathcal{P}}$ is a lattice iff every pair of elements $\{a,b\}\subseteq{\mathcal{P}}$ has a join (i.e., a supremum) $a\vee b$ and a meet (i.e., an infimum) $a\wedge b$ in ${\mathcal{P}}$. Of course, the projections $P$ in $A$ form a lattice; also, if $A$ is commutative, it too is a lattice (Theorem 6.5 below).

A mapping $a\mapsto a^{\prime}$ on ${\mathcal{P}}$ is called an involution iff it is order reversing and of period two, i.e., for $a,b\in{\mathcal{P}}$, $a\leq b\Rightarrow b^{\prime}\leq a^{\prime}$ and $(a^{\prime})^{\prime}=a$. An involution $a\mapsto a^{\prime}$ on ${\mathcal{P}}$ provides a “duality” between existing suprema and infima of subsets of ${\mathcal{P}}$ as follows: An element $b\in{\mathcal{P}}$ is the supremum ($c\in{\mathcal{P}}$ is the infimum) of ${\mathcal{Q}}$ in ${\mathcal{P}}$ iff $b^{\prime}$ is the infimum in ${\mathcal{P}}$ ($c^{\prime}$ is the supremum in ${\mathcal{P}}$) of the set $\{q^{\prime}:q\in{\mathcal{Q}}\}$. For instance, for the poset $A$, $a\mapsto-a$ is an involution. Also, the orthosupplementation mapping $p\mapsto p^{\perp}:=1-p$ on the OML $P$ is an involution on $P$.

Now we assume that the poset ${\mathcal{P}}$ hosts an involution. Thus we shall formulate the following conditions in terms of upper bounds and suprema only, since the dual conditions for lower bounds and infima are then automatic consequences.

The poset ${\mathcal{P}}$ is: (1) $\sigma$-complete iff every sequence in ${\mathcal{P}}$ has a supremum in ${\mathcal{P}}$, (2) complete iff every subset of ${\mathcal{P}}$ has a supremum in ${\mathcal{P}}$, (3) Dedekind complete iff every nonempty subset of ${\mathcal{P}}$ that has an upper bound in ${\mathcal{P}}$ has a supremum in ${\mathcal{P}}$, (4) Dedekind $\sigma$-complete iff every sequence in ${\mathcal{P}}$ that is bounded above in ${\mathcal{P}}$ has a supremum in ${\mathcal{P}}$, (5) monotone $\sigma$-complete iff every increasing sequence $a_{1}\leq a_{2}\leq a_{3}\leq\cdots$ that is bounded above in ${\mathcal{P}}$ has a supremum in ${\mathcal{P}}$, (6) monotone complete iff every nonempty upward directed subset of ${\mathcal{P}}$ that has an upper bound in ${\mathcal{P}}$ has a supremum in ${\mathcal{P}}$.

We now turn our attention back to the synaptic algebra $A$ and the OML $P$ in $A$. Let $p,q\in P$. If $pCq$, then $p\wedge q=pq$ and $p\vee q=p+q-pq$. Also, $p\leq q$ iff ($pCq$ and $p=pq=qp=p\wedge q$). The projections $p$ and $q$ are orthogonal, in symbols $p\perp q$, iff $p\leq q^{\perp}$, in which case $q\perp p$, $pCq$, $p\wedge q=pq=qp=0$ and $p\vee q=p+q$. It can be shown that $pCq$ iff $p=(p\wedge q)\vee(p\wedge q^{\perp})$ iff there exist $p_{1},q_{1},d\in P$ such that $p_{1}\perp q_{1}$, $(p_{1}+q_{1})\perp d$, $p_{1}+q_{1}+d=1$, $p=p_{1}+d$, and $q=q_{1}+d$.

The commutant and bicommutant have the following obvious properties. For all $M,N\subseteq A$: (i) $M\subseteq N\Rightarrow C(N)\subseteq C(M)$. (ii) $M\subseteq CC(M)$. From (i) and (ii), it follows that (iii) $M\subseteq N\Rightarrow CC(M)\subseteq CC(N)$ and (iv) $CC(C(M))=C(M)$. A subset $T\subseteq A$ is commutative iff $aCb$ for all $a,b\in T$, i.e., iff $T\subseteq C(T)$. If $T$ is commutative, then so is $CC(T)$ and $T\subseteq CC(T)\subseteq C(T)$.

If $D\subseteq P$, then $D$ is an orthogonal set iff $p\perp q$ for all $p,q\in D$. The OML $P$ is orthocomplete ($\sigma$-orthocomplete) iff every orthogonal subset (every countable orthogonal subset) has a supremum in $P$. Clearly, every orthogonal subset of $P$ is commutative.

6.1 Theorem**.**

(i)* The OML $P$ is orthocomplete ($\sigma$-orthocomplete*)* iff it is complete ($\sigma$-complete*). (ii) If $D\subseteq P$ is a maximal orthogonal set, then the supremum of $D$ in $P$ is $1$.

Proof.

See [24] for the proof of (i). To prove (ii), let $D\subseteq P$ be a maximal orthogonal set and suppose that $p$ is an upper bound for $D$ in $P$. Then $d\in D\Rightarrow d\leq p\Rightarrow d\perp p^{\perp}$, so $p^{\perp}=1-p\in D$ by maximality. But then, $1-p\leq p$, $1-p=(1-p)p=0$, i.e., $p=1$. Therefore, $1$ is the only upper bound of $D$ in $P$, so it is the supremum of $D$ in $P$. ∎

A subset $S\subseteq A$ is called a sub-synaptic algebra of $A$ iff $S$ is a linear subspace of $A$, $1\in S$, and $S$ is closed under the formation of squares, square roots, carriers, and inverses. A sub-synaptic algebra $S$ of $A$ is a synaptic algebra in its own right under the restrictions to $S$ of the partial order and the operations on $A$ and with the same enveloping algebra as $A$. For instance, if $T\subseteq A$, then $C(T)$ is a norm-closed sub-synaptic algebra of $A$. The commutative norm-closed sub-synaptic algebra $C(A)$ of $A$ is called the center of $A$ and the synaptic algebra $A$ is commutative iff $A=C(A)$.

By definition, a symmetry in $A$ is an element $s\in A$ such that $s^{2}=1$ [12]. There is a bijective correspondence $s\leftrightarrow p$ between symmetries $s$ and projections $p$ according to $p=\frac{1}{2}(1+s)$ and $s=2p-1$. If $p\in P$, $s:=2p-1$ is the corresponding symmetry, and $a\in A$, then clearly $aCp\Leftrightarrow aCs\Leftrightarrow a=sas$.

Spectral theory for $A$ is developed in [5, §8] based on the notion of the spectral resolution of an element $a\in A$, which is the one-parameter family of projections $\{p_{\lambda}:\lambda\in{\mathbb{R}}\}$ defined by $p_{\lambda}:=(((a-\lambda)^{+})^{\text{\rm o}})^{\perp}$ for all $\lambda\in{\mathbb{R}}$. We shall refer to the projections $p_{\lambda}$, $\lambda\in{\mathbb{R}}$, as the spectral projections of (or for) $a$. Note that $p_{\lambda}\in CC(a)$ for all $\lambda\in{\mathbb{R}}$. Moreover, for all $b\in A$, $bCa$ iff $bCp_{\lambda}$ for every $\lambda\in{\mathbb{R}}$ [5, Theorem 8.10]. We use the latter fact in the proofs of the next two theorems.

6.2 Theorem**.**

Let $T\subseteq A$ and suppose that $T$ has a supremum $b$ in $A$. Then $b\in CC(T)$.

Proof.

(i) Let $a\in C(T)$. We have to prove that $bCa$. Let $\{p_{\lambda}:\lambda\in{\mathbb{R}}\}$ be the spectral resolution of $a$ and let $t\in T$. Then $tCp_{\lambda}$ for all $\lambda\in{\mathbb{R}}$. It will be sufficient to prove that $bCp_{\lambda}$ for all $\lambda\in{\mathbb{R}}$. Let $\lambda\in{\mathbb{R}}$ and let $s:=2p_{\lambda}-1$ be the corresponding symmetry. Then $tCs$ for all $t\in T$ and as $t\leq b$, we have $t=sts=U_{s}t\leq U_{s}b=sbs$, whence $b\leq sbs$. Applying the quadratic mapping $U_{s}$ again to the latter inequality, we obtain $sbs\leq s^{2}bs^{2}=b$, whence $sbs=b$, so $bCs$, and therefore $bCp_{\lambda}$. Consequently, $bCa$. ∎

Following Kaplansky [26, p. 237], we may regard the next theorem as expressing “continuity properties” of a supremum in the OML $P$.

6.3 Theorem**.**

Let $V\subseteq P$ and suppose that $V$ has a supremum $p$ in the OML $P$. Then: (i)* $p\in CC(V)$. (ii) If $a\in A$ then $va=0$ for all $v\in V$ iff $pa=0$.*

Proof.

(i) We cannot use Theorem 6.2 because $p$ is not necessarily the supremum of $V$ in $A$. However, a similar argument does work: Let $a\in C(V)$ and let $s$ be the symmetry corresponding to any spectral projection of $a$. Then, for all $v\in V$, $vCs$ so $v=svs\leq sps$. But $(sps)^{2}=sps^{2}ps=sp^{2}s=sps$, so $sps\in P$, and therefore $p\leq sps$. Hence, $sps\leq p$, so $sps=p$, and consequently $pCs$, whereupon $pCa$.

(ii) Assume that $a\in A$ and $va=0$ for all $v\in V$. Then, for all $v\in V$, $va^{\text{\rm o}}=0$, i.e., $v\leq 1-a^{\text{\rm o}}\in P$, whence $p\leq 1-a^{\text{\rm o}}$, so $pa^{\text{\rm o}}=0$, and therefore $pa=0$. Conversely, suppose that $pa=0$. If $v\in V$, then $v\leq p$, whence $v=vp$, and we have $va=vpa=0$. ∎

The next lemma is the converse of Theorem 6.3 (ii).

6.4 Lemma**.**

Let $V\subseteq P$ and suppose there exists $p\in P$ such that, for all $a\in A$, $va=0$ for all $v\in V$ iff $pa=0$. Then $p$ is the supremum of $V$ in $P$.

Proof.

Since $p(1-p)=0$, we have $v(1-p)=0$, i.e., $v\leq p$ for all $v\in V$. Suppose that $v\leq r\in P$ for all $v\in V$. Then, for all $v\in V$, we have $v(1-r)=0$, so $p(1-r)=0$, i.e., $p\leq r$, and therefore $p$ is the supremum of $V$ in $P$. ∎

6.5 Theorem**.**

[16, Theorem 5.12]* The following conditions are mutually equivalent*: (i)* $A$ is commutative. (ii) As a partially ordered real linear space, $A$ is a lattice (i.e., $A$ is a vector lattice). (iii) The OML $P$ is a Boolean algebra.*

6.6 Theorem**.**

Suppose that the synaptic algebra $A$ is commutative. Then: (i)* $A$ is monotone complete (monotone $\sigma$-complete) iff $A$ is Dedekind complete (Dedekind $\sigma$-complete). (ii) If $V\subseteq P$ and $a$ is the supremum of $V$ in $A$, then $a\in P$ and $a$ is the supremum of $V$ in $P$. (iii) If $A$ is Dedekind complete (Dedekind $\sigma$-complete), then every subset $V$ of $P$ (every sequence $(p_{n})_{n\in{\mathbb{N}}}$ in $P$) has a supremum $p$ in $A$; moreover, $p\in P$, $p$ is also the supremum of $V$ in $P$ ($p$ is also the supremum of $(p_{n})_{n\in{\mathbb{N}}}$ in $P$), and $P$ is a complete (a $\sigma$-complete) Boolean algebra.*

Proof.

(i) Suppose that $A$ is monotone complete and that the nonempty set $S\subseteq A$ is bounded above in $A$. Since $A$ is commutative, it is a lattice. Let $T$ be the subset of $S$ obtained by appending to $S$ all suprema of finite nonempty subsets of $S$. Then $S$ and $T$ have the same upper bounds in $A$ and $T$ is upward directed, so it has a supremum $b$ in $A$, whence $b$ is also the supremum of $S$ in $A$. The converse is obvious. A similar argument holds for the monotone $\sigma$-complete case. Indeed, for a sequence $(a_{n})_{n\in{\mathbb{N}}}$ that is bounded above in $A$, the sequence $(b_{n})_{n\in{\mathbb{N}}}$ defined by $b_{n}:=a_{1}\vee a_{2}\vee\cdots\vee a_{n}$ is monotone increasing and has the same set of upper bounds in $A$ as $(a_{n})_{n\in{\mathbb{N}}}$.

(ii) For each $v\in V$, we have $0\leq v\leq a$ and $vCa$, whence by [18, Theorem 3.9], $v=v^{2}\leq a^{2}$ and $v=v^{1/2}\leq a^{1/2}$. Therefore, $a\leq a^{2}$ and $a\leq a^{1/2}$. Again by [18, Theorem 3.9], from $a\leq a^{1/2}$, we infer that $a^{2}\leq a$, and it follows that $a=a^{2}$, so $a\in P$. Clearly then, $a$ is the supremum of $V$ in $P$.

(iii) Suppose that $A$ is Dedekind complete and that $V\subseteq P$. Then $V$ is bounded above by $1\in P$, whence it has a supremum $p$ in $A$. Thus, by (ii), $p\in P$ and $p$ is the supremum of $V$ in $P$. By duality, $V$ has an infimum in $P$, whence $P$ is a complete Boolean algebra. The proof in the case that $A$ is Dedekind $\sigma$-complete is similar. ∎

6.7 Theorem**.**

Let $A$ be a commutative Banach synaptic algebra. Then $P$ is a $\sigma$-complete Boolean algebra and $A$ is both Dedekind and monotone $\sigma$-complete.

Proof.

Assume that $A$ is commutative and Banach. By Corollary 5.4 and Theorem 6.5, $P$ is a $\sigma$-complete Boolean algebra. Let $X$ be the Stone space of $P$, and let $C(X,{\mathbb{R}})$ be the lattice-ordered commutative associative unital Banach algebra under pointwise partial order and pointwise operations of all continuous real-valued functions on $X$. By [10, Theorem 4.1], there is a norm-dense subalgebra $F$ of $C(X,{\mathbb{R}})$ such that $F$ is a synaptic algebra, and there is a synaptic isomorphism $\Psi\colon A\to F$. Since $A$ is norm-complete, so is $F$, and since $F$ is norm-dense in $C(X,{\mathbb{R}})$, it follows that $F=C(X,{\mathbb{R}})$. Therefore, $C(X,{\mathbb{R}})$ is a commutative Banach synaptic algebra, and $\Psi\colon A\to C(X,{\mathbb{R}})$ is a synaptic isomorphism. By [17, Theorem 6.3], $C(X,{\mathbb{R}})$ is both Dedekind and monotone $\sigma$-complete; hence $A$ also has these properties. ∎

6.8 Theorem**.**

If $A$ is a commutative Banach synaptic algebra, then the following conditions are mutually equivalent: (i)* $P$ is a complete Boolean algebra. (ii) $A$ is Dedekind complete. (iii) $A$ is monotone complete.*

Proof.

(i) $\Leftrightarrow$ (ii). Suppose that $P$ is a complete Boolean algebra. Then the Stone space $X$ of $P$ is extremally disconnected, i.e., the closure of every open subset of $X$ remains open [21, Chapter 38]. As in the proof of Theorem 6.7, $C(X,{\mathbb{R}})$ is a Banach synaptic algebra isomorphic to $A$. By [33, Theorem 14], $C(X,{\mathbb{R}})$, hence also $A$, is Dedekind complete, and we have (i) $\Rightarrow$ (ii). The converse implication follows from Theorem 6.6 (iii).

By Theorem 6.6 (i), we have (ii) $\Leftrightarrow$ (iii). ∎

7 Blocks and C-blocks in a synaptic algebra

A block in the OML $P$ is a maximal commutative subset of $P$. Clearly, every block $Q$ in $P$ is closed under the formation of meets, joins, and orthocomplements in $P$, and under these operations it is a Boolean algebra. Evidently, $Q$ is a block in $P$ iff $Q=P\cap C(Q)$. By Zorn’s lemma, every commutative subset of $P$, and in particular, every singleton subset $\{p\}$ of $P$, can be enlarged to a block in $P$. Therefore, $P$ is covered by its own blocks.

By analogy with the notion of a block $Q$ in $P$, a maximal commutative subset $B$ of $A$ is called a C-block. Evidently, $B\subseteq A$ is a C-block iff $B=C(B)$, in which case $B=C(B)=CC(B)$. Every commutative subset $T$ of $A$ can be enlarged to a C-block $B\supseteq T$ (Zorn). In particular, if $a\in A$, then the singleton set $\{a\}$ is commutative, the bicommutant $CC(a)$ is commutative, $a\in CC(a)$, and $CC(a)$ can be enlarged to a C-block $B$ with $a\in B$. Therefore, $A$ is covered by its own C-blocks.

Let $B$ be a C-block in $A$. Then $B=C(B)$ is a commutative norm-closed sub-synaptic algebra of $A$ and $B\cap P$ is the projection lattice of $B$. By Theorem 6.5, $B$ is a vector lattice and $B\cap P$ is a Boolean algebra. If $p,q\in B\cap P$, then $p\vee q=p+q-pq$ is the supremum of $p$ and $q$ and $p\wedge q=pq=qp$ is the infimum of $p$ and $q$ both in $A$ and in the synaptic algebra $B$.

7.1 Theorem**.**

If $Q$ is a block in $P$, then there is a unique C-block $B$ with $Q\subseteq B$, namely, $B=C(Q)$; moreover, $B=C(B)=CC(B)=C(Q)=CC(Q)$. Conversely, if $B$ is a C-block in $A$, then $Q:=P\cap B$ is the unique block in $P$ such that $Q\subseteq B$.

Proof.

Since $Q$ is a commutative subset of $A$, there exists a C-block $B$ of $A$ with $Q\subseteq B$ (Zorn). Let $b\in B$ and let $p_{\lambda}$, $\lambda\in{\mathbb{R}}$, be a spectral projection of $b$. Then, for all $q\in Q\subseteq B$, we have $bCq$, whence $p_{\lambda}Cq$, and by the maximality of $Q$, $p_{\lambda}\in Q$. Thus, the spectral resolution of each element $b\in B$ is contained in $Q$, whence $C(Q)\subseteq C(B)=CC(B)=B$. Also, since $Q\subseteq B$, we have $B=C(B)\subseteq C(Q)$, so $B=C(Q)$. Therefore, $C(B)=CC(Q)$, whence $B=C(B)=CC(B)=C(Q)=CC(Q)$.

Conversely, suppose that $B$ is a C-block in $A$ and put $Q:=P\cap B$. As $Q\subseteq B$, we have $Q\subseteq C(Q)$, so $Q\subseteq P\cap C(Q)$. Let $b\in B$ and let $p_{\lambda}$, $\lambda\in{\mathbb{R}}$, be a spectral projection of $b$. Then $p_{\lambda}\in CC(b)\subseteq CC(B)=B$, so the spectral resolution of $b$ is contained in $P\cap B=Q$. Therefore, if $a\in C(Q)$, then $a\in C(b)$, and by the maximality of $B$, we have $a\in B$; whence $C(Q)\subseteq B$. Consequently, $P\cap C(Q)\subseteq P\cap B=Q$, so $Q=P\cap C(Q)$, and thus $Q$ is a block in $P$. Suppose $Q_{1}$ is a second block in $P$ such that $Q_{1}\subseteq B$. Then $Q_{1}\subseteq B\cap P=Q$ and it follows that $Q_{1}=Q$. ∎

7.2 Lemma**.**

Let $Q$ be a block in $P$, let $V\subseteq Q$, let $B$ be a C-block in $A$ and let $T\subseteq B$. Then: (i)* If $V$ has a supremum $p$ in $P$, then $p\in Q$ and $p$ is the supremum of $V$ in $Q$. Also, if $V$ has a supremum $a$ in $A$, then $a\in Q$ and $a$ is the supremum of $V$ both in $Q$ and in $P$. (ii) If $T$ has a supremum $b$ in $A$, then $b\in B$ and $b$ is the supremum of $T$ in $B$. (iii) If $P$ is a complete OML, then $P\cap B$ is a complete Boolean algebra.*

Proof.

(i) Since $Q$ is commutative, we have $Q\subseteq C(Q)$, whence $CC(Q)\subseteq C(Q)$. Thus, by Theorem 6.3 (i), $p\in CC(V)\subseteq CC(Q)\subseteq C(Q)$, and it follows from the maximality of $Q$ that $p\in Q$, whence $p$ is the supremum of $V$ in $Q$. Similarly, by Theorem 6.2, $a\in CC(V)\subseteq C(Q)$, so $a\in Q\subseteq P$, and $a$ is the supremum of $V$ both in $Q$ and in $P$.

(ii) By Theorem 6.2, $b\in CC(T)\subseteq CC(B)=B$.

(iii) Suppose that $P$ is a complete OML. By Theorem 7.1, $Q:=P\cap B$ is a block in $P$, and by (i), $Q$ is a complete Boolean algebra. ∎

7.3 Theorem**.**

If $A$ is a Banach synaptic algebra, then every C-block $B$ in $A$ is a commutative Banach synaptic algebra that is both Dedekind and monotone $\sigma$-complete and the Boolean algebra $B\cap P$ of projections in $B$ is $\sigma$-complete.

Proof.

Let $B$ be a C-block in $A$. Then $B=C(B)$ is norm-closed, and since $A$ is Banach, so is $B$. Therefore $B$ is a commutative Banach synaptic algebra, whence the theorem follows from Theorem 6.7 ∎

7.4 Theorem**.**

(i)* If every C-block in $A$ is Dedekind complete (Dedekind $\sigma$-complete), then every block in $P$ is a complete (a $\sigma$-complete) Boolean algebra. (ii) Every block in $P$ is a complete (a $\sigma$-complete) Boolean algebra iff $P$ is a complete (a $\sigma$-complete) OML.*

Proof.

(i) Assume that every C-block in $A$ is Dedekind complete and that $Q$ is a block in $P$. By Theorem 7.1, there is a unique C-block $B$ in $A$ with $Q\subseteq B$ and $Q=P\cap B$ is the Boolean algebra of projections in $B$. Thus, by Theorem 6.6 with $A$ replaced by $B$, it follows that $Q$ is a complete Boolean algebra. The proof for the Dedekind $\sigma$-complete case is similar.

(ii) If $P$ is complete, then so is every block in $P$ by Lemma 7.2 (i). Conversely, suppose that every block in $P$ is complete. To prove that $P$ is complete, it will be sufficient to prove that it is orthocomplete (Theorem 6.1 (i)), so let $D$ be an orthogonal subset of $P$. Note that $D$ is a commutative subset of $P$. We have to prove that $D$ has a supremum in $P$. Without loss of generality, we can assume that $0\in D$. If $D$ is a maximal orthogonal set, then $1$ is the supremum of $D$ in $P$ (Theorem 6.1 (ii)), so we can assume that $D$ is not a maximal orthogonal set. Therefore the set $V:=\{v\in P:v\not=0\text{\ and\ }d\in D\Rightarrow v\perp d\}$ is nonempty. By Zorn’s lemma, there is a maximal commutative subset $V_{1}$ of $V$. For $v\in V_{1}$ and $d\in D$, we have $v\perp d$, so $vCd$, and it follows that $V_{1}\cup D$ is a commutative subset of $P$ and that $d^{\perp}$ is an upper bound for $V_{1}$ in $P$. By Zorn’s lemma again, there is a block $Q$ in $P$ with $V_{1}\cup D\subseteq Q$.

By hypothesis, $Q$ is complete; hence, $V_{1}$ has a supremum $p$ in $Q$. For $d\in D$, we have $d\in Q$, whence $d^{\perp}\in Q$, so $d^{\perp}$ is an upper bound for $V_{1}$ in $Q$. Therefore, $p\leq d^{\perp}$, so $d\leq p^{\perp}$, whereupon $p^{\perp}$ is an upper bound for $D$ in $Q$. We claim that, in fact, $p^{\perp}$ is the supremum of $D$ in $P$, which will complete the proof.

Thus, suppose that $b$ is an upper bound for $D$ in $P$ and put $c:=p^{\perp}\wedge(p\vee b^{\perp})$. Then, $c\leq p^{\perp}$, so $p\leq c^{\perp}$, and $v\in V_{1}\Rightarrow v\leq p$, whence $v\in V_{1}\Rightarrow v\leq c^{\perp}$. In particular, $c\in C(V_{1})$. Also, $d\in D\Rightarrow d\leq p^{\perp}\wedge b\leq p\vee(p^{\perp}\wedge b)=c^{\perp}\Rightarrow c\perp d$. If $c\not=0$, it follows that $c\in V_{1}$, whence $c\leq c^{\perp}$, and so $c=0$, a contradiction. Therefore, $c=p^{\perp}\wedge(p\vee b^{\perp})=0$, and since $p$, hence also $p^{\perp}$ commutes with $p\vee b^{\perp}$, we have $p^{\perp}\leq(p\vee b^{\perp})^{\perp}=p^{\perp}\wedge b\leq b$. Thus, $P$ is complete. A similar argument takes care of the $\sigma$-complete case. ∎

7.5 Corollary**.**

If every C-block in $A$ is Dedekind complete (Dedekind $\sigma$-complete), then $P$ is a complete (a $\sigma$-complete) OML.

8 The self-adjoint part of an AW*∗*-algebra

In this section, we present some conditions that are equivalent to the requirement that the synaptic algebra $A$ is isomorphic to the self-adjoint part of an AW*∗*-algebra.

I. Kaplansky introduced AW*∗-algebras in [26] as algebraic generalizations of W∗* (i.e., von Neumann) algebras. According to Kaplansky’s original definition, an AW*∗-algebra is a C∗*-algebra ${\mathcal{C}}$ such that (1) the OML of projections in ${\mathcal{C}}$ is orthocomplete and (2) any maximal commutative $\ast$-subalgebra of ${\mathcal{C}}$ is norm-generated by its own projections. Nowadays, the following equivalent definition [27, p. 853] (which we shall use) is often given. In the definition $Sc:=\{sc:s\in S\}$ and $p_{S}{\mathcal{C}}:=\{p_{S}a:a\in{\mathcal{C}}\}$.

8.1 Definition**.**

An AW∗-algebra is a C*∗-algebra ${\mathcal{C}}$ that is a Baer∗* ring [4], i.e., for every subset $S\subseteq{\mathcal{C}}$, there is a projection $p_{S}\in{\mathcal{C}}$ such that the right annihilator of $S$, namely $\{c\in{\mathcal{C}}:Sc=\{0\}\}$, is the principal right ideal $p_{S}{\mathcal{C}}$ of ${\mathcal{C}}$ generated by the projection $p_{S}$.

If ${\mathcal{C}}$ is an AW*∗-algebra, then it is a Rickart C∗-algebra; indeed, if $c\in{\mathcal{C}}$, take $S:=\{c\}$, and we have $ca=0\Leftrightarrow a=p_{S}a$ for all $a\in A$. Therefore (Theorem 5.3), the self-adjoint part of an AW∗*-algebra is a Banach synaptic algebra.

8.2 Definition**.**

In this definition, if $T\subseteq A$, $a\in A$ and $p\in P$, then $Ta:=\{ta:t\in T\}$ and $pA:=\{pa:a\in A\}$ for $p\in P$.

1. (1)

The synaptic algebra $A$ has the Baer property iff, for every subset $T\subseteq A$, there is a projection $p_{T}\in P$ such that $\{a\in A:Ta=\{0\}\}=p_{T}A$. 2. (2)

$A$ has the complete carrier property iff, for every $T\subseteq A$, there is a projection $q_{T}\in P$ such that, for all $a\in A$, $Ta=\{0\}\Leftrightarrow q_{T}a=0$.

8.3 Theorem**.**

The following conditions are mutually equivalent:**

1. (i)

$A$* has the Baer property.* 2. (ii)

$A$* has the complete carrier property.* 3. (iii)

$P$* is a complete OML.*

Proof.

(i) $\Leftrightarrow$ (ii). With the notation of Definition 8.2, if $T\subseteq A$ and $A$ has the Baer property, put $q_{T}:=1-p_{T}$, and if $A$ has the complete carrier property, put $p_{T}:=1-q_{T}$.

(ii) $\Rightarrow$ (iii). Assume (ii) and let $V\subseteq P$. Then there is a projection $q_{V}\in P$ such that, for all $a\in A$, $Va=\{0\}\Leftrightarrow q_{V}a=0$, and by Lemma 6.4, $q_{V}$ is the supremum of $V$ in the OML $P$. By duality, every subset of the OML $P$ has an infimum.

(iii) $\Rightarrow$ (ii). Assume (iii), let $T\subseteq A$, put $S:=\{t^{\text{\rm o}}:t\in T\}\subseteq P$, and let $r$ be the supremum of $S$ in $P$. By Theorem 6.3 (ii), for every $a\in A$, $Sa=0\Leftrightarrow ra=0$. Now $ra=0\Leftrightarrow t^{\text{\rm o}}a=0,\forall t\in T\Leftrightarrow ta=0,\forall t\in T\Leftrightarrow Ta=\{0\}$, whence $A$ satisfies the condition in Definition 8.2 (2) with $q_{T}=r$. ∎

8.4 Theorem**.**

Let ${\mathcal{C}}$ be a Rickart C∗ algebra and organize the self-adjoint part ${\mathcal{C}}^{sa}$ of ${\mathcal{C}}$ into a Banach synaptic algebra (Theorem 5.3). Then ${\mathcal{C}}$ is an AW∗-algebra iff ${\mathcal{C}}^{sa}$ has the Baer property.

Proof.

Suppose that ${\mathcal{C}}$ is an AW*∗*-algebra and, $T\subseteq{\mathcal{C}}^{sa}$, and let $a\in{\mathcal{C}}^{sa}$. Then $T\subseteq{\mathcal{C}}$ and $a\in C$, whence, there is a projection $p_{T}$ in ${\mathcal{C}}^{sa}$ such that, $Ta=\{0\}\Leftrightarrow a=p_{T}a$. Therefore ${\mathcal{C}}^{sa}$ has the Baer property.

Conversely, suppose that ${\mathcal{C}}^{sa}$ has the Baer property, $S\subseteq{\mathcal{C}}$, and $c\in{\mathcal{C}}$. Then $T:=\{s^{\ast}s:s\in S\}\subseteq{\mathcal{C}}^{sa}$, so there exists a projection $p_{T}\in{\mathcal{C}}^{sa}$ such that, for all $a\in{\mathcal{C}}^{sa}$, $Ta=\{0\}\Leftrightarrow a=p_{T}a$. Putting $a=cc^{\ast}$, we obtain $Tcc^{\ast}=\{0\}\Leftrightarrow cc^{\ast}=p_{T}cc^{\ast}$, whence by Lemma 2.1 (ii), for all $c\in{\mathcal{C}}$,

[TABLE]

To prove that ${\mathcal{C}}$ is an AW*∗*-algebra, it will be sufficient to prove that $sc=0\text{\ for all\ }s\in S\Leftrightarrow c=p_{T}c$. If $sc=0$ for all $s\in S$, then $s^{\ast}scc^{\ast}=0$ for all $s\in S$, so $c=p_{T}c$ by (1). Conversely, suppose that $c=p_{T}c$. Putting $c=p_{T}$ in (1) and observing that $p_{T}=p_{T}p_{T}$, we find that, for all $s\in S$, $s^{\ast}sp_{T}=s^{\ast}sp_{T}p_{T}^{\ast}=0$, whence $(sp_{T})^{\ast}sp_{T}=p_{T}s^{\ast}sp_{T}=0$, and it follows that $sp_{T}=0$. Therefore, $sc=sp_{T}c=0$. ∎

8.5 Theorem**.**

The synaptic algebra $A$ is isomorphic to the self-adjoint part of an AW∗-algebra iff it is Banach and satisfies any one (hence all) of the following equivalent conditions: (i)* $A$ has the Baer property. (ii) $A$ has the complete carrier property. (iii) The OML $P$ of projections in $A$ is complete.*

Proof.

Conditions (i)–(iii) are mutually equivalent by Theorem 8.3. Suppose that $A$ is isomorphic to the self-adjoint part ${\mathcal{C}}^{sa}$ of an AW*∗-algebra ${\mathcal{C}}$. Than $A$ is Banach and by Theorem 8.4, ${\mathcal{C}}^{sa}$, hence also $A$, has the Baer property. Conversely, suppose that $A$ is Banach and satisfies any one of the equivalent conditions (i), (ii), or (iii). Then it satisfies (i), and since it is Banach, it is isomorphic to the self-adjoint part ${\mathcal{C}}^{sa}$ of a Rickart C∗-algebra ${\mathcal{C}}$ by Theorem 5.3. But then, ${\mathcal{C}}^{sa}$ has the Baer property, so ${\mathcal{C}}$ is an AW∗*-algebra by Theorem 8.4. ∎

In [30], K. Saitô and J.D.M. Wright proved that a C*∗-algebra ${\mathcal{C}}$ is an AW∗*-algebra iff every maximal abelian ${\ast}$-subalgebra of ${\mathcal{C}}$ is monotone complete. The equivalence of parts (i) and (v) in the following theorem can be regarded as an analogue for synaptic algebras of the Saitô-Wright theorem.

8.6 Theorem**.**

Let $A$ be a Banach synaptic algebra. Then the following conditions are mutually equivalent: (i)* Every C-block in $A$ is monotone complete. (ii) Every C-block in $A$ is Dedekind complete. (iii) Every block in $P$ is a complete Boolean algebra. (iv) $P$ is a complete OML. (v) $A$ is isomorphic to the self-adjoint part of an AW∗-algebra*

Proof.

That (i) $\Leftrightarrow$ (ii) follows from Theorem 6.6 (i). By Theorem 7.4, we have (ii) $\Rightarrow$ (iii) $\Leftrightarrow$ (iv). We claim that (iv) $\Rightarrow$ (i). Thus, assume that $A$ is Banach, $P$ is a complete OML, and $B$ is a C-block in $A$. Then $B$ is a norm-closed, hence Banach, commutative synaptic algebra under the restrictions of the partial order and operations on $A$, and $P\cap B$ is the Boolean algebra of projections in $B$. By Lemma 7.2 (iii), $P\cap B$ is a complete Boolean algebra, and, applying Theorem 6.8 to the commutative Banach synaptic algebra $B$, we infer that $B$ is monotone complete. Thus, conditions (i)–(iv) are mutually equivalent, and by Theorem 8.5, (iv) $\Leftrightarrow$ (v). ∎

9 Synaptic algebras and GH-algebras

Axioms for a generalized Hermitian algebra (GH-algebra) can be found in [7, Definition 2.1]. By the discussion in [5, §6] and Theorem 6.2, we have the following.

9.1 Theorem**.**

A GH-algebra is the same thing as a synaptic algebra $A$ such that every bounded monotone increasing sequence $a_{1}\leq a_{2}\leq\cdots$ of pairwise commuting elements in $A$ has a supremum in $A$.

9.2 Corollary**.**

If $A$ is monotone $\sigma$-complete, then $A$ is a Banach GH-algebra.

Proof.

If $A$ is monotone $\sigma$-complete, then $A$ is a GH-algebra by Theorem 9.1 and $A$ is Banach by [19, Theorem 2.4]. ∎

9.3 Theorem**.**

Suppose that $A$ is a commutative synaptic algebra. Then the following conditions are mutually equivalent: (i)* $A$ is Dedekind $\sigma$-complete. (ii) $A$ is monotone $\sigma$-complete. (iii) $A$ is Banach. (iv) $A$ is a GH-algebra.*

Proof.

Obviously (i) $\Rightarrow$ (ii) and by [19, Theorem 2.4], (ii) $\Rightarrow$ (iii). By Theorem 6.7, (iii) $\Rightarrow$(i), whence (i), (ii), and (iii) are mutually equivalent. Also, Since $A$ is commutative, the equivalence (ii) $\Leftrightarrow$ (iv) is obvious from Theorem 9.1. ∎

In the following corollary of Theorem 9.3, note that every C-block $B\subseteq A$ and every bicommutant $CC(M)$ of a commutative subset $M\subseteq A$ qualifies as a norm-closed commutative sub-synaptic algebra of $A$.

9.4 Corollary**.**

Let $A$ be a Banach synaptic algebra and let $H$ be a norm-closed commutative sub-synaptic algebra of $A$. Then $H$ is a Banach GH-algebra.

Proof.

As $A$ is norm complete and $H$ is norm-closed, it follows that $H$ is norm-complete. Therefore, we can apply Theorem 9.3 to the commutative synaptic algebra $H$ and conclude that $H$ is a GH-algebra. ∎

9.5 Theorem**.**

If $A$ is a GH-algebra, then every C-block in $A$ is a commutative monotone $\sigma$-complete Banach GH-algebra and $P$ is a $\sigma$-complete OML.

Proof.

Suppose $A$ is a GH-algebra, let $B$ be a C-block of $A$, and let $(b_{n})_{{n\in{\mathbb{N}}}}$ be a bounded monotone increasing sequence in the commutative synaptic algebra $B$. Then $(b_{n})_{{n\in{\mathbb{N}}}}$ is a bounded monotone increasing sequence of pairwise commuting elements in $A$, so it has a supremum $a\in A$, whence by Theorem 6.2, $a\in CC((b_{n})_{n\in{\mathbb{N}}})\subseteq CC(B)=B$. Thus, $B$ is a commutative monotone $\sigma$-complete synaptic algebra, hence it is a GH-algebra. By Corollary 7.5, $P$ is $\sigma$-complete. ∎

9.6 Remark**.**

Even if every C-block $B$ in the synaptic algebra $A$ is monotone $\sigma$-complete, we cannot conclude that $A$ is a GH-algebra because the supremum in a C-block $B\subseteq A$ of a sequence $(b_{n})_{{n\in{\mathbb{N}}}}$ in $B$ is not necessarily the supremum of the sequence in $A$.

We conclude this section by reviewing two functional representation theorems for a commutative GH-algebra $A$. By Theorem 9.3, $A$ is a Dedekind $\sigma$-complete Banach synaptic algebra and $P$ is a $\sigma$-complete Boolean algebra. Thus, as is well-known, the Stone space $X$ of $P$ is not only totally disconnected, but it is basically disconnected, i.e., the closure of any open Fσ subset of $X$ remains open. As in the proof of Theorem 6.7, the lattice ordered commutative associative unital Banach algebra $C(X,{\mathbb{R}})$ with pointwise partial order and pointwise operations, and with the supremum norm, of all continuous real-valued functions on $X$ is a GH-algebra; moreover, $A$ is isomorphic (as a synaptic algebra) to $C(X,{\mathbb{R}})$. A disadvantage of this functional representation is that the supremum of an increasing sequence in $C(X,{\mathbb{R}})$ is not necessarily the pointwise supremum.

A second functional representation of the commutative GH-algebra $A$ avoids the disadvantage mentioned above. This representation involves a so-called gh-tribe on the Stone space $X$ of $P$. By [19, Definition 6.3], a gh-tribe on $X$ is a set ${\mathcal{T}}$ of bounded real-valued functions on $X$ such that: (1) the constant functions 0 and 1 belong to ${\mathcal{T}}$; (2) ${\mathcal{T}}$ is closed under pointwise sums and real multiples of its elements; and (3) ${\mathcal{T}}$ is closed under pointwise suprema of sequences of its elements which are bounded above by an element of ${\mathcal{T}}$. It turns out that such a gh-tribe ${\mathcal{T}}$ is closed under pointwise multiplication ([19, Lemma 6.4]) and it is a GH-algebra. Moreover, by [19, Theorem 6.6], there exists a gh-tribe ${\mathcal{T}}$ on $X$ such that $C(X,{\mathbb{R}})\subseteq{\mathcal{T}}$ and there exists a surjective synaptic morphism $h$ of GH-algebras from ${\mathcal{T}}$ onto $A$. The morphism $h$ has the property that, for $f,g\in{\mathcal{T}}$, $h(f)=h(g)$ iff $\{x\in X:f(x)\not=g(x)\}$ is a meager subset of $X$, i.e., it is a countable union of nowhere dense sets. This representation by a gh-tribe can be regarded as an analogue of the classical Loomis-Sikorski theorem for $\sigma$-complete Boolean algebras.