# Spectral order unit spaces and JB-algebras

Anna Jenčová and Sylvia Pulmannová111 Mathematical Institute, Slovak Academy
of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia;
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Order unit spaces with comparability and spectrality properties as introduced
by Foulis are studied. We define continuous functional calculus for order unit
spaces with the comparability property and Borel functional calculus for
spectral order unit spaces. Applying the conditions of Alfsen and Schultz, we
characterize order unit spaces with comparability property that are JB-
algebras. We also prove a characterization of Rickart JB-algebras as those JB-
algebras for which every maximal associative subalgebra is monotone
$\sigma$-complete, extending an analogous result of Saitô and Wright for
C*-algebras.

Keywords: order unit space, spectrality, functional calculus, JB-algebra,
generalized spin factor

##  1 Introduction

We continue the study of spectrality of order unit spaces in the sense
introduced by Foulis ([12, 13, 15, 16, 14, 17, 18]), started in [25], where we
compared this approach with the well known notion of spectral duality of
Alfsen and Schultz [1, 2]. It was proved that the purely algebraic approach of
Foulis is strictly more general than that of Alfsen and Schultz which is based
on the geometry of dual pairs of order unit and base normed spaces. This was
illustrated on the examples of JB-algebras and of order unit spaces
constructed from Banach spaces. Moreover, the structure of spectral order unit
spaces was described in detail.

Spectrality in the sense of Foulis requires the existence of a distinguished
family of positive idempotent mappings, called a compression base. A
compression base is introduced as a generalization of the set of compressions
on a von Neumann algebra $\mathcal{A}$, defined by a projection
$p\in\mathcal{A}$ as

$a\mapsto pap,\qquad a\in\mathcal{A}.$

Spectrality of the order unit space is then characterized by two properties of
the chosen compression base: the comparability property (that can be
interpreted as existence of a unique orthogonal decomposition for each element
into its positive and negative parts) and the projection cover property
(generalizing the existence of supports of self-adjoint elements in
$\mathcal{A}$).

In the present paper, we show that for an order unit space $A$ with the
comparability property, we can define a continuous functional calculus, which
can be extended to Borel functional calculus if $A$ is spectral. Further, we
characterize the order unit spaces with comparability property that are JB-
algebras. For order unit spaces in spectral duality, Alfsen and Shultz proved
several equivalent conditions for being JB-algebras. We show that those
conditions, formulated in terms of compressions and by functional calculus,
can be used also in our more general situation. This result is applied to the
example of order unit spaces obtained from Banach spaces (which we here call
generalized spin factors) and it is proved that such an order unit space is a
JB-algebra if and only if it is a spin factor (that is, the underlying Banach
space is a Hilbert space).

In the last section, we concentrate on spectral JB-algebras. It was proved in
[25] that a JB-algebra is (Foulis) spectral if and only if it is Rickart
(recall that a JB-algebra is Alfsen-Schultz spectral if and only if it is a
JBW-algebra). Rickart JB-algebras were introduced as a generalization of
Rickart C*-algebras [5, 4]. Using recent results in [28], we prove for JB-
algebras an analogue of the following characterization by Saitô and Wright
[29]: A C*-algebra is Rickart if and only if every maximal commutative self-
adjoint subalgebra is monotone $\sigma$-complete.

##  2 Spectrality in order unit spaces

Recall that an _order unit space_ is an archimedean partially ordered real
vector space with a distinguished order unit. An order unit space will be
denoted by a triple $(A,A^{+},1)$, where $A^{+}:=\\{a\in A:0\leq a\\}$ is the
positive cone and $1\in A^{+}$ denotes the order unit. The unit interval in
$A$ will be denoted by $E:=\\{e\in A:0\leq e\leq 1\\}$. Notice that $E$ is an
effect algebra in the sense of [11]. The _order unit norm_ $\|.\|_{1}$ on $A$
is defined by

$\|a\|_{1}:=\inf\\{\lambda\in{\mathbb{R}}^{+}:-\lambda\leq a\leq\lambda\\},\
a\in A.$

The theory of spectral duality in order unit spaces was developed by Alfsen
and Shultz, [1, 2]. In this paper, we will deal with another approach to
spectrality developed in [18], based on the works by Foulis [12, 13, 15, 16,
17, 14]. A comparison of these two approaches can be found in [25]. In the
sequel we briefly describe some details of the latter approach.

######  2.1 Definition.

A positive linear mapping $J:A\to A$ is a _compression with focus_ $p$ on $A$
if for all $e\in E$,

  1. (F1)

$J(1)=p\in E$ (that is, $J$ is normalized),

  2. (F2)

$e\leq p\ \implies\ J(e)=e$,

  3. (F3))

$J(e)=0\ \implies\ e\leq 1-p$.

If $J$ satisfies (F1) and (F2) but not necessarily (F3), we say that $J$ is a
_retraction_.

Note that the maps defined above were called _F-compressions_ in [25], in
order to distinguish them from the stronger notion of compressions by Alfsen
and Schultz. Since we will deal with only one type of maps, we will call them
compressions throughout this work.

Let us denote

$\operatorname{Ker}^{+}(J):=\\{a\in
A^{+}:J(a)=0\\},\quad\operatorname{Im}^{+}(J):=\\{a\in A^{+}:J(a)=a\\}.$ (1)

The compressions $J$ and $J^{\prime}$ are _complementary_ if
$\operatorname{Ker}^{+}(J)=\operatorname{Im}^{+}(J^{\prime})$ and
$\operatorname{Ker}^{+}(J^{\prime})=\operatorname{Im}^{+}(J)$. Let $J(1)=p$
and $J^{\prime}(1)=p^{\prime}$. It turns out that $J$ and $J^{\prime}$ are
complementary iff $p^{\prime}=1-p$, [25, Lemma 3.6].

Recall that an element $p\in E$ is _sharp_ if $p\wedge(1-p)=0$ (i.e., if
$x\leq p,x\leq 1-p$ then $x=0$), and an element $p\in E$ is _principal_ if
$e,f\leq p$ and $e+f\leq 1$, then $e+f\leq p$. It is easy to see that a
principal element is sharp.

######  2.2 Lemma.

[25, Lemma 3.7]. Let $p$ be the focus of a retraction $J$. Then $p$ is
principal.

Elements $a,b$ in an effect algebra $E$ are _Mackey compatible_ (denoted
$a\leftrightarrow b$) in $E$, if there are elements $c,a_{1},b_{1}\in E$ such
that $c+a_{1}+b_{1}\in E$ and $a=c+a_{1},b=c+b_{1}$. If $F$ is a subset of
$E$, we say that $a,b\in F$ are Mackey compatible in $F$, if they are
compatible in $E$, and elements $c,a_{1},b_{1}$ exist in $F$.

Let $E$ be an effect algebra and let $P$ be a subalgebra of $E$. Then $P$ is a
_normal subalgebra_ if for all $d,e,f\in E$ such that $d+e+f\leq 1$ and
$d+e,d+f\in P$ we have $d\in P$. Notice that if $P$ is normal, then elements
$p,q$ in $P$ are Mackey compatible in $E$ if and only if they are Mackey
compatible in $P$.

######  2.3 Definition.

A _compression base_ for $A$ is a family $(J_{p})_{p\in P}$ of compressions on
$A$ , indexed by their own foci, such that $P$ is a normal subalgebra of $E$
and whenever $p,q,r\in P$ and $p+q+r\leq 1$, then $J_{p+r}\circ
J_{q+r}=J_{r}$. Elements of $P$ will be called _projections_.

######  2.4 Example.

An important example is the order unit space $A=B_{sa}(\mathcal{H})$ of
bounded self-adjoint operators on a Hilbert space $\mathcal{H}$, or, more
generally, the self-adjoint part $A=\mathcal{A}_{sa}$ of a C*-algebra
$\mathcal{A}$, with the set of maps $U_{p}:a\mapsto pap$ for a projection
$p\in P(\mathcal{H})$. It can be seen that each $U_{p}$ is a compression with
focus $p$ and $(U_{p})_{p\in P(\mathcal{H})}$ is a compression base. Moreover,
any compression on $A$ has this form for some projection $p$, [13].

In the sequel, we fix a compression base $(J_{p})_{p\in P}$ for $A$. Notice
that since $P$ is a subalgebra, any $J_{p}$ has a (fixed) complementary
compression $J_{1-p}$. Using the fact that all projections are principal
elements, it is easily seen, that if $p,q\in P$ and $p+q\leq 1$, then
$p+q=p\vee q$, so that $P$ with the orthocomplementation $p\mapsto 1-p$ is an
_orthomodular poset_ (OMP) [27, 24]. It follows that if $p,q\in P$ are Mackey
compatible, then $p\wedge q$ and $p\vee q$ exist in $P$. It was shown that $P$
is moreover a _r_ egular OMP, which means that any pairwise Mackey compatible
subset in $P$ is contained in a Boolean subalgebra of $P$ [24]. A maximal
subset of pairwise Mackey compatible elements is a maximal Boolean subalgebra
in $P$ and is called a _block_ of $P$. The set $P$ is covered by its blocks.

The following notion for $p\in P$ and $a\in A$ was introduced in [18, Def.
1.5].

######  2.5 Definition.

For $p\in P$ and $a\in A$, we say that $a$ _commutes_ with $p$ if

$a=J_{p}(a)+J_{1-p}(a).$

The set of all $a$ commuting with $p\in P$ will be denoted by $C(p)$ and
called the _commutant_ of $p$. For a subset $Q\subseteq P$, we denote
$C(Q):=\bigcap_{p\in Q}C(p)$. If $B\subseteq P$ is a block of $P$, the set
$C(B)$ is called a _C-block_ of $A$.

The following facts are easily checked (see also [18, Lemma 1.3]).

######  2.6 Lemma.

Let $a\in A$, $p\in P$. Then

  1. (i)

If $J_{p}(a)\leq a$, then $a\in C(p)$.

  2. (ii)

If $a\in A^{+}$, then $a\in C(p)$ if and only if $J_{p}(a)\leq a$.

  3. (iii)

If $a\in E$, then $a\in C(p)$ if and only if $a$ and $p$ are Mackey
compatible. In this case, $J_{p}(a)=a\wedge p$.

  4. (iv)

If $q\in P$, then $q\in C(p)$ if and only if $p\in C(q)$ if and only if

$J_{p}J_{q}=J_{q}J_{p}=J_{p\wedge q}$

We denote the set of all projections commuting with $a\in A$ by $PC(a)$ and
for $B\subseteq A$, $PC(B):=\bigcap_{a\in B}PC(a)$.

The following subset was called a _bicommutant_ of $a$ in $P$ in [25]. To
avoid confusion with other notions to be introduced below, we prefer here to
call it a _P-bicommutant_ of $a$:

$P(a)=PC(PC(a)\cup\\{a\\}).$

######  2.7 Definition.

We say that the compression base $(J_{p})_{p\in P}$ has the _projection cover
property_ if for every effect $e\in E$ there is an element $p\in P$ such that
for $q\in P$, we have $e\leq q$ iff $p\leq q$. Such an element is unique and
is called the _projection cover_ of $e$, denoted by $e^{o}$.

It turns out that for every $e\in E$, $e^{o}\in P(e)$, [25, Lemma 3.16]. If
$(J_{p})_{p\in P}$ has the projection cover property, then $P$ is an
orthomodular lattice (OML), [15, Theorem 6.4], [25, Theorem 3.17].

######  2.8 Definition.

[18, Def. 1.6] We say that the compression base $(J_{p})_{p\in P}$ in $A$ has
the
_comparability property_ if, for every $a\in A$, $P^{\pm}(a)\neq\emptyset$,
where

$P^{\pm}(a):=\\{p\in P(a)\ \mbox{and}\ J_{1-p}(a)\leq 0\leq J_{p}(a)\\}.$

Notice that if a compression base $(J_{p})_{p\in P}$ has the comparability
property, then $p\in P$ iff $p$ is sharp [25, Lemma 3.20].

Let $p\in P^{\pm}(a)$ and put $b:=J_{p}(a)$, $c=-J_{1-p}(a)$, then we have

$a=b-c,\ b,c\in A^{+},\ J_{p}(b)=b,J_{p}(c)=0.$ (2)

Any decomposition of $a$ of the form (2) for some $p\in P$ is called a _$P$
-orthogonal decomposition_ of $a$.

By [25, Proposition 3.19], if $A$ has the comparability property, then every
element $a\in A$ admits a unique orthogonal decomposition denoted by
$a=a^{+}-a^{-}$, $a^{+}=J_{p}(a)$, $a^{-}=-J_{1-p}(a)$. Moreover, we have
$a^{+},a^{-},|a|:=a^{+}+a^{-}\in C(PC(a))$.

######  2.9 Theorem.

[25, Corollary 3.26] Assume that $(J_{p})_{p\in P}$ has the comparability
property.

  1. (i)

For any $a\in A$ and $p\in P$, $a\in C(p)$ if and only if $P(a)\subseteq
C(p)$.

  2. (ii)

For any $a\in A$, there is some block $B\subseteq P$ such that $a\in C(B)$.

  3. (iii)

For any block $B$, $C(B)=\overline{\mathrm{span}}(B)$.

The statement (ii) above shows that if $A$ has the comparability property,
then it is covered by C-blocks. We next describe the structure of the C-blocks
in more detail.

######  2.10 Lemma.

[25, Lemma 3.24] Assume that $(J_{p})_{p\in P}$ has the comparability property
and let $B$ be a block of $P$. Then $C(B)$ is an order unit space and
$(\bar{J_{p}})_{p\in B}$ where ${\bar{J}_{p}}:=J_{p}|C(B)$, is a compression
base in $C(B)$ with the comparability property.

######  2.11 Theorem.

[25, Theorem 3.25] Let $(J_{p})_{p\in P}$ be a compression base in $A$ with
the comparability property. Let $B\subseteq P$ be a block. Then there is a
totally disconnected compact Hausdorff space $X$ such that

  1. (i)

$B$ is isomorphic (as a Boolean algebra) to the Boolean algebra
${\mathcal{P}}(X)$ of all clopen subsets in $X$.

  2. (ii)

$C(B)$ is isomorphic (as an order unit space) to a norm-dense order unit
subspace in $C(X,{\mathbb{R}})$.

  3. (iii)

If $A$ is norm-complete, then $C(B)\simeq C(X,{\mathbb{R}})$ (as an order unit
space).

In the comparability case, we can extend the notion of commutativity to all
pairs of elements in $A$.

######  2.12 Definition.

[25, Definition 3.26] Let $(J_{p})_{p\in P}$ be a compression base in $A$ with
the comparability property. We say that $a,b\in A$ _commute_ , in notation
$aCb$, if $p$ and $q$ are compatible for all $p\in P(a)$ and $q\in P(b)$.

By [25, Proposition 3.27], two elements in $A$ commute if and only if they are
in the same C-block. In addition, the C-blocks of $A$ are precisely the
maximal sets of mutually commuting elements.

Now we introduce the notion of spectrality of order unit spaces in the sense
of [18].

######  2.13 Definition.

[18, Definition 1.7] The compression base $(J_{p})_{p\in P}$ in an order unit
space is _spectral_ if it has both the projection cover and the comparability
property.

######  2.14 Theorem.

[25, Theorem 3.33] Assume that $A$ is norm complete and let $(J_{p})_{p\in P}$
be a compression base with the comparability property. The following are
equivalent:

  1. (i)

$(J_{p})_{p\in P}$ is spectral.

  2. (ii)

For any block $B$ of $P$, $({\bar{J}_{p}}:=J_{p}|C(B))_{p\in B}$ is a spectral
compression base in $C(B)$.

  3. (iii)

Any C-block in $A$ is monotone $\sigma$-complete.

  4. (iv)

Any C-block in $A$ is isomorphic to $C(X,{\mathbb{R}})$ for some basically
disconnected compact Hausdorff space $X$.

  5. (v)

$P$ is monotone $\sigma$-complete.

If $(J_{p})_{p\in P}$ is a spectral compression base in $A$, then there is a
unique mapping $*:A\to P$, called the _Rickart mapping_ , such that for all
$a\in A$ and $p\in P$,

$p\leq a^{*}\ \Leftrightarrow\ a\in C(p)\ \mbox{and}\ J_{p}(a)=0.$ (3)

If a compression base has the comparability property, then the projection
cover property is equivalent to the existence of the Rickart mapping [18, Thm
2.1]. In particular, for $a\in E$ we have $a^{*}=1-a^{o}$. Moreover, by [25,
Proposition 3.35], for any $a\in A$, $(a^{+})^{**}$ is the least element in
$P^{\pm}(a)$.

We may now define the _spectral resolution_ of $a$ as the family
$(p_{a,\lambda})_{\lambda\in{\mathbb{R}}}\subseteq P$ defined by [18, Def. 2]

$p_{a,\lambda}:=((a-\lambda)^{+})^{*}.$

We have the following characterization of compatibility via spectral
resolution:

######  2.15 Lemma.

[25, Cor 3.36] If $p\in P$, then $a\in C(p)\Leftrightarrow p_{a,\lambda}\in
C(p)$ for all $\lambda\in{\mathbb{R}}$.

Moreover, every element in $A$ can be written as a norm-convergent integral of
Riemann-Stieltjes type [18, Rem. 3.1]

$a=\int^{U_{a}}_{L_{a}-0}\lambda dp(\lambda)$

where the _spectral lower and upper bounds_ for $a$ are defined by
$L_{a}:=\sup\\{\lambda\in{\mathbb{R}}:\lambda\leq a\\}$ and
$U_{a}:=\inf\\{\lambda\in{\mathbb{R}}:a\leq\lambda\\}$, respectively.

######  2.16 Definition.

We say that an order unit space is spectral (has the comparability property)
if there exists a compression base on $A$ that is spectral (has the
comparability property).

Note that it follows from [26, Prop. 4.7 and Thm. 5.10] that if $A$ is
spectral (has the comparability property), then any compression base such that
$P$ is the set of all sharp elements must be spectral (have the comparability
property). This can be seen from the fact that both the comparability and the
projection cover property are obtained from the behaviour of the restrictions
of the compression base to C-blocks. The C-blocks in $A$ only depend on the
blocks in $P$ (Theorem 2.9), moreover, any compression with focus $p\in B$ for
a block $B\subseteq P$ restricts to the unique compression with focus $p$ on
$C(B)$, since $C(B)$ is isomorphic to a dense subspace $C(X)$ (Theorem 2.11)
which is the self-adjoint part of an abelian C*-algebra (cf. Example 2.4). By
the results of [26], the spectral resolution of an element $a\in A$ coincides
with the (unique) spectral resolution in any C-block containing $a$, so that
the spectral resolution does not depend on the choice of the spectral
compression base.

##  3 Functional calculus in order unit spaces

In this section, we introduce functional calculus in an order unit Banach
space with the comparability or spectrality property.

###  3.1 Commutants

Assume that the compression base $(J_{p})_{p\in P}$ in $A$ has the
comparability property. Let us denote (cf. Definition 2.12)

$C(a):=\\{b\in A:aCb\\}.$

The set $C(a)$ will be called the _commutant_ of $A$. For a subset
$A_{0}\subset A$, we denote $C(A_{0})=\bigcap_{a\in A_{0}}C(a)$. Note that for
$A_{0}\subseteq P$, we get the same notion of a commutant as before.

######  3.1 Proposition.

Let the compression base $(J_{p})_{p\in P}$ in $A$ have the comparability
property. Let $a\in A$ and let $A_{0}\subseteq A$. Then

  1. (i)

$C(a)=C(P(a))$.

  2. (ii)

$a\in C(a)$.

  3. (iii)

For any $b\in C(A_{0})$, we have $P(b)\subseteq C(A_{0})$.

  4. (iv)

$C(A_{0})$ is a norm-closed subspace containing 1 and
$\\{J_{p}|_{C(A_{0})}\\}_{p\in P\cap C(A_{0})}$ is a compression base in
$C(A_{0})$ with the comparability property.

###### Proof.

(i) By definition and Theorem 2.9,

$C(a)=\\{b:\ aCb\\}=\\{b:\ P(b)\subseteq C(P(a))\\}=\\{b:\ b\in
C(P(a))\\}=C(P(a)).$

This also implies (ii) since by the definition of $P(a)$, we have $a\in
C(P(a))$. The statement (iii) follows from the definition of the commutant:
let $a\in A_{0}$, then $bCa$ means that $a\in C(P(b))$ so that $P(b)\subseteq
C(a)$. To prove (iv), note that for $p\in P$, $C(p)$ is clearly a norm-closed
subspace and $1\in C(p)$. Since $C(a)=\cap_{p\in P(a)}C(p)$ by (i), the same
holds true for $C(a)$ and clearly also for $C(A_{0})$. Let $F$ be the interval
$[0,1]$ in $C(A_{0})$, we show that $Q:=C(A_{0})\cap P$ is a normal subalgebra
in $F$. So let $d,e,f\in F$ such that $d+e+f\leq 1$ and $d+e,d+f\in Q$. This
implies that $d\in P\cap C(A_{0})=Q$, since $P$ is a normal subalgebra in
$E=A[0,1]$. Let $q\in Q$ and $b\in C(A_{0})$. Then for any $a\in A_{0}$ and
$p\in P(a)$ we have $q,b\in C(p)$ and therefore

$J_{q}(b)=J_{q}(J_{p}(b)+J_{1-p}(b))=J_{p}(J_{q}(b))+J_{1-p}(J_{q}(b))$

so that $J_{q}(b)\in C(p)$ for all $p\in P(a)$, hence $J_{q}(b)\in
C(P(a))=C(a)$. This implies that all the compressions $J_{q}$, $q\in Q$
preserve $C(A_{0})$, so they define compressions on $C(A_{0})$. It is clear
that $\\{J_{q}\\}_{q\in Q}$ is a compression base. The comparability property
follows by comparability in $A$ and the fact that by (iii),
$P^{\pm}(b)\subseteq P(b)\subseteq Q$ for any $b\in C(A_{0})$.

∎

We will also denote by $CC(A_{0})=C(C(A_{0}))$ the _bicommutant_ of
$A_{0}\subset A$.

######  3.2 Proposition.

  1. (i)

If $A_{0}\subseteq A$ is commutative then $CC(A_{0})$ is commutative.

  2. (ii)

For $a\in A$, $P(a)=CC(a)\cap P$.

  3. (iii)

For $a\in A$, $CC(a)=\overline{\mathrm{span}}(P(a))$ (the norm-closed linear
span of $P(a)$).

###### Proof.

Since $A_{0}$ is commutative, we have $A_{0}\subseteq C(A_{0})$ so that
$CC(A_{0})\subseteq C(A_{0})=C(CC(A_{0}))$, which implies that $CC(A_{0})$ is
commutative. For (ii), we clearly have $P(a)\subseteq CC(P(a))=CC(a)$ by Prop.
3.1 (i). Conversely, let $p\in CC(a)\cap P$, then $a\in C(p)$ since $a\in
C(a)$. Moreover, we have $PC(a)=C(a)\cap P$, so that $p\in
PC(PC(a)\cup\\{a\\})=P(a)$. For (iii), note that by Theorem [25, Thm. 3.22],
we have for any $b\in A$ that $b\in\overline{\mathrm{span}}(P(b))$. Further,
we obtain from (ii) and Prop. 3.1 (iii) that for $b\in CC(a)$

$P(b)\subseteq CC(a)\cap P=P(a),$

so that $CC(a)\subseteq\overline{\mathrm{span}}(P(a))$. The other inclusion
follows from (ii) and Prop. 3.1 (iv).

∎

The following theorem is similar to [25, Thm. 3.25] ( Theorem 2.11), the proof
is exactly the same, using the fact that $P(a)$ is a Boolean algebra.

######  3.3 Theorem.

Let $A$ have the comparability property and let $a\in A$. There is a totally
disconnected compact Hausdorff space $X$ such that

  1. (i)

$P(a)$ is isomorphic (as a Boolean algebra) to the Boolean algebra
$\mathcal{P}(X)$ of all clopen subsets in $X$.

  2. (ii)

$CC(a)$ is isomorphic (as an order unit space) to a norm-dense order unit
subspace in $C(X,\mathbb{R})$.

  3. (iii)

If $A$ is norm-complete, then $CC(a)\cong C(X,\mathbb{R})$ (as an order unit
space).

###  3.2 Functional calculus

We can now define a continuous functional calculus in the case when $A$ has
the comparability property and is norm-complete, so it is an order unit Banach
space. In what follows, for a (compact Hausdorff) topological space $X$ we
will use the notation $C(X)=C(X,\mathbb{R})$, since we will work with real
functions only.

For $a\in A$, let $\Phi:CC(a)\to C(X)$ be the isomorphism as in Theorem 3.3
(iii) and let $f=\Phi(a)\in C(X)$ be the element corresponding to $a$. Since
$C(X)$ is a commutative C*-algebra, there is a continuous functions calculus
in $C(X)$: let $sp(a):=sp(f)$ be the spectrum of $f$ (that is, the range of
$f$ in $\mathbb{R}$). Then $sp(a)$ is a compact subset of $\mathbb{R}$ and the
subalgebra $A(f)$ of $C(X)$ generated by $f$ and $1$ is isomorphic to
$C(sp(a))$. In this isomorphism, $f$ is associated with the identity function
$t\mapsto t$ and any element in $A(f)$ is of the form $g\circ f$ for some
$g\in C(sp(a))$. The set $sp(a)$ will be called the spectrum of $a$ and the
subalgebra $A(f)$ is isomorphic to a subspace in $CC(a)\subseteq A$,
isomorphic to $C(sp(a))$. The element corresponding to $g\in C(sp(a))$ in this
isomorphism will be denoted by $g(a):=\Phi^{-1}(g\circ f)$.

Assume that $A$ is spectral. Our next aim is to show that in this case we may
define a Borel functional calculus.

######  3.4 Proposition.

Let $A$ be spectral, $a\in A$. Then $CC(a)$ is a spectral order unit space.

###### Proof.

We already know that $CC(a)$ is an order unit space with the comparability
property with respect to the restricted compression base $\\{J_{p}\\}_{p\in
P(a)}$ (cf. Proposition 3.1). Let $b\in CC(a)$, then $b^{\circ}\in
P(b)\subseteq CC(a)$. It follows that $CC(a)$ has also the projection cover
property and is therefore spectral.

∎

We now have by Thm. 3.3, Prop. 3.4 and [17], that $X$ is basically
disconnected, equivalently, $C(X)$ is monotone $\sigma$-complete. The next
result shows that in this case the bounded Baire functions are ‘close’ to
continuous functions. This fact was used in the proof of the Loomis-Sikorski
theorem for $\sigma$-MV algebras, [9, 10]. We give the proof in our setting
below for convenience of the reader. Below, $\mathcal{B}_{0}(X)$ denotes the
set of all bounded Baire functions on $X$.

######  3.5 Proposition.

Let $X$ be a basically disconnected compact Hausdorff space. There is a
mapping $\Theta$ of $\mathcal{B}_{0}(X)$ onto $C(X)$ such that

  1. (i)

for $h\in\mathcal{B}_{0}(X)$, $\\{x\in X:\ h(x)\neq\Theta(h)(x)\\}$ is a
meager set,

  2. (ii)

$\Theta$ preserves the products and the linear structure:
$\Theta(h_{1}h_{2})=\Theta(h_{1})\Theta(h_{2})$, $\Theta(\lambda h_{1}+\mu
h_{2})=\lambda\Theta(h_{1})+\mu\Theta(h_{2})$,

  3. (iii)

$\Theta$ is positive: $\Theta(h)\geq 0$ if $h\geq 0$,

  4. (iv)

$\Theta(\max\\{h_{1},h_{2}\\})=\Theta(h_{1})\vee\Theta(h_{2})$, for
$h_{1},h_{2}\in\mathcal{B}_{0}(X)$,

  5. (v)

if $h_{n}\in\mathcal{B}_{0}(X)$ is a nondecreasing sequence with pointwise
supremum $h$, then $\bigvee_{n}\Theta(h_{n})=\Theta(h)$ in $C(X)$.

###### Proof.

Since $\mathcal{B}_{0}(X)$ is the set of functions measurable with respect to
the $\sigma$-algebra of Baire sets, it is closed under pointwise suprema and
pointwise limits. We will first show that it is the smallest algebra of
bounded functions with this property, containing $C(X)$.

So let this smallest algebra of functions be denoted by $\mathcal{T}$.
Clearly, $\mathcal{T}\subseteq\mathcal{B}_{0}(X)$. Let

$\mathcal{T}_{0}:=\\{A\subset X:\ \chi_{A}\in\mathcal{T}\\}.$

Then $\mathcal{T}_{0}$ is a $\sigma$-algebra of subsets of $X$. Indeed,
clearly $\chi_{X}=1_{X}\in C(X)\subseteq\mathcal{T}$, so that
$X\in\mathcal{T}_{0}$, and if $\chi_{A}\in\mathcal{T}$ then also
$\chi_{X\setminus A}=1_{X}-\chi_{A}\in\mathcal{T}$, so that $\mathcal{T}_{0}$
is closed under complements. Let $A_{n}\in\mathcal{T}_{0}$, then

$\chi_{\cup_{n}A_{n}}(x)=\min\\{1,\sum_{n}\chi_{A_{n}}(x)\\}=-\max\\{-1,\sum_{n}-\chi_{A_{n}}\\}$

so that $\chi_{\cup_{n}A_{n}}\in\mathcal{T}$, hence
$\cup_{n}A_{n}\in\mathcal{T}_{0}$.

We next show that $\mathcal{T}$ is the set of all bounded measurable functions
with respect to $\mathcal{T}_{0}$. So let $f\in\mathcal{T}$, we have to prove
that for all $t\in\mathbb{R}$, $F_{t}:=\\{x:\ f(x)\geq
t\\}\in\mathcal{T}_{0}$. Since $\mathcal{T}$ is a linear subspace containing
the constant $1_{X}$, and $1_{X}$ is clearly $\mathcal{T}_{0}$-measurable, we
may assume that $f$ has values in $[0,1]$. So let $t\leq 0$, then
$F_{t}=X\in\mathcal{T}_{0}$. Similarly, if $t>1$, then
$F_{t}=\emptyset\in\mathcal{T}_{0}$. For $t=1$, we have $F_{1}=\\{x:f(x)=1\\}$
and

$\chi_{F_{1}}=1-\lim_{n}\min\\{n(1-f),1\\}\in\mathcal{T}$

We now use the fact that the binary fractions of the form
$\sum_{i=1}^{n}a_{i}2^{-i}$, $a_{i}\in\\{0,1\\}$ are dense in $(0,1)$ and we
have $F_{t}=\cap_{n}F_{r_{n}}$ for $r_{n}\nearrow t$, so that is enough to
assume that $t$ is a binary fraction. So let $t=\sum_{i=1}^{n}a_{i}2^{-i}$, we
will proceed by induction on $n$. So assume that $n=1$, then we have for
$t=1/2$

$F_{1/2}=\\{x:\ \min\\{2f(x),1\\}=1\\}\in\mathcal{T}_{0}$

by the previous considerations, since $\min\\{2f,1_{X}\\}\in\mathcal{T}$ has
values in $[0,1]$. Assume the assertion holds for $n$ and let
$t=\sum_{i=1}^{n+1}a_{i}2^{-i}$. Put $s=\sum_{i=2}^{n+1}a_{i}2^{1-i}$. If
$a_{1}=0$, then $t=s/2$ and we have

$F_{t}=\\{x:\ 2f(x)\geq s\\}=\\{x:\min\\{2f(x),1\\}\geq
s\\}\in\mathcal{T}_{0},$

by the induction assumption and the fact that
$\min\\{2f,1_{X}\\}\in\mathcal{T}$ has values in $[0,1]$. Similarly, if
$a_{1}=1$, then $t=(1+s)/2$ and

$F_{t}=\\{x:\ 2f(x)-1\geq s\\}=\\{x:\max\\{2f(x)-1,0\\}\geq
s\\}\in\mathcal{T}_{0}.$

Hence all functions in $\mathcal{T}$ are $\mathcal{T}_{0}$-measurable.
Conversely, since any $\mathcal{T}_{0}$-measurable function is a pointwise
limit of simple functions which are linear combinations of characteristic
functions in $\mathcal{T}$, it is clear that any such function belongs to
$\mathcal{T}$. Since $C(X)\subseteq\mathcal{T}$, all continuous functions are
$\mathcal{T}_{0}$-measurable, and since Baire $\sigma$-algebra is smallest
with this property, it must be contained in $\mathcal{T}_{0}$. This implies
that $\mathcal{B}_{0}(X)\subseteq\mathcal{T}\subseteq\mathcal{B}_{0}(X)$ (cf.
the proof of [10, Thm. 7.1.7]).

We now introduce an equivalence relation in $\mathcal{B}_{0}(X)$ as follows:
$f\sim g$ if $\\{x:\ f(x)\neq g(x)\\}$ is a meager set. Note that $f\sim g$
implies $f=g$ for $f,g\in C(X)$, so that every equivalence class may contain a
unique continuous function (if any). Let

$\mathcal{T}^{\prime}:=\\{f\in\mathcal{B}_{0}(X):\ f\sim g,\text{ for some
}g\in C(X)\\}.$

We will show that $\mathcal{T}^{\prime}=\mathcal{B}_{0}(X)$. It is easily seen
that $\mathcal{T}^{\prime}$ is a subalgebra in $\mathcal{B}_{0}(X)$ and
$\max\\{f,g\\}\in\mathcal{T}^{\prime}$ for $f,g\in\mathcal{T}^{\prime}$. It is
now enough to prove that $\mathcal{T}^{\prime}$ is closed under pointwise
limits of non-decreasing sequences, since then $\mathcal{T}^{\prime}$ is also
closed under all pointwise limits and this implies
$\mathcal{T}^{\prime}=\mathcal{B}_{0}(X)$ by minimality of
$\mathcal{B}_{0}(X)$ in the first part of the proof.

So let $f_{n}\in\mathcal{T}^{\prime}$, $f_{n}\leq f_{n+1}$, $f_{n}\nearrow f$
and let $b_{n}\in C(X)$ be such that $f_{n}\sim b_{n}$. Then $b_{n}\vee
b_{n+1}\sim f_{n}\vee f_{n+1}=f_{n+1}\sim b_{n+1}$, so that $b_{n}\vee
b_{n+1}$ and $b_{n+1}$ are continuous functions that differ only on a meager
set, so that we must have $b_{n}\vee b_{n+1}=b_{n+1}$ and $b_{n}\leq b_{n+1}$.
Further, since $f_{n}$ is bounded, $b_{n}$ is a bounded increasing sequence.
Since $X$ is basically disconnected, $C(X)$ is monotone $\sigma$-complete, so
that there is some $b=\bigvee_{n}b_{n}$ in $C(X)$. Let us also denote
$b_{0}:=\lim_{n}b_{n}$, the pointwise limit. Our aim is to prove that $f\sim
b_{0}$ and $b_{0}\sim b$, so that $f\in\mathcal{T}^{\prime}$. We have

$\\{x:\ f(x)\neq b_{0}(x)\\}=\\{x:\ f(x)<b_{0}(x)\\}\cup\\{x:\
f(x)>b_{0}(x)\\}.$

If $f(x)<b_{0}(x)=\lim_{n}b_{n}(x)$, then there is some $n$ such that
$f_{n}(x)\leq f(x)<b_{n}(x)\leq b_{0}(x)$. Similarly, if $f(x)>b_{0}(x)$ then
$f(x)\geq f_{n}(x)>b_{0}(x)\geq b_{n}(x)$. It follows that

$\\{x:\ f(x)\neq b_{0}(x)\\}\subseteq\bigcup_{n}\\{x:\ f_{n}(x)\neq
b_{n}(x)\\},$

which is a meager set. Hence $f\sim b_{0}$. To prove $b_{0}\sim b$, we invoke
the construction from [19, Lemma 9.1]: put

$\tilde{b}_{0}(x):=\inf_{U\in\mathcal{N}(x)}\sup_{y\in U}b_{0}(y),$

where $\mathcal{N}(x)$ is the collection of open neighborhoods of $x$ in $X$,
then $\tilde{b}_{0}$ is continuous if $b_{0}^{-1}(\alpha,\infty)$ is an open
$F_{\sigma}$ subset for all $\alpha\in\mathbb{R}$. This last condition is
satisfied, since

$b_{0}^{-1}(\alpha,\infty)=\\{x:\ b_{0}(x)>\alpha\\}=\\{x:\ \exists
n,b_{n}(x)>\alpha\\}=\cup_{n}b_{n}^{-1}(\alpha,\infty)$

is an open $F_{\sigma}$-set, since $b_{n}$ are continuous. We also have
$\tilde{b}_{0}(x)=b_{0}(x)$ in all $x$ where $b_{0}$ is continuous, so that

$\\{x:\ \tilde{b}_{0}(x)\neq b_{0}(x)\\}\subseteq\\{x:\ b_{0}\text{ is
discontinuous in }x\\}.$

Since $b_{0}$ is a pointwise limit of a sequence of continuous functions, this
last set is meager, hence $f\sim b_{0}\sim\tilde{b}_{0}$ and
$f\in\mathcal{T}^{\prime}$. Note also that by definition, $\tilde{b}_{0}\geq
b_{0}\geq b_{n}$, for all $n$, so that $b_{0}\leq
b=\bigvee_{n}b_{n}\leq\tilde{b}_{0}$. This implies $b\sim
b_{0}\sim\tilde{b}_{0}$, so that $b=\tilde{b}_{0}$.

It follows that $\mathcal{T}^{\prime}=\mathcal{B}_{0}(X)$, so that every
bounded Baire function is equivalent to a unique continuous function. Let
$\Theta:\mathcal{B}_{0}(X)\to C(X)$ be the corresponding map. Then it is
easily seen by construction that $\Theta$ has the properties (i)-(v) (cf. the
proof of [10, Thm. 7.1.22]).

∎

We can now define a Borel function calculus on $A$. For a subset
$S\subseteq\mathbb{R}$, we will denote by $\mathcal{B}(S)$ the set of bounded
Borel functions $S\to\mathbb{R}$.

######  3.6 Theorem (Borel functional calculus).

Let $A$ be a spectral order unit Banach space. Then for every $a\in A$, there
is a positive unital linear map $\Psi_{a}:\mathcal{B}(sp(a))\to CC(a)\subseteq
A$ such that:

  1. (i)

$\Psi_{a}(id)=a$, where $id$ is the identity function $t\mapsto t$,

  2. (ii)

$\Psi_{a}(\chi_{B})\in P(a)\subseteq P$ for the characteristic function
$\chi_{B}$ of any Borel subset $B\subseteq sp(a)$,

  3. (iii)

$\Psi_{a}(\max\\{h_{1},h_{2}\\})=\Psi_{a}(h_{1})\vee\Psi_{a}(h_{2})$ in
$CC(a)$,

  4. (iv)

If $h_{n}$ is a nondecreasing sequence of functions in $\mathcal{B}(sp(a))$
with a pointwise supremum $h$, then $\bigvee_{n}\Psi_{a}(h_{n})=\Psi(h)$ in
$CC(a)$.

We will use the notation $g(a):=\Psi_{a}(g)$ for any $g\in\mathcal{B}(sp(a))$.

Let us stress that the suprema in the above theorem are taken in $CC(a)$ and
they are not necessarily suprema in $A$.

###### Proof.

As before, there is an isomorphism $\Phi:CC(a)\to C(X)$ for some basically
disconnected compact Hausdorff space $X$. Let $f=\Phi(a)$, so that
$sp(a)=sp(f)$. Notice that for $g\in\mathcal{B}(sp(a))$, $g\circ f$ is in
$\mathcal{B}_{0}(X)$, so we may put

$\Psi_{a}(g):=\tilde{\Phi}(g\circ f)\in CC(a),$

where

$\tilde{\Phi}:=\Phi^{-1}\circ\Theta:\mathcal{B}_{0}(X)\to C(X)\to CC(a).$

It follows by Proposition 3.5 (ii) and (iii) that $\Psi_{a}$ is positive and
linear, further, we have

$\Psi_{a}(1_{sp(a)})=\tilde{\Phi}(1_{sp(f)}\circ f)=\tilde{\Phi}(1_{X})=1,$

since $\Theta(1_{X})=1_{X}$ and $\Phi$ is unital. Let us check the properties
(i) - (iv). First, we have

$\Psi_{a}(id)=\tilde{\Phi}(f)=a,$

since clearly $\Theta(f)=f$. Further, let $\chi_{B}$ be the characteristic
function of a Borel subset $B\subseteq sp(a)$, then

$\Psi_{a}(\chi_{B})=\tilde{\Phi}(\chi_{B}\circ
f)=\tilde{\Phi}(\chi_{f^{-1}(B)}).$

By Proposition 3.5 (ii),
$\Theta(\chi_{f^{-1}(B)})^{2}=\Theta(\chi_{f^{-1}(B)}^{2})=\Theta(\chi_{f^{-1}(B)})$,
which means that $\Theta(\chi_{f^{-1}(B)})$ is a continuous functions with
values in $\\{0,1\\}$, that is, a characteristic function of a clopen subset.
By construction, such functions in $C(X)$ correspond to elements in $P(a)$.
This proves (ii). The property (iii) follows easily from Proposition 3.5 (iv).
Let now $h_{n}\in\mathcal{B}(sp(a))$ be a nondecreasing sequence with
pointwise supremum $h$, then $h_{n}\circ f$ is a nondecreasing sequence in
$\mathcal{B}_{0}(X)$ with pointwise supremum $h\circ f$ and we have by
Proposition 3.5 (v) that

$\Psi_{a}(h)=\tilde{\Phi}(h\circ f)=\bigvee_{n}\tilde{\Phi}(h_{n}\circ
f)=\bigvee_{n}\Psi_{a}(h_{n}).$

∎

We now use the above Borel functional calculus to describe the spectral
resolution of $a$. Let us define

$\xi_{a}(B):=\chi_{B}(a),\qquad B\subseteq sp(a)\text{ a Borel subset.}$

Then by Theorem 3.6, it is easily seen that $\xi_{a}$ is a
$\sigma$-homomorphism from the Borel subsets in $sp(a)$ into the Boolean
$\sigma$-algebra $P(a)$. Further, if $g\in\mathcal{B}(sp(a))$, then it is not
difficult to see that $\xi_{g(a)}=\xi_{a}\circ g^{-1}$.

Let $h\in C(X)$ and let $\\{x\in X,h(x)\neq 0\\}^{-}$ be the support of $h$.
Since $X$ is basically disconnected, the support of any continuous function is
a clopen set. It is easily seen that the characteristic function of the
support of $h$ is the projection cover $h^{\circ}$ and for $b\in CC(a)$,
$\Phi(b^{\circ})=\Phi(b)^{\circ}$. Let now $\lambda\in\mathbb{R}$ and let
$f=\Phi(a)$, then

$p_{a,\lambda}=1-(a-\lambda)_{+}^{\circ}=\Phi^{-1}(1-(f-\lambda)_{+}^{\circ})=\Phi^{-1}(1-\chi_{f^{-1}((\lambda,\infty))^{-}})=\Phi^{-1}(\chi_{f^{-1}((-\infty,\lambda])^{i}})$

where for $Y\subseteq X$, $Y^{i}$ denotes the interior of $Y$. Now note that
the difference of the closed set $f^{-1}((-\infty,\lambda])$ and its interior
is a meager set, hence we obtain

$\xi_{a}((-\infty,\lambda])=\tilde{\Phi}(\chi_{f^{-1}((-\infty,\lambda])})=\Phi^{-1}(\chi_{f^{-1}((-\infty,\lambda])^{i}})=p_{a,\lambda}.$

Recall that $a$ has an integral expression with respect to its spectral
resolution:

$a=\int\lambda dp_{a,\lambda},$

in the sense that the integral sums
$\sum_{i=1}^{n}\lambda_{i}(p_{a,\lambda_{i}}-p_{a,\lambda_{i-1}})$ for
$\lambda_{0}\leq L_{a}\leq\lambda_{1}\leq\dots\leq U_{a}\leq\lambda_{n}$
converge to $a$ in norm as $\max_{i}|\lambda_{i}-\lambda_{i-1}|\to 0$. If
$g\in C(sp(a))$, then we have

$g(a)=\int g(\lambda)dp_{a,\lambda}.$ (4)

Indeed, let $u_{1},\dots,u_{n}$ be mutually orthogonal projections in $P(a)$,
(that is, the corresponding clopen subsets of $X$ are mutually disjoint) and
let $b=\sum_{i}\lambda_{i}u_{i}$, such elements are called simple. Then it is
easy to see that $b^{k}=\sum_{i}\lambda_{i}^{k}u_{i}$ for any $k\in\mathbb{N}$
and hence $p(b)=\sum_{i}p(\lambda_{i})u_{i}$ for any polynomial $p$. By Stone-
Weierstrass theorem and continuity of the functional calculus, we obtain that
$g(b)=\sum_{i}g(\lambda_{i})u_{i}$ for any $g\in C(sp(a))$. Since $a$ is the
norm limit of integral sums $b_{n}$ that are simple, it follows that $g(a)$ is
the norm limit of $g(b_{n})$. Hence $g(a)$ is the norm limit of integral sums
of the form $\sum_{i}g(\lambda_{i})(p_{a,\lambda_{i}}-p_{a,\lambda_{i-1}})$ as
$\max_{i}|\lambda_{i}-\lambda_{i-1}|\to 0$, this implies the statement.

For a Borel function $g:sp(a)\to\mathbb{R}$ we do not have the above integral
as a norm limit in general, but only in some weaker sense: For any
$\sigma$-additive state $\rho$ on $CC(a)$, $\rho\circ\xi_{a}$ defines a
probability measure on Borel subsets of $sp(a)$ (and can be extended to
$\mathbb{R}$), with distribution function
$F_{a}(\lambda)=\rho(p_{a,\lambda})$. Using the integral representation of $a$
we obtain

$\rho(a)=\int\lambda dF_{a}(\lambda)=\int\lambda\rho\circ\xi_{a}(d\lambda).$

For a Borel measurable function $g:sp(a)\to{\mathbb{R}}$ we have
$\rho\circ\xi_{g(a)}=(\rho\circ\xi_{a})\circ g^{-1}$ and so

$\rho(g(a))=\int\lambda(\rho\circ\xi_{a})(g^{-1}(d\lambda))=\int
g(\lambda)(\rho\circ\xi_{a})(d\lambda)$

by the integral transformation theorem. If there are enough $\sigma$-additive
states on $CC(a)$ (or on $A$) to separate points, then we can define the
integral (4) in a weak sense. In general, we only have the relation of the
spectral resolutions as

$\xi_{g(a)}=\xi_{a}\circ g^{-1}\ \implies\
p_{g(a),\lambda}=\xi_{a}(g^{-1}(-\infty,\lambda])).$

##  4 Order unit spaces that are JB-algebras

We recall that a _Jordan algebra_ over $\mathbb{R}$ is a vector space $A$ over
$\mathbb{R}$ equipped with a commutative bilinear product $\circ$ that
satisfies the identity

$(a^{2}\circ b)\circ a=a^{2}\circ(b\circ a)\ \mbox{for all}\ a,b\in A,$ (5)

here $a^{2}=a\circ a$. A _JB-algebra_ is a Jordan algebra $A$ over
$\mathbb{R}$ with identity element $1$ equipped with a complete norm
satisfying the following requirements for $a,b\in A$ ([1]):

  1. (JBi)

$\|a\circ b\|\leq\|a\|\|b\|$,

  2. (JBii)

$\|a^{2}\|\leq\|a\|^{2}$,

  3. (JBiii)

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

Notice that by (JBi) the Jordan product is norm-continuous. A JB-algebra that
is the dual of a Banach space is called a JBW-algebra. For more information
see e.g. [23]. We will use [1] as a general reference.

The following theorem shows relations between JB-algebras and order unit
spaces.

######  4.1 Theorem.

[1, Theorem 1.11] If $A$ is a JB-algebra, then $A$ with its given norm is an
order unit Banach space with $A^{+}=\\{a^{2}:a\in A\\}$ and the identity $1$
as distinguished order unit. Furthermore, for each $a\in A$

$-1\leq a\leq 1\ \implies\ 0\leq a^{2}\leq 1.$ (6)

Conversely, if $A$ is a order unit Banach space equipped with a Jordan product
for which the distinguished order unit is the identity and satisfies (6), then
$A$ is a JB-algebra with the order unit norm.

In any Jordan algebra, a _triple product_ $\\{abc\\}$ is defined by

$\\{abc\\}=(a\circ b)\circ c+(b\circ c)\circ a-(a\circ c)\circ b.$ (7)

The special case of the triple product $\\{abc\\}$ with $a=c$ is denoted

$U_{a}b=\\{aba\\}=2a\circ(a\circ b)-a^{2}\circ b.$ (8)

We will also need the Jordan multiplication operator

$T_{a}:A\to A,\quad T_{a}b=a\circ b,\ b\in A.$

It is clear that $T_{a}$ defines a bounded linear operator on $A$ for all
$a\in A$.

An element $p$ in a JB-algebra $A$ is called a _projection_ if $p^{2}=p$. In
this case, we have

$U_{p}=2T_{p}^{2}-T_{p},\qquad T_{p}=\frac{1}{2}(I+U_{p}-U_{1-p}),$ (9)

here $I$ is the identity map $A\to A$.

Let $P$ be the set of all projections in $A$. It was shown in [25] that every
compression on a JB-algebra $A$ is of the form

$J_{p}(a)=U_{p}a=\\{pap\\}$ (10)

for some $p\in P$ and the set $(U_{p})_{p\in P}$ forms a compression base in
$A$ [25, Corollary 5.11]. Note that this is a unique compression base in $A$
which is maximal, that is, is not contained in any other compression base. In
what follows, we always assume this compression base in a JB-algebra $A$.

In this section, we will assume that $A$ is an order unit Banach space with a
compression base $(J_{p})_{p\in P}$ satisfying the comparability property. We
will show some additional conditions under which $A$ becomes a JB-algebra.
These conditions are the same as in [1] and the proofs follow the same steps,
we only show that our (weaker) assumption of comparability is sufficient.

######  4.2 Theorem.

$A$ is a JB-algebra if and only if for all $p,q\in P$, we have

$J_{p}(q)+J_{1-p}(1-q)=J_{q}(p)+J_{1-q}(1-p).$ (11)

###### Proof.

([1, Theorem 9.43]) For $p\in P$, we define the operator $T_{p}:A\to A$ by
$T_{p}=\frac{1}{2}(I+J_{p}-J_{1-p})$. The condition (11) is equivalent to

$T_{p}(q)=T_{q}(p).$ (12)

Note that in the case that $A$ is a JB algebra, $T_{p}$ is precisely the
multiplication operator, so that the conditions (12) are necessary.

For the converse, let $A_{0}=\mathrm{span}(P)$, then $A_{0}$ is a subspace in
$A$, which is dense in $A$ by comparability (c.f. Theorem 2.9). Let $a,b\in
A_{0}$, $a=\sum_{i}\alpha_{i}p_{i}$ and $b=\sum_{j}\beta_{j}q_{j}$ and define

$a\circ
b=\sum_{i}\alpha_{i}T_{p_{i}}(b)=\sum_{i,j}\alpha_{i}\beta_{j}T_{p_{i}}(q_{j})=\sum_{i,j}\alpha_{i}\beta_{j}T_{q_{j}}(p_{i})=\sum_{j}\beta_{j}T_{q_{j}}(a)=b\circ
a.$

It is clear that this does not depend on the representation of $a$ and $b$ as
linear combinations of projections. Moreover, $a\circ b$ is bilinear,
commutative and separately norm continuous. Since $A_{0}$ is dense in $A$,
$a\circ b$ uniquely extends to a bilinear, commutative and separately
continuous product on $A$. To show that $A$ is a JB-algebra, it is now enough
to show that $a\circ b$ is a Jordan product satisfying the condition

$-1\leq a\leq 1\quad\implies\quad 0\leq a\circ a\leq 1,\qquad a\in A,$ (13)

cf. [1, Theorem 1.11].

By Theorem 3.3, we have $a\in CC(a)=\overline{\mathrm{span}}(P(a))\simeq C(X)$
for some totally disconnected compact Hausdorff space $X$. Let $\Phi:CC(a)\to
C(X)$ be the isomorphism. Note that for $p,q\in P(a)$, we have $p\circ
q=T_{p}(q)=J_{p}(q)=p\wedge q$, so that we have

$\Phi(p\circ q)=\Phi(p\wedge q)=\Phi(p)\wedge\Phi(q)=\Phi(p)\Phi(q),$

since $\Phi(p)$ and $\Phi(q)$ are characteristic functions. For a simple
element $a_{0}=\sum_{i}\alpha_{i}p_{i}$ where $p_{i}\in P(a)$, we obtain

$a_{0}\circ
a_{0}=\sum_{i,j}\alpha_{i}\alpha_{j}J_{p_{i}}(p_{j})=\sum_{i,j}\alpha_{i}\alpha_{j}p_{i}\wedge
p_{j},$

which implies $\Phi(a_{0}\circ a_{0})=\Phi(a_{0})^{2}$. Since $a$ is the norm
limit of simple elements of this form, we obtain by continuity of $\circ$ and
the product of functions that $\Phi(a\circ a)=\Phi(a)^{2}$. Since $\Phi(a)$ is
a function and the isomorphism preserves the order, this implies (13).

To finish the proof, we now have to prove the Jordan identity

$a^{2}\circ(b\circ a)=(a^{2}\circ b)\circ a,$

note that we may now safely write $a\circ a=a^{2}$ since by the previous
paragraph it is in agreement with the continuous functional calculus obtained
in the previous section. Note first that for commuting projections $p,q\in P$
we have (using Lemma 2.6 (iii))

$(p\circ b)\circ q=T_{q}T_{p}(b)=T_{p}T_{q}(b)=p\circ(b\circ q).$

Let now $a_{0}=\sum_{i}\alpha_{i}p_{i}$, where $p_{1},\dots,p_{n}$ are
mutually orthogonal projections (i.e. $p_{i}\wedge p_{j}=0$). Then
$a^{2}=\sum_{i}\alpha^{2}_{i}p_{i}$ and we have

$a^{2}\circ(b\circ a)=\sum_{i,j}\alpha_{i}^{2}\alpha_{j}p_{i}\circ(b\circ
p_{j})=\sum_{i,j}\alpha_{i}^{2}\alpha_{j}(p_{i}\circ b)\circ p_{j}=(a^{2}\circ
b)\circ a.$

Now note that any simple element of the form $\sum_{i}\beta_{i}q_{i}$ with
$q_{i}\in P(a)$ can be written as a finite linear combination of mutually
orthogonal projections in $P(a)$. Since each $a\in A$ is a norm limit of such
simple elements, the statement follow by continuity of the product.

∎

There is another characterization of JB-algebras using the functional calculus
on the space $A$. Define the product

$a\circ b=\frac{1}{4}[(a+b)^{2}+(a-b)^{2}],\qquad a,b\in A.$

######  4.3 Theorem.

The product $a\circ b$ is bilinear if and only if (11) holds. In this case,
$A$ is a JB-algebra and $a\circ b$ is the Jordan product in $A$.

###### Proof.

(cf. [1, Cor. 9.44]) Assume that (11) holds. By Theorem 4.2 and its proof, $A$
is then a JB-algebra with Jordan product $*$ such that $a*a=a^{2}$ agrees with
the continuous functional calculus on $A$. By bilinearity of $*$,

$a\circ b=\frac{1}{4}[(a+b)*(a+b)+(a-b)*(a-b)]=a*b$

is bilinear as well.

Conversely, assume that $a\circ b$ is bilinear. It is clear by the definition
that $a\circ b$ is commutative and continuous in both variables. We will use
the fact that a JB-algebra can be characterized as a complete order unit space
with a commutative power associative product for which the order unit 1 is the
identity and (13) holds, [1, Thm. 2.49 and A.51]. It is clear that $a\circ
a=a^{2}$ coincides with the squares defined by the continuous functional
calculus on $A$, so that (13) holds as before, since it holds in $C(X)\simeq
CC(a)$. Similarly, $\circ$ is power associative since so is the usual product
of functions in $C(X)$. It follows that $A$ with respect to this product is a
JB-algebra. Note that since $A$ has comparability, $P$ must contain all the
sharp elements [25, Lemma 3.20], By [25, Theorem 5.10], for each sharp element
$p$ there is a unique retraction with focus $p$, namely $U_{p}=\\{p\cdot
p\\}$. It follows that for all $p\in P$, $J_{p}=U_{p}$ for which the condition
(11) holds.

∎

###  4.1 Generalized spin factors

We now consider the following example of an order unit space obtained from a
normed space $(X,\|\cdot\|)$, cf. [25, 6, 7]. In this case,
$A=\mathbb{R}\times X^{*}$, with the positive cone

$A^{+}=\\{(\alpha,y),\ \|y\|\leq\alpha\\}$

here $(X^{*},\|\cdot\|)$ is the dual space to $(X,\|\cdot\|)$, and the order
unit is $(1,0)$. By [25, Thm. 6.5], there exists a spectral compression base
$(J_{p})_{p\in P}$ in $A$ if and only if $X$ is a reflexive and smooth Banach
space. This means that $X$ is reflexive and every nonzero element $x\in X$
attains its norm at a unique element of the dual unit ball that is,

$\partial_{x}:=\\{y\in X^{*}:\ \|y\|=1,y(x)=\|x\|\\}$

is a singleton for all $0\neq x\in X$. Equivalently, $X$ is reflexive and the
dual space $X^{*}$ is strictly convex, which means that every boundary point
of the unit ball is an extreme point. By [7], see also [25, Thm. 6.6], $A$ is
spectral in the Alfsen-Shultz sense if and only if $X$ is reflexive, smooth,
and strictly convex as well. In this case, the space $A$ was called a
generalized spin factor in [7]. In the present work, by a generalized spin
factor we mean any space $A$ obtained in the above way from a Banach space
$(X,\|\cdot\|)$.

In the special case when $X$ is a Hilbert space with inner product
$(\cdot,\cdot)$, $A$ is a JBW-algebra called a spin factor, [1, Def. 3.33],
with the Jordan product

$(\alpha,y)\circ(\beta,z)=(\alpha\beta+(y,z),\alpha z+\beta y).$

######  4.4 Theorem.

Let $A$ be a generalized spin factor obtained from a Banach space
$(X,\|\cdot\|)$. The following are equivalent.

  1. (i)

$A$ is spectral and satisfies the condition (11);

  2. (ii)

$A$ is a JB-algebra;

  3. (iii)

$A$ is a spin factor;

  4. (iv)

$A$ is a JBW-algebra.

###### Proof.

By what was said above and Theorem 4.2, it is enough to show that if $A$ is a
JB-algebra, then $A$ must be a spin factor. So assume that $A$ is a JB-
algebra. Then any sharp element $p\in A$ is a projection [25, Lemma 5.7], and
hence the focus of the compression $U_{p}$, moreover, the compressions satisfy
condition (11). Let $y\in X^{*}$, $\|y\|=1$. By [25, Lemma 6.1], the element
$p=1/2(1,y)\in A$ is sharp. But then $p$ must be the focus of a compression,
hence by [25, Prop. 6.4], $y$ must be an extremal point in the dual unit ball
such that

$\partial^{*}_{y}=\\{x\in X,\ \|x\|=1,\ \langle y,x\rangle=1\\}\neq\emptyset.$

This implies that any nonzero element in $X^{*}$ attains its norm on the unit
ball in $X$, so that $X$ is reflexive. Moreover, we also see that any boundary
point of the dual unit ball is an extremal point, hence $X^{*}$ is strictly
convex. Note that by [25, Thm. 6.5], this shows that $A$ is a spectral order
unit space.

For a sharp element $p=1/2(1,y)$ with $y\in X^{*}$, $\|y\|=1$, we have by [25,
Prop. 6.4] that the compression with focus $p$ must be of the form

$U_{p}((\alpha,w))=(\alpha+\langle w,x_{y}\rangle)p,$

for some (fixed) choice of $x_{y}\in\partial_{y}^{*}$. Note also that we have
$1-p=1/2(1,-y)$. Let us check the condition (11) in this case. After some
computations, we obtain

$x_{-y}=-x_{y}\ \text{ and }\ \langle z,x_{y}\rangle=\langle
y,x_{z}\rangle,\qquad\forall y,z\in X^{*},\ \|y\|=\|z\|=1.$ (14)

Let us define a mapping $\psi:X^{*}\to X$ as

$\psi(y)=\|y\|x_{\|y\|^{-1}y},\ y\neq 0,\quad\psi(0)=0.$

Then $\psi(ty)=t\psi(y)$ holds for all $t\in\mathbb{R}$. Further, for any
$z,y,w\in X^{*}$ with $\|w\|=1$, we have by (14)

$\displaystyle\langle w,\psi(y+z)\rangle$ $\displaystyle=\|y+z\|\langle
w,x_{\|y+z\|^{-1}(y+z)}\rangle=\langle
y+z,x_{w}\rangle=\|y\|\langle\frac{y}{\|y\|},x_{w}\rangle+\|z\|\langle\frac{z}{\|z\|},x_{w}\rangle$
$\displaystyle=\|y\|\langle w,x_{\|y\|^{-1}y}\rangle+\|z\|\langle
w,x_{\|z\|^{-1}z}\rangle=\langle w,\psi(y)+\psi(z)\rangle,$

so that $\psi$ is a linear map. Further, for $y,z\in X^{*}$,

$\langle y,\psi(z)\rangle=\|z\|\langle y,x_{\|z\|^{-1}z}\rangle=\|y\|\langle
z,x_{\|y\|^{-1}y}\rangle=\langle z,\psi(y)\rangle$

and $\langle y,\psi(y)\rangle=\|y\|^{2}$. It follows that $(y,z):=\langle
y,\psi(z)\rangle$ defines a scalar product in $X^{*}$ such that
$\|y\|=\sqrt{(y,y)}$. Hence $X^{*}$, and therefore also $X=X^{**}$, is a
Hilbert space, so that $A$ is a spin factor.

∎

##  5 Rickart JB-algebras

Let $A$ be a JB-algebra. In [4], the following symbols were defined for a
subset $S$ of $A$:

$\displaystyle S^{\perp}=\\{a\in A:U_{a}(x)=0,\forall x\in S\\},$ (15)
${}^{\perp}S=\\{x\in A:U_{a}(x)=0,\forall a\in A\\},$ (16)
${}^{\perp}S^{+}={{}^{\perp}S}\bigcap A^{+}.$ (17)

The following notion of a _Rickart JB-algebra_ was introduced by Ayupov and
Arzikulov [5].

######  5.1 Definition.

A JB-algebra $A$ is Rickart if one of the following equivalent statements is
true

  1. (A1)

For every element $x\in A^{+}$ there is a projection $p\in A$ such that

$\\{x\\}^{\perp}=U_{p}(A)$

  2. (A2)

For every element $x\in A$ there is a projection $p\in A$ such that

${}^{\perp}\\{x\\}^{+}=U_{p}(A)^{+}.$

It was proved in [5] that the self-adjoint part of a C*-algebra $\mathcal{A}$
is a Rickart JB-algebra if and only if $\mathcal{A}$ is a Rickart C*-algebra,
so that this notion generalizes Rickart C*-algebras. The aim of this section
is to prove the following generalization of a result proved for C*-algebras in
[29].

######  5.2 Theorem.

Let $A$ be a JB-algebra. Then $A$ is Rickart if and only if every maximal
associative subalgebra in $A$ is monotone $\sigma$-complete.

The proof will be based on the following result.

######  5.3 Theorem.

[25, Theorem 5.16] Let $A$ be a JB-algebra. Then $A$ is Rickart if and only if
$A$ is spectral.

We will need some further preparations. For elements $a,b\in A$ we say that
$a$ and $b$ _operator commute_ if $T_{a}T_{b}=T_{b}T_{a}$. If one of the
elements is a projection, then we have the following characterizations of
operator commutativity.

######  5.4 Proposition.

[1, Prop. 1.47] Let $a\in A$ and $p\in P$. Then the following are equivalent.

  1. (i)

$a$ and $p$ operator commute;

  2. (ii)

$T_{p}(a)=U_{p}(a)$;

  3. (iii)

$a=U_{p}(a)+U_{1-p}(a)$;

  4. (iv)

$a$ and $p$ are contained in an associative subalgebra.

The equivalence of (i) and (iv) was generalized to all pairs of elements in
[28]. Notice that by Lemma 2.6 (iii), (iii) is equivalent to $a\leftrightarrow
p$.

######  5.5 Lemma.

Let $A$ be a JB-algebra and let $C\subseteq A$ be a subalgebra. Then $C$ is
associative if and only if the elements in $C$ mutually operator commute.

###### Proof.

Assume that $C$ is associative and let $a,b\in C$. Then $a$ and $b$ generate
an associative JB-algebra, so that $a$ and $b$ operator commute, [28, Thm.
3.13]. Conversely, assume all elements in $C$ mutually operator commute, then
for $a,b,c\in C$, we have

$a\circ(b\circ c)=a\circ(c\circ b)=T_{a}T_{c}(b)=T_{c}T_{a}(b)=c\circ(a\circ
b)=(a\circ b)\circ c.$

∎

For each element $a$ in a JB-algebra $A$, $A(a,1)$ denotes the norm-closed
subalgebra generated by $a$ and $1$. By [1, Corollary 1.4], and continuity of
Jordan product, $A(a,1)$ is associative.

######  5.6 Theorem.

[1, Proposition 1.12] If $A$ is a JB-algebra and $B$ is a norm-closed
associative subalgebra containing $1$, in particular if $B=A(a,1)$ for $a\in
A$, then $B$ is isometrically (order- and algebra-) isomorphic to
$C(X,{\mathbb{R}})$ for some compact Hausdorff space $X$.

Assume that $A$ has the comparability property. Recall that (cf. Definition
2.12)

$aCb\iff P(a)\leftrightarrow P(b).$

If $A$ is spectral, this is also equivalent to $P_{sp}(a)\leftrightarrow
P_{sp}(b)$, where $P_{sp}(a)$ denotes the Boolean subalgebra in $P$ generated
by spectral projections of $a$ (Lemma 2.15). If $b\in P$, then $aCb$ is
equivalent to $a\in C(b)$ and Proposition 5.4 shows that for $b\in P$, $a\in
C(b)$ is the same as operator commutativity. We next show how this can be
extended to arbitrary $b\in A$.

######  5.7 Lemma.

Let $A$ be a JB-algebra with the comparability property and let $a,b\in A$.
Then $aCb$ implies that $a$ and $b$ operator commute.

###### Proof.

Assume $aCb$, so that $a\leftrightarrow P(b)$, which means that $a$ operator
commutes with all elements of $P(b)$. By Proposition 3.2 (iii), $b$ is a norm
limit of a sequence of linear combinations of elements in $P(b)$. Since the
map $b\mapsto T_{b}$ is linear and norm-continuous, we obtain that $a$
operator commutes with $b$.

∎

######  5.8 Corollary.

Assume that $A$ is a JB-algebra with the comparability property. Then any
C-block $C$ of $A$ is a maximal associative subalgebra.

###### Proof.

Let $B\subseteq P$ be a block of $P$. We show that $C=C(B)$ is a subalgebra,
which means that it is closed under the Jordan product of its elements. Let
$a\in C$, $p\in B$, then by Proposition 5.4, $p\circ a=T_{p}a=U_{p}a\in C$.
This can be extended to all $a,b\in C$ using Theorem 2.9(iii), continuity of
the map $b\mapsto T_{b}$ and the fact that $C$ is a norm-closed linear
subspace in $A$. Since we have $aCb$ for all $a,b\in C$, we see from Lemmas
5.7 and 5.5 that $C$ is associative, hence it is contained in some maximal
associative subalgebra $C_{0}$. It follows that all elements of $C_{0}$
mutually operator commute, which by Proposition 5.4 implies that all elements
of $C_{0}$ commute with all projections in $B$, so that $C_{0}\subseteq
C(B)=C$ and $C_{0}=C$.

∎

Assume now that $A$ is Rickart, then each $a\in A$ has a _carrier_ $s(a)\in
P$, which is the smallest projection such that $p\circ a=a$ [5, Prop. 1.8]. It
was shown in [25, Lemma 5.14] that if $0\leq a\leq 1$, then $s(a)$ is the
projection cover of $a$. We also have the following relation to the Rickart
mapping $a\mapsto a^{*}$ (see (3)).

######  5.9 Lemma.

For every $a\in A$, $a^{*}=1-s(a)$.

###### Proof.

Let $a\in A$, $p\in P$. By (9) we have

$p\circ a=0\ \Leftrightarrow\ U_{p}(a)=0,\qquad p\circ a=a\ \Leftrightarrow\
U_{p}(a)=a.$

As $p\circ a=a$ iff $(1-p)\circ a=0$ by distributivity of the Jordan product,
in both the above cases we have $a\in C(p)$.

Put $p:=s(a)$. Then $p\circ a=a$ implies $a\in C(p)$ and $U_{1-p}(a)=0$. Let
$q\in P$ be such that $a\in C(q)$ and $U_{q}(a)=0$. Then $(1-q)\circ a=a$,
which entails $p\leq 1-q$. Then $q\leq 1-p$, and by definition of the Rickart
mapping, $a^{*}=1-p$. ∎

######  5.10 Lemma.

Let $A$ be a Rickart JB-algebra and let $a\in A^{+}$. Then $s(a)\in A(a,1)$.

###### Proof.

By replacing $a$ with $\|a\|^{-1}a$, we may clearly assume that $0\leq a\leq
1$. Let $C$ be a C-block containing $a$. Since $A$ is spectral, $C\cong C(X)$
for a basically disconnected compact Hausdorff space $X$. Let $f\in C(X)$ be a
function corresponding to $a$ and let $Y\subset X$ be the support of $f$, then
$s(a)$ corresponds to the characteristic function $\chi_{Y}\in C(X)$. Then
$0\leq f(x)\leq 1$ and $f_{n}(x):=f(x)^{1/n}$ is a nondecreasing sequence of
continuous functions pointwise converging to $\chi_{Y}$. By Dini’s theorem,
$f_{n}$ converges to $\chi_{Y}$ in norm, so that the corresponding sequence
$a_{n}:=a^{1/n}$ norm-converges to $s(a)$. Since all $a_{n}$ are contained in
$A(a,1)$, so is $s(a)$.

∎

######  5.11 Corollary.

Let $A$ be a Rickart JB-algebra and let $a\in A$, then $A(a,1)$ contains all
the spectral projections of $a$.

###### Proof.

By Lemma 5.9, the spectral projections can be obtained as complements of
supports of elements of the form $(a-\lambda 1)^{+}$, $\lambda\in\mathbb{R}$.
Since $(a-\lambda 1)^{+}$ belong to $A(a,1)$ for all $\lambda$, all such
supports belong to $A(a,1)$ by Lemma 5.10.

∎

######  5.12 Lemma.

Let $A$ be a Rickart JB-algebra. Then $aCb$ if and only if $a$ and $b$
operator commute.

###### Proof.

Assume that $A$ is Rickart. Note that if $b$ operator commutes with $a$ then
it operator commutes with $a^{2}$ [28, Thm. 3.13] and hence with all
polynomials in $a$ [28, Cor. 2.11]. Consequently, $b$ operator commutes with
all of $A(a,1)$. By Corollary 5.11, $A(a,1)$ includes all spectral projections
of $a$, which implies that $aCb$. The converse follows from Lemma 5.7 and the
fact that $A$ is spectral.

∎

######  5.13 Proposition.

Let $A$ be a Rickart JB-algebra. Then the C-blocks of $A$ are precisely the
maximal associative subalgebras in $A$.

###### Proof.

We already know by Corollary 5.8 that any C-block is a maximal associative
subalgebra. Let $C\subseteq A$ be a maximal associative subalgebra in $A$.
Then by Lemmas 5.5 and 5.12, all elements of $C$ are mutually commuting so
that $C$ is contained in some C-block $C_{0}$. By Corollary 5.8, $C_{0}$ is a
maximal associative subalgebra, hence $C=C_{0}$ is a C-block.

∎

Finally, we will need the following characterization of a spectral order unit
space, proved in [25].

######  5.14 Theorem.

[25] Let $A$ be a complete order unit space with a compression base having the
comparability property. Then the following are equivalent.

  1. (i)

$A$ is spectral;

  2. (ii)

Every C-block of $A$ is spectral;

  3. (iii)

Every C-block of $A$ is monotone $\sigma$-complete.

###### Proof of Theorem 5.2.

Assume that $A$ is Rickart, so the compression base $(U_{p})_{p\in P}$ in $A$
is spectral. Since $A$ is complete, we have by Theorem 5.14 that every C-block
in $A$ is monotone $\sigma$-complete. The statement now follows by Proposition
5.13.

Conversely, assume that every maximal associative subalgebra in $A$ is
monotone $\sigma$-complete. We will first show that $A$ with the compression
base $(U_{p})_{p\in P}$ has the comparability property. Any element $a\in A$
is contained in some maximal associative subalgebra $A_{0}$, which is norm-
closed by maximality. By Theorem 5.6, $A_{0}$ is isometrically isomorphic to
$C(X)$ for some compact Hausdorff space $X$. By the assumption, $A_{0}$ is
monotone $\sigma$-complete, so that $X$ is basically disconnected. Let $f\in
C(X)$ be the element corresponding to $a$ and let $f^{+}$ be the positive
part, then $f^{+}=f\vee 0$ belongs to the subalgebra $A(f,1)$ in $C(X)$
generated by $f$ and 1. Similarly as in the proof of Lemma 5.10, we obtain
that the characteristic function of the support of $f^{+}$ is in $A(f,1)$.
Hence there is a corresponding projection $p\in A(a,1)$. Observe that
$A(a,1)\subseteq C(PC(a))$ ([25, Eq. (10)]), so that $p\in P(a)$ and it is
easily seen by definition of $p$ that $p\in P^{\pm}(a)$. Hence $A$ has
comparability.

Using Theorem 5.14 once again, it is enough to show that any C-block $C$ of
$A$ is a maximal associative subalgebra, which is precisely Corollary 5.8.

∎

##  6 Concluding remarks

We studied spectral order unit spaces in the sense of Foulis [18]. We proved
that for order unit spaces with comparability property, a continuous
functional calculus can be introduced and there is a Borel functional calculus
for spectral order unit spaces.

We then concentrated on order unit spaces which are JB-algebras. As it turned
out, the condition of Alfsen and Shultz characterizing JB-algebras in the
setting of spectral duality [1] can be applied to the much more general
situation of order unit spaces having the comparability property. In
particular, in the case of order unit spaces derived from Banach spaces (here
we call them generalized spin factors), such an order unit space is a JB-
algebra if and only if it is a spin factor.

Finally, we obtained a generalization of a characterization of Rickart
C*-algebras due to Saitô and Wright [29] to Rickart JB-algebras, using the
connection to spectrality of order unit spaces and recent results in [28].

There is a number of problems related to spectrality of order unit spaces,
which are left to future work. For example, if $A$ is a JB-algebra, then
$A\subseteq A^{**}$ and by [1, Cor. 2.50], the second dual is a JBW-algebra,
hence it is spectral (in the sense of Alfsen and Shultz) [1, Thm. 2.20, Prop.
8.76], so that in particular, any element $a\in A$ has a spectral resolution
in $A^{**}$. If $A$ is Rickart, then it follows from Theorem 5.3 that $a$ has
a spectral resolution in $A$. It is a question what is the connection between
these spectral resolutions. Another question is the relation of the spectral
resolutions of an element $a$ and $g(a)$ for a Borel function $g$, here we may
ask for conditions when the integrals of the form (4) are defined, see the end
of Section 3.

Another possible research direction is the study of convex effect algebras.
The interval $[0,1]$ in an order unit space is an archimedean convex effect
algebra [20] and, conversely, any such effect algebra can be represented as a
unit interval in an order unit space. Spectrality in effect algebras was
studied in [26] and it was proved that an archimedean convex effect algebra is
spectral (has the comparability property) if and only if the representing
order unit space is spectral (has the comparability property). We may apply
our results to some questions for convex effect algebras, for example, for the
study of convex sequential effect algebras, their spectrality and their
relation to JB-algebras.

## Acknowledgements

The research was supported by the grants VEGA 1/0142/20 and the Slovak
Research and Development Agency grant APVV-20-0069.

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# On logarithmic coefficients of certain starlike functions related to the
vertical strip
The Journal Of Analysis, DOI: 10.1007/s41478-018-0157-7

Rahim Kargar  Young Researchers and Elite Club, Ardabil Branch, Islamic Azad
University, Ardabil, Iran<EMAIL_ADDRESS>
###### Abstract.

In the present paper two certain subclasses of the starlike functions
associated with the vertical strip are considered. The main aim of this paper
is to investigate some basic properties of these classes such as,
subordination relations, sharp inequalities for sums involving logarithmic
coefficients and estimate of logarithmic coefficients for functions belonging
to these subclasses.

###### Key words and phrases:

Univalent; Starlike; Vertical strip; Logarithmic coefficients; Subordination;
Hadamard product

###### 2010 Mathematics Subject Classification:

30C50; 30C45

##  1\. Introduction

Throughout this paper $\Delta$ is the open unit disc on the complex plane
$\mathbb{C}$. Let $\mathcal{H}$ be the class of all analytic functions in
$\Delta$ and $\mathcal{A}$ be a subclass of $\mathcal{H}$ with the
normalization $f(0)=f^{\prime}(0)-1=0$. The subclass of $\mathcal{A}$
consisting of all univalent functions $f$ in $\Delta$ is denoted by
$\mathcal{S}$. Let $\mathcal{S}^{*}$ and $\mathcal{K}$ denote the subclasses
of $\mathcal{S}$ consisting of the normalized starlike and convex functions in
$\Delta$, respectively. Also, we say that a function $f\in\mathcal{A}$ is
close–to–convex, if there is a convex function $g$ such that

${\rm
Re}\left\\{\frac{f^{\prime}(z)}{g^{\prime}(z)}\right\\}>0\quad(z\in\Delta).$

Let $\mathcal{U}(\lambda)$ denote the set of all $f\in\mathcal{A}$ in $\Delta$
satisfying the condition

$\left|\left(\frac{z}{f(z)}\right)^{2}f^{\prime}(z)-1\right|<\lambda\quad(z\in\Delta),$

where $0<\lambda\leq 1$. For more details and interesting properties of the
family $\mathcal{U}(\lambda)$, the reader may refer to [18]. Also, let
$\mathcal{G}(a)$ denote the class of locally univalent normalized analytic
functions $f$ in $\Delta$ satisfying the condition

${\rm
Re}\left\\{1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right\\}<1+\frac{a}{2}\quad(a>0,\,z\in\Delta).$

The class $\mathcal{G}(a)$ has been studied extensively by Kargar et al. [11],
Maharana et al. [16], Obradović et al. [17], and Ponnusamy and Sahoo [20].

It is well–known that the logarithmic coefficients have had great impact in
the development of the theory of univalent functions. For example, de Branges
by use of this concept, was able to prove the famous Bieberbach’s conjecture
[1]. The logarithmic coefficients $\gamma_{n}:=\gamma_{n}(f)$ of
$f\in\mathcal{A}$ are defined by

(1.1)
$\log\left\\{\frac{f(z)}{z}\right\\}=\sum_{n=1}^{\infty}2\gamma_{n}z^{n}\quad(z\in\Delta).$

As an example consider the rotation of Koebe function

$k_{\varepsilon}(z)=\frac{z}{(1-\varepsilon z)^{2}}\quad(|\varepsilon|=1).$

Then a simple calculation gives that

$\gamma_{n}(k_{\varepsilon})=\frac{\varepsilon^{n}}{n}\quad(n\geq 1).$

The inequality $|\gamma_{n}(f)|\leq 1/n$ holds for each starlike function
$f\in\mathcal{S}$ and the equality is attained for the rotation of Koebe
function, but it is false for the full class $\mathcal{S}$, even in order of
magnitude. Let $f\in\mathcal{S}$ and $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$.
Then by (1.1), it follows that

$\gamma_{1}=\frac{a_{2}}{2}\quad{\rm
and}\quad\gamma_{2}=\frac{1}{2}\left(a_{3}-\frac{a_{2}^{2}}{2}\right)$

and thus the following sharp estimates hold

$|\gamma_{1}|\leq 1\quad{\rm
and}\quad|\gamma_{2}|\leq\frac{1}{2}(1+2e^{-2})\approx 0.635.$

However, the sharp estimate of $|\gamma_{n}|$ when $n\geq 3$ and
$f\in\mathcal{S}$ it is still open. For more explanation of this issue, it is
necessary to point out that there is a bounded and univalent function with
logarithmic coefficients $\gamma_{n}$ such that $\gamma_{n}\neq O(n^{-0.83})$
[5, p. 242]. Also, there exists a close–to–convex function $f$ such that
$|\gamma_{n}(f)|>1/n$, [7]. In completing this entry Ye showed that the
logarithmic coefficients $\gamma_{n}$ of each close–to–convex function $f$ in
$\mathcal{S}$ satisfy $|\gamma_{n}(f)|\leq(A\log n)/n$, where $A$ is an
absolute constant, see [30].

Sharp inequalities are known for sums involving logarithmic coefficients. For
instance, the logarithmic coefficients $\gamma_{n}$ of every function
$f\in\mathcal{S}$ satisfy the sharp inequality

(1.2) $\sum_{n=1}^{\infty}|\gamma_{n}|^{2}\leq\frac{\pi^{2}}{6}$

and the equality is attained for the Koebe function (see [6, Theorem 4]).
Also, for each $f\in\mathcal{S}$ the sharp inequality

$\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{2}|\gamma_{n}|^{2}\leq
4\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{2}\frac{1}{n^{2}}=\frac{2\pi^{2}-12}{3}$

holds (see [24]). Recently, Obradović et al. [19] proved that the logarithmic
coefficients $\gamma_{n}$ of any $f\in\mathcal{U}(\lambda)$ satisfy the sharp
inequality

$\sum_{n=1}^{\infty}|\gamma_{n}|^{2}\leq\frac{1}{4}\left(\frac{\pi^{2}}{6}+2Li_{2}(\lambda)+Li_{2}(\lambda^{2})\right),$

where $Li_{2}$ denotes the dilogarithm function and that the logarithmic
coefficients $\gamma_{n}$ of $f\in\mathcal{U}(1)$ satisfy the inequality
(1.2). Also, they proved that (see [19, Theorem 2]) the logarithmic
coefficients $\gamma_{n}$ of $f\in\mathcal{G}(a)$ satisfy the inequalities

$\sum_{n=1}^{\infty}n^{2}|\gamma_{n}|^{2}\leq\frac{a}{4(a+2)}\quad(0<a\leq
1),$
$\sum_{n=1}^{\infty}|\gamma_{n}|^{2}\leq\frac{a^{2}}{4}Li_{2}\left((1+a)^{-2}\right)\quad(0<a\leq
1)$

and

(1.3) $|\gamma_{n}|\leq\frac{a}{2(a+1)n}\quad(0<a\leq 1,n=1,2,\ldots).$

It’s worth mentioning, that the above inequality (1.3) is not sharp. Very
recently the sharp estimates for the initial logarithmic coefficients
$\gamma_{n}$ of $f\in\mathcal{G}(a)$ where $0<a\leq 1$ and $n=1,2,3$ were
obtained by Ponnusamy et al. (see [22, Theorem 2.10]). They, also studied the
logarithmic inverse coefficients, denoted by $\Gamma_{n}(F)$, of
$f\in\mathcal{G}(a)$, where $F$ is the inverse function of
$f\in\mathcal{G}(a)$. For more details see [21].

In the sequel, we recall two certain subclasses of the starlike functions. Let
$\mathcal{S}(\alpha,\beta)$ denote the class of all functions
$f\in\mathcal{A}$ which satisfy the following two–sided inequality

$\alpha<{\rm
Re}\left\\{\frac{zf^{\prime}(z)}{f(z)}\right\\}<\beta\quad(\alpha<1,\,\beta>1).$

The class $\mathcal{S}(\alpha,\beta)$ was introduced in [14] and studied in
[15] and [29]. Also, we expanded the class $\mathcal{S}(\alpha,\beta)$ in
[10]. By definition of subordination, $f\in\mathcal{S}(\alpha,\beta)$ if, and
only if

(1.4) $\frac{zf^{\prime}(z)}{f(z)}\prec P_{\alpha,\beta}(z)\quad(z\in\Delta),$

where

(1.5)
$P_{\alpha,\beta}(z):=1+\frac{\beta-\alpha}{\pi}i\log\left(\frac{1-e^{2\pi
i\frac{1-\alpha}{\beta-\alpha}}z}{1-z}\right).$

The function $P_{\alpha,\beta}(z)$ is convex univalent in $\Delta$ and has the
form

(1.6) $P_{\alpha,\beta}(z)=1+\sum_{n=1}^{\infty}B_{n}z^{n},$

where

(1.7) $B_{n}=\frac{\beta-\alpha}{n\pi}i\left(1-e^{2n\pi
i\frac{1-\alpha}{\beta-\alpha}}\right)\quad(n=1,2,\ldots)$

and maps $\Delta$ onto a convex domain

$\Omega_{\alpha,\beta}:=\\{w\in\mathbb{C}:\alpha<{\rm Re}\,w<\beta\\}$

conformally. Recently, the function $P_{\alpha,\beta}(z)$ has been studied by
many works, see for example [8, 10, 14, 15, 28, 29].

Also, we say that a function $f\in\mathcal{A}$ belongs to the class
$\mathcal{M}(\delta)$, if $f$ satisfies

$1+\frac{\delta-\pi}{2\sin\delta}<{\rm
Re}\left\\{\frac{zf^{\prime}(z)}{f(z)}\right\\}<1+\frac{\delta}{2\sin\delta}\quad(z\in\Delta),$

where $\pi/2\leq\delta<\pi$. The class $\mathcal{M}(\delta)$ was introduced by
Kargar et al. [9]. Moreover, by definition of subordination,
$f\in\mathcal{M}(\delta)$ if, and only if

$\left(\frac{zf^{\prime}(z)}{f(z)}-1\right)\prec\mathcal{B}_{\delta}(z)\quad(z\in\Delta),$

where

(1.8)
$\mathcal{B}_{\delta}(z):=\frac{1}{2i\sin\delta}\log\left(\frac{1+ze^{i\delta}}{1+ze^{-i\delta}}\right)\quad(z\in\Delta).$

The function $\mathcal{B}_{\delta}(z)$ due to Dorff [2] and studied in [3],
[4], [12] and [13]. The function $\mathcal{B}_{\delta}(z)$ is convex univalent
in $\Delta$ and has the form

(1.9)
$\mathcal{B}_{\delta}(z)=\sum_{n=1}^{\infty}A_{n}z^{n}\quad(z\in\Delta),$

where

(1.10) $A_{n}=\frac{(-1)^{(n-1)}\sin
n\delta}{n\sin\delta}\quad(n=1,2,\ldots).$

The following lemma due to Ruscheweyh and Stankiewicz, will be useful in this
paper.

######  Lemma 1.1.

(see [27]) Let $\phi,\varphi\in\mathcal{H}$ be any convex univalent functions
in $\Delta$. If $f(z)\prec\phi(z)$ and $g(z)\prec\varphi(z)$, then

$f(z)*g(z)\prec\phi(z)*\varphi(z)\quad(z\in\Delta),$

where ”*” denotes the Hadamard product.

In this paper, some subordination relations among the classes
$\mathcal{S}(\alpha,\beta)$ and $\mathcal{M}(\delta)$ are presented. These
relations are then used to obtain sharp estimates for sums involving their
logarithmic coefficients. Also, the estimate of logarithmic coefficients for
functions belonging to these subclasses are determined.

##  2\. Main Results

One of the aims of this paper is the following theorem which will be useful in
order to estimate of sums involving logarithmic coefficients of functions in
the class $\mathcal{S}(\alpha,\beta)$.

######  Theorem 2.1.

Let $f(z)\in\mathcal{A}$, $\alpha<1$ and $\beta>1$. Also let
$P_{\alpha,\beta}(z)$ be defined by (1.5). If
$f(z)\in\mathcal{S}(\alpha,\beta)$, then

(2.1) $\log\left\\{\frac{f(z)}{z}\right\\}\prec\widehat{P}_{\alpha,\beta}(z),$

where

(2.2)
$\widehat{P}_{\alpha,\beta}(z):=\int_{0}^{z}\frac{P_{\alpha,\beta}(t)-1}{t}{\rm
d}t,$

and $\widehat{P}_{\alpha,\beta}$ is convex univalent.

###### Proof.

Let $f(z)\in\mathcal{A}$. If we define $p(z):=f(z)/z$, then $p(z)$ is analytic
in $\Delta$ and $p(0)=1$. Also, since $f(z)\in\mathcal{S}(\alpha,\beta)$,
therefore by (1.4), we have

(2.3) $\frac{zp^{\prime}(z)}{p(z)}=\frac{zf^{\prime}(z)}{f(z)}-1\prec
P_{\alpha,\beta}(z)-1\quad(z\in\Delta),$

where $P_{\alpha,\beta}$ is of the form (1.5). On the other hand, it is
well–known that (see [25]) the function

$\widehat{h}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n}$

is convex univalent in $\Delta$ and

$\psi(z)\ast\widehat{h}(z)=\int_{0}^{z}\frac{\psi(t)}{t}{\rm
d}t\quad(\psi\in\mathcal{H}).$

Now by Lemma 1.1 and from (2.3) we get

(2.4)
$\frac{zp^{\prime}(z)}{p(z)}\ast\widehat{h}(z)\prec(P_{\alpha,\beta}(z)-1)\ast\widehat{h}(z)\quad(z\in\Delta).$

Moreover by (2.4), we can obtain (2.1). On the other hand, since
$P_{\alpha,\beta}(z)$ and $\widehat{h}(z)$ are convex univalent functions, by
the Pòlya–Schoenberg conjecture (this conjecture states that the class of
convex univalent functions is preserved under the convolution) that is proved
by Ruscheweyh and Sheil–Small (see [26]), the function
$\widehat{P}_{\alpha,\beta}(z)$ is convex univalent, too. ∎

Because $\widehat{P}_{\alpha,\beta}(z)$ is convex univalent, thus we get.

######  Corollary 2.1.

Let $f(z)\in\mathcal{S}(\alpha,\beta)$. Then

$\frac{f(z)}{z}\prec\exp\widehat{P}_{\alpha,\beta}(z)\quad(z\in\Delta),$

where $\widehat{P}_{\alpha,\beta}(z)$ is given by (2.2).

######  Theorem 2.2.

For $\alpha<1$ and $\beta>1$, the logarithmic coefficients of
$f\in\mathcal{S}(\alpha,\beta)$ satisfy the following inequality

(2.5)
$\sum_{n=1}^{\infty}|\gamma_{n}|^{2}\leq\frac{(\beta-\alpha)^{2}}{4\pi^{2}}\left(\frac{\pi^{4}}{45}-Li_{4}\left(e^{-2\pi
i\frac{1-\alpha}{\beta-\alpha}}\right)-Li_{4}\left(e^{2\pi
i\frac{1-\alpha}{\beta-\alpha}}\right)\right),$

where $Li_{4}$ is defined as following

(2.6)
$Li_{4}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{4}}=-\frac{1}{2}\int_{0}^{1}\frac{\log^{2}(1/t)\log(1-tz)}{t}{\rm
d}t.$

The result is sharp.

###### Proof.

Let $f\in\mathcal{S}(\alpha,\beta)$. Then by Theorem 2.1, we have

(2.7) $\log\left\\{\frac{f(z)}{z}\right\\}\prec\widehat{P}_{\alpha,\beta}(z),$

where $\widehat{P}_{\alpha,\beta}(z)$ is defined in (2.2). By using (1.6) and
(1.7), one can rewrite $\widehat{P}_{\alpha,\beta}(z)$ as the following

(2.8)
$\widehat{P}_{\alpha,\beta}(z)=\sum_{n=1}^{\infty}\frac{\beta-\alpha}{\pi
n^{2}}i\left(1-e^{2\pi ni\frac{1-\alpha}{\beta-\alpha}}\right)z^{n}.$

With placement of (1.1) and (2.8) into (2.7), we get

$\sum_{n=1}^{\infty}2\gamma_{n}z^{n}\prec\sum_{n=1}^{\infty}\frac{\beta-\alpha}{\pi
n^{2}}i\left(1-e^{2\pi ni\frac{1-\alpha}{\beta-\alpha}}\right)z^{n}.$

Applying Rogosinski’s theorem (see [23] or [5, Theorem 6.2]), we obtain

$\displaystyle 4\sum_{n=1}^{\infty}|\gamma_{n}|^{2}$
$\displaystyle\leq\sum_{n=1}^{\infty}\frac{(\beta-\alpha)^{2}}{\pi^{2}n^{4}}\left|i\left(1-e^{2\pi
ni\frac{1-\alpha}{\beta-\alpha}}\right)\right|^{2}$
$\displaystyle=\frac{2(\beta-\alpha)^{2}}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{4}}\left(1-\cos
2\pi n\frac{1-\alpha}{\beta-\alpha}\right)$
$\displaystyle=\frac{2(\beta-\alpha)^{2}}{\pi^{2}}\left(\frac{\pi^{4}}{90}-\frac{1}{2}\left[Li_{4}\left(e^{-2\pi
i\frac{1-\alpha}{\beta-\alpha}}\right)+Li_{4}\left(e^{2\pi
i\frac{1-\alpha}{\beta-\alpha}}\right)\right]\right)$

and we get the inequality (2.5). The inequality is sharp for the logarithmic
coefficients of the function

$\mathfrak{F}_{\alpha,\beta}(z)=z\exp\widehat{P}_{\alpha,\beta}(z),$

where $\widehat{P}_{\alpha,\beta}(z)$ is given by (2.2). A simple check gives
us

$\gamma_{n}(\mathfrak{F}_{\alpha,\beta}(z))=\frac{\beta-\alpha}{2\pi
n^{2}}i\left(1-e^{2\pi ni\frac{1-\alpha}{\beta-\alpha}}\right)$

and concluding the proof. ∎

######  Theorem 2.3.

Let $f\in\mathcal{A}$ belongs to the class $\mathcal{S}(\alpha,\beta)$ and
$\gamma_{n}$ be the logarithmic coefficients of $f$. Then

(2.9)
$|\gamma_{n}|\leq\frac{\beta-\alpha}{n\pi}\left|\sin\frac{\pi(1-\alpha)}{\beta-\alpha}\right|\quad(n\geq
1,\alpha<1<\beta).$

###### Proof.

If $f\in\mathcal{A}$ belongs to the class $\mathcal{S}(\alpha,\beta)$, then by
(1.4) we have

$z\frac{f^{\prime}(z)}{f(z)}-1=z\left(\log\left\\{\frac{f(z)}{z}\right\\}\right)^{\prime}\prec
P_{\alpha,\beta}(z)-1,$

where $P_{\alpha,\beta}$ is defined in (1.5). Moreover, in terms of the
logarithmic coefficients $\gamma_{n}$ of $f$ defined by (1.1) and (1.6), is
equivalent to

$\sum_{n=1}^{\infty}2n\gamma_{n}z^{n}\prec\sum_{n=1}^{\infty}B_{n}z^{n}.$

Now by Rogosinski’s theorem (see [23, Theorem X]), we get
$2n|\gamma_{n}|\leq|B_{1}|$. Therefore the inequality (2.9) follows. This
completes the proof. ∎

It is clear that if $\beta\rightarrow+\infty$, then
$\mathcal{S}(\alpha,\beta)\rightarrow\mathcal{S}^{*}(\alpha)$ (the class of
starlike functions of order $\alpha$, where $0\leq\alpha<1$). Thus we have the
following result (see [19, Remark 1]).

######  Corollary 2.2.

If $f\in\mathcal{S}(\alpha,\beta)$ when $\beta\rightarrow+\infty$, then

$|\gamma_{n}|\leq\frac{\beta-\alpha}{n\pi}\left|\sin\frac{\pi(1-\alpha)}{\beta-\alpha}\right|\leq\frac{\beta-\alpha}{n\pi}\times\frac{\pi(1-\alpha)}{\beta-\alpha}=\frac{1-\alpha}{n}\quad(n\geq
1).$

Indeed, if $f\in\mathcal{S}^{*}(\alpha)$ $(0\leq\alpha<1)$ and $\gamma_{n}$ is
the corresponding logarithmic coefficients, then we have
$|\gamma_{n}|\leq(1-\alpha)/n$ for $n\geq 1$.

Next, we have the following.

######  Theorem 2.4.

Let $\pi/2\leq\delta<\pi$. Also let $\mathcal{B}_{\delta}(z)$ and $A_{n}$ be
defined by (1.8) and (1.10), respectively. If $f(z)\in\mathcal{M}(\delta)$,
then

$\log\left\\{\frac{f(z)}{z}\right\\}\prec\int_{0}^{z}\frac{\mathcal{B}_{\delta}(t)}{t}{\rm
d}t.$

Moreover,

(2.10)
$\mathcal{\widetilde{B}}_{\delta}(z):=\int_{0}^{z}\frac{\mathcal{B}_{\delta}(t)}{t}{\rm
d}t=\sum_{n=1}^{\infty}\frac{A_{n}}{n}z^{n}$

is a convex univalent function.

###### Proof.

The proof is similar to the proof of the Theorem 2.1, and thus we omit the
details. ∎

Since $\mathcal{\widetilde{B}}_{\delta}(z)$ is a convex univalent function,
thus we have.

######  Corollary 2.3.

If $f(z)\in\mathcal{M}(\delta)$, then

$\frac{f(z)}{z}\prec\exp\mathcal{\widetilde{B}}_{\delta}(z)\quad(z\in\Delta),$

where $\mathcal{\widetilde{B}}_{\delta}(z)$ is of the form (2.10).

######  Theorem 2.5.

Let $f\in\mathcal{A}$ belongs to the class $\mathcal{M}(\delta)$ and
$\pi/2\leq\delta<\pi$. Then the logarithmic coefficients of $f$ satisfy the
inequality

(2.11)
$\sum_{n=1}^{\infty}|\gamma_{n}|^{2}\leq\frac{1}{16\sin^{2}\delta}\left[\frac{\pi^{4}}{45}-Li_{4}\left(e^{-2i\delta}\right)-Li_{4}\left(e^{2i\delta}\right)\right],$

where $Li_{4}$ is defined in (2.6). The result is sharp.

###### Proof.

Let $f\in\mathcal{M}(\delta)$. Then by Theorem 2.4, we have

(2.12) $\displaystyle\log\left\\{\frac{f(z)}{z}\right\\}$
$\displaystyle\prec\mathcal{\widetilde{B}}_{\delta}(z)\quad(z\in\Delta).$

By using (1.1) and (1.9), the relation (2.12) implies that

$\sum_{n=1}^{\infty}2\gamma_{n}z^{n}\prec\sum_{n=1}^{\infty}\frac{A_{n}}{n}z^{n}\quad(z\in\Delta).$

Now by Rogosinski’s theorem (see [23] or [5, Theorem 6.2]), we get

$\displaystyle 4\sum_{n=1}^{\infty}|\gamma_{n}|^{2}$
$\displaystyle\leq\sum_{n=1}^{\infty}\frac{1}{n^{2}}|A_{n}|^{2}$
$\displaystyle=\frac{1}{\sin^{2}\delta}\sum_{n=1}^{\infty}\frac{\sin^{2}n\delta}{n^{4}}$
$\displaystyle=\frac{1}{\sin^{2}\delta}\sum_{n=1}^{\infty}\left(\frac{1}{180}\left[\pi^{4}-45Li_{4}\left(e^{-2i\delta}\right)-45Li_{4}\left(e^{2i\delta}\right)\right]\right),$

where $Li_{4}$ is defined by (2.6). Therefore the desired inequality (2.11)
follows. For the sharpness of (2.11), consider

$F_{\delta}(z)=z\exp\mathcal{\widetilde{B}}_{\delta}(z),$

where $\mathcal{\widetilde{B}}_{\delta}(z)$ is defined by (2.10). It is easy
to see that $F_{\delta}(z)\in\mathcal{M}(\delta)$ and
$\gamma_{n}(F_{\delta})=A_{n}/2n$, where $A_{n}$ is given by (1.10).
Therefore, we have the equality in (2.11). This is the end of proof. ∎

######  Theorem 2.6.

Let $\pi/2\leq\delta<\pi$. If $f\in\mathcal{A}$ belongs to the class
$\mathcal{M}(\delta)$, then the logarithmic coefficients of $f$ satisfy

$|\gamma_{n}|\leq\frac{1}{2n}\quad(n\geq 1).$

###### Proof.

The proof is similar to the proof of the Theorem 2.3, and thus the details are
omitted ∎

Acknowledgements This work is supported by Young Researchers and Elite Club,
Ardabil branch. The author would like to thank the anonymous referee(s) for
their careful readings, valuable suggestions and comments, which helped to
improve the presentation of the paper.
Compliance with ethical standards
Research involving human participants and/or animals This research does not
contain any studies with human participants and/or animals performed by the
author.
Conflict of interest The author declares there is no conflict of interest
related to this article.

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