Abstract
Let \({{\mathscr {M}}}\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau \). We prove that every continuous mhomogeneous polynomial P from \(L^p({{\mathscr {M}}},\tau )\), with \(0<p<\infty \), into each topological linear space X with the property that \(P(x+y)=P(x)+P(y)\) whenever x and y are mutually orthogonal positive elements of \(L^p({{\mathscr {M}}},\tau )\) can be represented in the form \(P(x)=\varPhi (x^m)\)\((x\in L^p({{\mathscr {M}}},\tau ))\) for some continuous linear map \(\varPhi :L^{p/m}({{\mathscr {M}}},\tau )\rightarrow X\).
Introduction
In [16], the author succeeded in providing a useful representation of the orthogonally additive homogeneous polynomials on the spaces \(L^p([0,1])\) and \(\ell ^p\) with \(1\le p<\infty \). In [12] (see also [6]), the authors obtained a similar representation for the space C(K), for a compact Hausdorff space K. These results were generalized to Banach lattices [4] and Riesz spaces [9]. Further, the problem of representing the orthogonally additive homogeneous polynomials has been also considered in the context of Banach function algebras [1, 19] and noncommutative Banach algebras [2, 3, 11]. Notably, [11] can be thought of as the natural noncommutative analogue of the representation of orthogonally additive polynomials on C(K)spaces, and the purpose to this paper is to extend the results of [16] on the representation of orthogonally additive homogeneous polynomials on \(L^p\)spaces to the noncommutative \(L^p\)spaces.
The noncommutative \(L^p\)spaces that we consider are those associated with a von Neumann algebra \({{\mathscr {M}}}\) equipped with a normal semifinite faithful trace \(\tau \). From now on, \(S({{\mathscr {M}}},\tau )\) stands for the linear span of the positive elements x of \({{\mathscr {M}}}\) such that \(\tau \bigl ({{\,\mathrm{supp}\,}}(x)\bigr )<\infty \); here \({{\,\mathrm{supp}\,}}(x)\) stands for the support of x. Then \(S({{\mathscr {M}}},\tau )\) is a \(*\)subalgebra of \({{\mathscr {M}}}\) with the property that \(\vert x\vert ^p\in S({{\mathscr {M}}},\tau )\) for each \(x\in S({{\mathscr {M}}},\tau )\) and each \(0<p<\infty \). For \(0<p<\infty \), we define \(\Vert \cdot \Vert _p:S({{\mathscr {M}}},\tau )\rightarrow {\mathbb {R}}\) by \(\Vert x\Vert _p=\tau (\vert x\vert ^p)^{1/p}\)\((x\in S({{\mathscr {M}}},\tau ))\). Then \(\Vert \cdot \Vert _p\) is a norm or a pnorm according to \(1\le p<\infty \) or \(0<p<1\), and the space \(L^p({{\mathscr {M}}},\tau )\) can be defined as the completion of \(S({{\mathscr {M}}},\tau )\) with respect to \(\Vert \cdot \Vert _p\). Nevertheless, for our purposes here, it is important to realize the elements of \(L^p({{\mathscr {M}}},\tau )\) as measurable operators. Specifically, the set \(L^0({{\mathscr {M}}},\tau )\) of measurable closed densely defined operators affiliated to \({{\mathscr {M}}}\) is a topological \(*\)algebra with respect to the strong sum, the strong product, the adjoint operation, and the topology of the convergence in measure. The algebra \({{\mathscr {M}}}\) is a dense \(*\)subalgebra of \(L^0({{\mathscr {M}}},\tau )\), the trace \(\tau \) extends to the positive cone of \(L^0({{\mathscr {M}}},\tau )\) in a natural way, and we can define
Also we set \(L^\infty ({{\mathscr {M}}},\tau )={{\mathscr {M}}}\) (with \(\Vert \cdot \Vert _\infty :=\Vert \cdot \Vert \), the operator norm). Operators \(x,y\in L^0({{\mathscr {M}}},\tau )\) are mutually orthogonal, written \(x\perp y\), if \(xy^*=y^*x=0\). This condition is equivalent to requiring that x and y have mutually orthogonal left, and right, supports. Further, for \(x,y\in L^p({{\mathscr {M}}},\tau )\) with \(0<p<\infty \), the condition \(x\perp y\) implies that \(\Vert x+y\Vert _p^p=\Vert x\Vert _p^p+\Vert y\Vert _p^p\), and conversely, if \(\Vert x\pm y\Vert _p^p=\Vert x\Vert _p^p+\Vert y\Vert _p^p\) and \(p\ne 2\), then \(x\perp y\) (see [14, Fact 1.3]). The orthogonal additivity considered in [16] for the spaces \(L^p([0,1])\) and \(\ell ^p\) can of course equally well be considered for the space \(L^p({{\mathscr {M}}},\tau )\). Let P be a map from \(L^p({{\mathscr {M}}},\tau )\) into a linear space X. Then P is:

(i)
orthogonally additive on a subset \({\mathscr {S}}\) of \(L^p({{\mathscr {M}}},\tau )\) if
$$\begin{aligned} x,y \in {\mathscr {S}}, \, x\perp y=0 \ \Rightarrow \ P(x+y)=P(x)+P(y); \end{aligned}$$ 
(ii)
an mhomogeneous polynomial if there exists an mlinear map \(\varphi \) from \(L^p({{\mathscr {M}}},\tau )^m\) into X such that
$$\begin{aligned} P(x)=\varphi \left( x,\dotsc ,x \right) \quad (x\in L^p({{\mathscr {M}}},\tau )). \end{aligned}$$Here and subsequently, \(m\in {\mathbb {N}}\) is fixed with \(m\ge 2\) and the superscript m stands for the mfold Cartesian product. Such a map is unique if it is required to be symmetric. Further, in the case where X is a topological linear space, the polynomial P is continuous if and only if the symmetric mlinear map \(\varphi \) associated with P is continuous.
Given a continuous linear map \(\varPhi :L^{p/m}({{\mathscr {M}}},\tau )\rightarrow X\), where X is an arbitrary topological linear space, the map \(P_\varPhi :L^p({{\mathscr {M}}},\tau )\rightarrow X\) defined by
is a natural example of a continuous mhomogeneous polynomial which is orthogonally additive on \(L^p({{\mathscr {M}}},\tau )_{\mathrm{sa}}\) (Theorem 4), and we will prove that every continuous mhomogeneous polynomial which is orthogonally additive on \(L^p({{\mathscr {M}}},\tau )_{\mathrm{sa}}\) is actually of this special form (Theorem 5). Here and subsequently, the subscripts “sa” and \(+\) are used to denote the selfadjoint and the positive parts of a given subset of \(L^0({{\mathscr {M}}},\tau )\), respectively.
We require a few remarks about the setting of our present work. Throughout the paper we are concerned with mhomogeneous polynomials on the space \(L^p({{\mathscr {M}}},\tau )\) with \(0<p\), and thus one might wish to consider polynomials with values in the space \(L^q({{\mathscr {M}}},\tau )\), especially with \(q\le p\). Further, in the case case where \(p/m<1\) and the von Neumann algebra \({{\mathscr {M}}}\) has no minimal projections, there are no nonzero continuous linear functionals on \(L^{p/m}({{\mathscr {M}}},\tau )\); since one should like to have nontrivial “orthogonally additive” polynomials on \(L^p({{\mathscr {M}}},\tau )\), some weakening of the normability must be allowed to the range space (see Corollary 2). For these reasons, throughout the paper, X will be a (complex and Hausdorff) topological linear space. In the case where the von Neumann algebra \({{\mathscr {M}}}\) is commutative, the prototypical polynomials \(P_\varPhi \) mentioned above are easily seen to be orthogonally additive on the whole domain. In contrast, we will point out in Propositions 1 and 3 that this is not the case for the von Neumann algebra \({{\mathscr {B}}}(H)\) of all bounded operators on a Hilbert space H whenever \(\dim H\ge 2\).
We assume a basic knowledge of \(C^*\)algebras and von Neumann algebras, tracial noncommutative \(L^p\)spaces, and polynomials on topological linear spaces. For the relevant background material concerning these topics, see [5, 7, 8, 10, 13, 17, 18].
\(C^*\)algebras and von Neumann algebras
Our approach to the problem of representing the orthogonally additive homogeneous polynomials on the noncommutative \(L^p\)spaces relies on the representation of those polynomials on the von Neumann algebras.
Recall that two elements x and y of a \(C^*\)algebra \({\mathscr {A}}\) are mutually orthogonal if \(xy^*=y^*x=0\), in which case the identity \(\Vert x+y\Vert =\max \{\Vert x\Vert ,\Vert y\Vert \}\) holds. The reader should be aware that we have chosen the standard definition of orthogonality in the setting of noncommutative \(L^p\)spaces. This definition is slightly different from the one used in [11], which is the standard one in the setting of Banach algebras. In [11] the orthogonality of two elements x and y is defined by the relation \(xy=yx=0\), and, further, the orthogonally additive polynomials on the selfadjoint part of a \(C^*\)algebra are automatically orthogonally additive on the whole algebra. The important point to note here is that both the definitions of orthogonality agree on the selfadjoint part of the \(C^*\)algebra. Thus, for a polynomial on a \(C^*\)algebra, the property of being orthogonally additive on the selfadjoint part according to our definition is the same as being orthogonally additive according to [11]. Nevertheless, in contrast to [11], there are no nonzero orthogonally additive polynomials from the von Neumann algebra \({{\mathscr {B}}}(H)\) into any topological Banach space according to our definition (Proposition 1).
Suppose that \({\mathscr {A}}\) is a linear space with an involution \(*\). Recall that for a linear functional \(\varPhi :{\mathscr {A}}\rightarrow {\mathbb {C}}\), the map \(\varPhi ^*:{\mathscr {A}}\rightarrow {\mathbb {C}}\) defined by \(\varPhi ^*(x)=\overline{\varPhi (x^{*})}\)\((x\in {\mathscr {A}})\) is a linear functional, and \(\varPhi \) is said to be hermitian if \(\varPhi ^*=\varPhi \). Similarly, for an mhomogeneous polynomial \(P:{\mathscr {A}}\rightarrow {\mathbb {C}}\), the map \(P^*:{\mathscr {A}}\rightarrow {\mathbb {C}}\) defined by \(P^*(x)=\overline{P(x^*)}\)\((x\in {\mathscr {A}})\) is an mhomogeneous polynomial, and we call P hermitian if \(P^*=P\).
Lemma 1
Let X and Y be linear spaces, and let \(P:X\rightarrow Y\) be an mhomogeneous polynomial. Suppose that P vanishes on a convex set \(C\subset X\). Then P vanishes on the linear span of C.
Proof
Set \(x_1,x_2,x_3,x_4\in C\). Let \(\eta :Y\rightarrow {\mathbb {C}}\) be a linear functional, and define \(f:{\mathbb {C}}^4\rightarrow {\mathbb {C}}\) by
Then f is a complex polynomial function in four complex variables that vanishes on the set
This implies that f is identically equal to 0 on \({\mathbb {C}}^4\), and, in particular,
for all \(\rho _1,\rho _2,\rho _3,\rho _4\ge 0\). Since this identity holds for each linear functional \(\eta \), it may be concluded that \(P(\rho _1x_1\rho _2x_2+i\rho _3x_3i\rho _4x_4)=0\) for all \(\rho _1,\rho _2,\rho _3,\rho _4\ge 0\). Thus P vanishes on the set
which is exactly the linear span of the set C. \(\square \)
Theorem 1
Let \({\mathscr {A}}\) be a \(C^*\)algebra, let X be a topological linear space, and let \(\varPhi :{\mathscr {A}}\rightarrow X\) be a continuous linear map. Then:

(i)
the map \(P_\varPhi :{\mathscr {A}}\rightarrow X\) defined by \(P_\varPhi (x)=\varPhi (x^m)\)\((x\in {\mathscr {A}})\) is a continuous mhomogeneous polynomial which is orthogonally additive on \({\mathscr {A}}_{\mathrm{sa}}\);

(ii)
the polynomial \(P_\varPhi \) is uniquely specified by the map \(\varPhi \).
Suppose, further, that X is a qnormed space, \(0<q\le 1\). Then:

(iii)
\(2^{1/q}\Vert \varPhi \Vert \le \Vert P_\varPhi \Vert \le \Vert \varPhi \Vert \).
Moreover, in the case where \(X={\mathbb {C}}\),

(iv)
the functional \(\varPhi \) is hermitian if and only if the polynomial \(P_\varPhi \) is hermitian, in which case \(\Vert P_\varPhi \Vert =\Vert \varPhi \Vert \).
Proof
(i) It is clear that the map \(P_\varPhi \) is continuous and that \(P_\varPhi \) is the mhomogeneous polynomial associated with the symmetric mlinear map \(\varphi :{\mathscr {A}}^m\rightarrow X\) defined by
here and subsequently, we write \(\mathfrak {S}_m\) for the symmetric group of order m.
Suppose that \(x,y\in {\mathscr {A}}_{\mathrm{sa}}\) are such that \(x\perp y\). Then \(xy=yx=0\), and so \((x+y)^m=x^m+y^m\), which gives
(ii) Assume that \(\Psi :{\mathscr {A}}\rightarrow X\) is a linear map with the property that \(P_\Psi =P_\varPhi \). If \(x\in {\mathscr {A}}_+\), then
By linearity we also get \(\Psi (x) = \varPhi (x)\) for each \(x \in {\mathscr {A}}\).
(iii) Next, assume that X is a qnormed space. For each \(x\in {\mathscr {A}}\), we have
which implies that \(\Vert P_\varPhi \Vert \le \Vert \varPhi \Vert \). Now take \(x\in {\mathscr {A}}\), and let \(\omega \in {\mathbb {C}}\) with \(\omega ^m=1\). Then \(x=\mathfrak {R}x+i\mathfrak {I}x\), where
and, further, \(\Vert \mathfrak {R}x\Vert ,\Vert \mathfrak {I}x\Vert \le \Vert x\Vert \). Moreover, \(\mathfrak {R}x=x_1x_2\) and \(\mathfrak {I}x=x_3x_4\), where \(x_1,x_2,x_3,x_4\in {\mathscr {A}}_+\), \(x_1\perp x_2\), and \(x_3\perp x_4\). Since \(x_1\perp x_2\) and \(x_3\perp x_4\), it follows that \(x_1^{1/m}\perp x_2^{1/m}\) and \(x_3^{1/m}\perp x_4^{1/m}\). Consequently,
and
Since
it follows, from (1) and (2), that
On the other hand, we have
and so
Hence, by (3),
This clearly forces \(\Vert \varPhi \Vert \le 2^{1/q}\Vert P_\varPhi \Vert \), as claimed.
(iv) It is straightforward to check that \(P_\varPhi ^*=P_{\varPhi ^*}\). Consequently, if \(\varPhi \) is hermitian, then \(P_\varPhi ^*=P_{\varPhi ^*}=P_\varPhi \) so that \(P_\varPhi \) is hermitian. Conversely, if \(P_\varPhi \) is hermitian, then \(P_{\varPhi ^*}=P_\varPhi ^*=P_\varPhi \) and (ii) implies that \(\varPhi ^*=\varPhi \). Finally, assume that \(\varPhi \) is a hermitian functional. For the calculation of \(\Vert P_\varPhi \Vert \) it suffices to check that \(\Vert \varPhi \Vert \le \Vert P_\varPhi \Vert \). For this purpose, let \(\varepsilon \in {\mathbb {R}}^+\), and choose \(x\in {\mathscr {A}}\) such that \(\Vert x\Vert =1\) and \(\Vert \varPhi \Vert \varepsilon <\vert \varPhi (x)\vert \). We take \(\alpha \in {\mathbb {C}}\) with \(\vert \alpha \vert =1\) and \(\vert \varPhi (x)\vert =\alpha \varPhi (x)\), so that
Note that \(\Vert \mathfrak {R}(\alpha x)\Vert \le 1\) and \(\Vert \varPhi \Vert \varepsilon <\varPhi (\mathfrak {R}(\alpha x))\). Now we consider the decomposition \(\mathfrak {R}(\alpha x)=x_1x_2\) with \(x_1,x_2\in {\mathscr {A}}_+\) and \(x_1\perp x_2\) and take \(\omega \in {\mathbb {C}}\) with \(\omega ^m=1\). As in (3), we see that \(\big \Vert x_1^{1/m}+\omega x_2^{1/m}\big \Vert =\Vert \mathfrak {R}(\alpha x)\Vert ^{1/m}\le 1\). Moreover, we have
which gives \(\Vert \varPhi \Vert \varepsilon <\Vert P_\varPhi \Vert \). \(\square \)
Lemma 2
Let \({\mathscr {A}}\) be a \(C^*\)algebra, let \({\mathscr {R}}\) be a \(*\)subalgebra of \({\mathscr {A}}\), let X be a topological linear space, and let \(\varPhi :{\mathscr {R}}\rightarrow X\) be a linear map. Suppose that the polynomial \(P:{\mathscr {R}}\rightarrow X\) defined by \(P(x)=\varPhi (x^m)\)\((x\in {\mathscr {R}})\) is continuous and that \({\mathscr {R}}\) satisfies the following conditions:

(i)
\(\vert x\vert \in {\mathscr {R}}\) for each \(x\in {\mathscr {R}}_{\mathrm{sa}}\);

(ii)
\(x^{1/m}\in {\mathscr {R}}\) for each \(x\in {\mathscr {R}}_+\).
Then \(\varPhi \) is continuous.
Proof
Let U be a neighbourhood of 0 in X. Let V be a balanced neighbourhood of 0 in X with \(V+V+V+V\subset U\). The set \(P^{1}(V)\) is a neighbourhood of 0 in \({\mathscr {R}}\), which implies that there exists \(r\in {\mathbb {R}}^+\) such that \(P(x)\in V\) whenever \(x\in {\mathscr {R}}\) and \(\Vert x\Vert <r\). Take \(x\in {\mathscr {R}}\) with \(\Vert x\Vert <r^m\). Since \({\mathscr {R}}\) is a \(*\)subalgebra of \({\mathscr {A}}\), we see that \(\mathfrak {R}x,\mathfrak {I}x\in {\mathscr {R}}_{\mathrm{sa}}\). We write \(\mathfrak {R}x=x_1x_2\) and \(\mathfrak {I}x=x_3x_4\), as in the proof of Theorem 1, where, on account of the condition (i),
For each \(j\in \{1,2,3,4\}\), condition (ii) gives \(x_j^{1/m}\in {\mathscr {R}}\), and, further, we have \(\Vert x_{j}^{1/m}\big \Vert =\Vert x_j\Vert ^{1/m}\le \Vert x\Vert ^{1/m}<r\). Hence
which establishes the continuity of \(\varPhi \). \(\square \)
Theorem 2
Let \({\mathscr {A}}\) be a \(C^*\)algebra, let X be a locally convex space, and let \(P:{\mathscr {A}}\rightarrow X\) be a continuous mhomogeneous polynomial. Then the following conditions are equivalent:

(i)
there exists a continuous linear map \(\varPhi :{\mathscr {A}} \rightarrow X\) such that \(P(x)=\varPhi (x^m)\)\((x\in {\mathscr {A}})\);

(ii)
the polynomial P is orthogonally additive on \({\mathscr {A}}_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \({\mathscr {A}}_+\).
If the conditions are satisfied, then the map \(\varPhi \) is unique.
Proof
Theorem 1 gives (i)\(\Rightarrow \)(ii), and obviously (ii)\(\Rightarrow \)(iii). The task is now to prove that (iii)\(\Rightarrow \)(i).
Suppose that (iii) holds. For each continuous linear functional \(\eta :X\rightarrow {\mathbb {C}}\), set \(P_\eta =\eta \circ P\). Then \(P_\eta \) is a complexvalued continuous mhomogeneous polynomial. We claim that \(P_\eta \) is orthogonally additive on \({\mathscr {A}}_{\mathrm{sa}}\). Take \(x,y\in {\mathscr {A}}_{\mathrm{sa}}\) with \(x\perp y\). Then we can write \(x=x_{+}x_{}\) and \(y=y_{+}y_{}\) with \(x_{+},x_{},y_{+},y_{}\in {\mathscr {A}}_{+}\) mutually orthogonal. Define \(f:{\mathbb {C}}^2\rightarrow {\mathbb {C}}\) by
Then f is a complex polynomial function in two complex variables. If \(\alpha ,\beta \in {\mathbb {R}}^+\), then \(x_++\alpha x_,y_++\beta y_\in {\mathscr {A}}_+\) are mutually orthogonal, and so, by hypothesis, \(P(x_++\alpha x_+y_++\beta y_)=P(x_++\alpha x_)+P(y_++\beta y_)\). This shows that \(f(\alpha ,\beta )=0\). Since f vanishes on \({\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\), it follows that f vanishes on \({\mathbb {C}}^2\), which, in particular, implies
Having proved that \(P_\eta \) is orthogonally additive on \({\mathscr {A}}_{\mathrm{sa}}\) we can apply [11, Theorem 2.8] to obtain a unique continuous linear functional \(\varPhi _\eta \) on \({\mathscr {A}}\) such that
Each \(x\in {\mathscr {A}}\) can be written in the form \(x_1^m+\cdots +x_k^m\) for suitable \(x_1,\ldots ,x_k\in {\mathscr {A}}\), and we define
Our next goal is to show that \(\varPhi \) is welldefined. Suppose that \(x_1,\ldots ,x_k\in {\mathscr {A}}\) are such that \(x_1^m+\cdots +x_k^m=0\). For each continuous linear functional \(\eta \) on X, (4) gives
Since X is locally convex, we conclude that \(\sum _{j=1}^kP(x_j)=0\).
It is a simple matter to check that \(\varPhi \) is linear and, by definition, \(P(x)=\varPhi (x^m)\)\((x\in {\mathscr {A}})\). The continuity of \(\varPhi \) then follows from Lemma 2.
The uniqueness of the map \(\varPhi \) follows from Theorem 1(ii). \(\square \)
The assumption that the space X be locally convex can be removed by requiring that the \(C^*\)algebra \({\mathscr {A}}\) be sufficiently rich in projections. The real rank zero is the most important existence of projections property in the theory of \(C^*\)algebras. We refer the reader to [5, Section V.3.2] and [7, Section V.7] for the basic properties and examples of \(C^*\)algebras of real rank zero. This class of \(C^*\)algebras contains the von Neumann algebras and the \(C^*\)algebras \({\mathscr {K}}(H)\) of all compact operators on any Hilbert space H. Let us remark that every \(C^*\)algebra of real rank zero has an approximate unit of projections (but not necessarily increasing).
Theorem 3
Let \({\mathscr {A}}\) be a \(C^*\)algebra of real rank zero, let X be a topological linear space, and let \(P:{\mathscr {A}}\rightarrow X\) be a continuous mhomogeneous polynomial. Suppose that \({\mathscr {A}}\) has an increasing approximate unit of projections. Then the following conditions are equivalent:

(i)
there exists a continuous linear map \(\varPhi :{\mathscr {A}} \rightarrow X\) such that \(P(x)=\varPhi (x^m)\)\((x\in {\mathscr {A}})\);

(ii)
the polynomial P is orthogonally additive on \({\mathscr {A}}_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \({\mathscr {A}}_+\).
If the conditions are satisfied, then the map \(\varPhi \) is unique.
Proof
Theorem 1 gives (i)\(\Rightarrow \)(ii), and it is clear that (ii)\(\Rightarrow \)(iii). We will henceforth prove that (iii)\(\Rightarrow \)(i).
We first note that such a map \(\varPhi \) is necessarily unique, because of Theorem 1(ii).
Suppose that (iii) holds and that \({\mathscr {A}}\) is unital. Let \(\varphi :{\mathscr {A}}^m\rightarrow X\) be the symmetric mlinear map associated with P and define \(\varPhi :{\mathscr {A}}\rightarrow X\) by
Let \(Q:{\mathscr {A}}\rightarrow X\) be the mhomogeneous polynomial defined by
We will prove that \(P=Q\). On account of Lemma 1, it suffices to show that \(P(x)=Q(x)\) for each \(x\in {\mathscr {A}}_{\mathrm{sa}}\).
First, consider the case where \(x\in {\mathscr {A}}_{\mathrm{sa}}\) has finite spectrum, say \(\{\rho _1,\ldots ,\rho _k\}\subset {\mathbb {R}}\). This implies that x can be written in the form
where \(e_1,\ldots ,e_k\in {\mathscr {A}}\) are mutually orthogonal projections (specifically, the projection \(e_j\) is defined by using the continuous functional calculus for x by \(e_j=\chi _{\{\rho _j\}}(x)\) for each \(j\in \{1,\ldots ,k\}\)). We also set \(e_0=1(e_1+\cdots +e_k)\), so that the projections \(e_0,e_1,\ldots ,e_k\) are mutually orthogonal, and \(\rho _0=0\). We claim that if \(j_1,\ldots ,j_m\in \{0,\ldots ,k\}\) and \(j_l\ne j_{l'}\) for some \(l,l'\in \{1,\ldots ,m\}\), then
Let \(\varLambda _1=\bigr \{n\in \{1,\ldots ,m\} : j_n=j_l\bigr \}\) and \(\varLambda _2=\bigr \{n\in \{1,\ldots ,m\} : j_n\ne j_l\bigr \}\). For each \(\alpha _1,\ldots ,\alpha _m\in {\mathbb {R}}^+\), the elements \(\sum _{n\in \varLambda _1}\alpha _{n}e_{j_n}\) and \(\sum _{n\in \varLambda _2}\alpha _{n}e_{j_n}\) are positive and mutually orthogonal, so that the orthogonal additivity of P on \({\mathscr {A}}_+\) gives
This implies that, for each linear functional \(\eta :X\rightarrow {\mathbb {C}}\), the function \(f:{\mathbb {C}}^m\rightarrow {\mathbb {C}}\) defined by
for all \(\alpha _1,\ldots ,\alpha _m\in {\mathbb {C}}\), is a complex polynomial function in m complex variables vanishing in \(\bigl ({\mathbb {R}}^+\bigr )^m\). Therefore f vanishes on \({\mathbb {C}}^m\). Moreover, we observe that the coefficient of the monomial \(\alpha _1\cdots \alpha _m\) is given by \(n!\eta \bigl (\varphi (e_{j_1},\ldots ,e_{j_m})\bigr )\), because both \(\varLambda _1\) and \(\varLambda _2\) are different from \(\{1,\ldots ,m\}\). We thus get
Since this identity holds for each linear functional \(\eta \), our claim follows. Property (5) now leads to
and
We thus get \(P(x)=Q(x)\).
Now suppose that \(x\in {\mathscr {A}}_{\mathrm{sa}}\) is an arbitrary element. Since \({\mathscr {A}}\) has real rank zero, it follows that there exists a sequence \((x_n)\) in \({\mathscr {A}}_{\mathrm{sa}}\) such that each \(x_n\) has finite spectrum and \(\lim x_n=x\). On account of the above case, we have \(P(x_n)=Q(x_n)\) (\(n\in {\mathbb {N}}\)), and the continuity of both P and Q now yields \(P(x)=\lim P(x_n)=\lim Q(x_n)=Q(x)\), as required.
We are now in a position to prove the nonunital case. By hypothesis, there exists an increasing approximate unit of projections \((e_\lambda )_{\lambda \in \varLambda }\). For each \(\lambda \in \varLambda \), set \({\mathscr {A}}_\lambda =e_\lambda {\mathscr {A}}e_\lambda \). Then \({\mathscr {A}}_\lambda \) is a unital \(C^*\)algebra (with identity \(e_\lambda \)) and has real rank zero (because \({\mathscr {A}}_\lambda \) is a hereditary \(C^*\)subalgebra of \({\mathscr {A}}\)). From what has previously been proved, it follows that there exists a unique continuous linear map \(\varPhi _\lambda :{\mathscr {A}}_\lambda \rightarrow X\) such that
Define
and, for each \(x\in {\mathscr {R}}\), set
where \(\lambda \in \varLambda \) is such that \(x\in {\mathscr {A}}_\lambda \). We will show that \(\varPhi \) is welldefined. Suppose \(\lambda ,\mu \in \varLambda \) are such that \(x\in {\mathscr {A}}_\lambda \cap {\mathscr {A}}_\mu \). Then there exists \(\nu \in \varLambda \) with \(\lambda ,\mu \le \nu \). Since the net \((e_\lambda )_{\lambda \in \varLambda }\) is increasing, we see that \(e_\lambda ,e_\mu \le e_\nu \) and therefore \({\mathscr {A}}_\lambda ,{\mathscr {A}}_\mu \subset {\mathscr {A}}_\nu \). The uniqueness of the representation of P on both \({\mathscr {A}}_\lambda \) and \({\mathscr {A}}_\mu \) implies that \(\varPhi _\nu \mid _{{\mathscr {A}}_\lambda }=\varPhi _\lambda \) and \(\varPhi _\nu \mid _{{\mathscr {A}}_\mu }=\varPhi _\mu \), which implies that \(\varPhi _\lambda (x)=\varPhi _\nu (x)=\varPhi _\mu (x)\). We now show that \({\mathscr {R}}\) is a \(*\)subalgebra of \({\mathscr {A}}\) and that \(\varPhi \) is linear. Take \(x,y\in {\mathscr {R}}\) and \(\alpha ,\beta \in {\mathbb {C}}\). We take \(\lambda ,\mu \in \varLambda \) such that \(x\in {\mathscr {A}}_{\lambda }\) and \(y\in {\mathscr {A}}_\mu \). Then \(x^*\in {\mathscr {A}}_\lambda \subset {\mathscr {R}}\). Now set \(\nu \in \varLambda \) with \(\lambda ,\mu \le \nu \). Hence \(x,y\in {\mathscr {A}}_\nu \), so that \(\alpha x+\beta y,xy\in {\mathscr {A}}_\nu \subset {\mathscr {R}}\), which shows that \({\mathscr {R}}\) is a subalgebra of \({\mathscr {A}}\). Further, we have
which shows that \(\varPhi \) is linear.
From (6) we deduce that \(P(x)=\varPhi (x^m)\) for each \(x\in {\mathscr {R}}\).
Our next goal is to show that \({\mathscr {R}}\) satisfies the conditions of Lemma 2. If \(x\in {\mathscr {R}}_{\mathrm{sa}}\) (\(x\in {\mathscr {R}}_+\)), then there exists \(\lambda \in \varLambda \) such that \(x\in \bigl ({\mathscr {A}}_\lambda \bigr )_{\mathrm{sa}}\) (\(x\in \bigl ({\mathscr {A}}_\lambda \bigr )_{+}\), respectively) and therefore \(\vert x\vert \in {\mathscr {A}}_\lambda \subset {\mathscr {R}}\) (\(x^{1/m}\in {\mathscr {A}}_\lambda \subset {\mathscr {R}}\), respectively). Since the polynomial \(P\mid _{\mathscr {R}}\) is continuous, Lemma 2 shows that the map \(\varPhi \) is continuous.
Since \((e_\lambda )_{\lambda \in \varLambda }\) is an approximate unit, it follows that \({\mathscr {R}}\) is dense in \({\mathscr {A}}\), and hence that the map \(\varPhi \) extends uniquely to a continuous linear map from \({\mathscr {A}}\) into the completion of X. By abuse of notation we continue to write \(\varPhi \) for this extension. Since both P and \(\varPhi \) are continuous, it may be concluded that \(P(x)=\varPhi (x^m)\) for each \(x\in {\mathscr {A}}\). We next prove that the image of \(\varPhi \) is actually contained in X. Of course, it suffices to show that \(\varPhi \) takes \({\mathscr {A}}_+\) into X. If \(x\in {\mathscr {A}}_+\), then
as required. \(\square \)
Since every von Neumann algebra is unital and has real rank zero, Theorem 3 applies in this setting and gives the following.
Corollary 1
Let \({{\mathscr {M}}}\) be a von Neumann algebra, let X be a topological linear space, and let \(P:{{\mathscr {M}}}\rightarrow X\) be a continuous mhomogeneous polynomial. Then the following conditions are equivalent:

(i)
there exists a continuous linear map \(\varPhi :{{\mathscr {M}}} \rightarrow X\) such that \(P(x)=\varPhi (x^m)\)\((x\in {{\mathscr {M}}})\);

(ii)
the polynomial P is orthogonally additive on \({{\mathscr {M}}}_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \({{\mathscr {M}}}_+\).
If the conditions are satisfied, then the map \(\varPhi \) is unique.
Proposition 1
Let H be a Hilbert space with \(\dim H \ge 2\), let X be a topological linear space, and let \(P:{\mathscr {B}}(H) \rightarrow X\) be a continuous mhomogeneous polynomial. Suppose that P is orthogonally additive in \({\mathscr {B}}(H)\). Then \(P=0\).
Proof
For each unitary \(v\in {\mathscr {B}}(H)\), the map \(P_v:{\mathscr {B}}(H)\rightarrow X\) defined by
is easily seen to be a continuous mhomogeneous polynomial that is orthogonally additive on \({\mathscr {B}}(H)\). In particular, \(P_v\) is orthogonally additive on \({\mathscr {B}}(H)_{\mathrm{sa}}\), and Corollary 1 then gives a unique continuous linear map \(\varPhi _v:{\mathscr {B}}(H)\rightarrow X\) such that
We claim that, if \(e,e'\in {\mathscr {B}}(H)\) are equivalent projections with \(e\perp e'\), then \(P(e)=P(e')=0\). Let \(u\in {\mathscr {B}}(H)\) be a partial isometry such that \(u^*u=e\) and \(uu^*=e'\). Then
which gives \(u^2=0\). From this we see that \(u\perp u^*\), and therefore
We now take \(\omega \in {\mathbb {C}}\) with \(\omega ^m=1\), and define
It is immediately seen that both v and \(v_\omega \) are unitary, and so applying (7) (and using the orthogonal additivity of P and that \(e\perp e'\)), we see that
By comparing both identities, we conclude that \(P(e)=P(e')=0\), as claimed.
Our next objective is to prove that \(P(e)=0\) for each projection \(e\in {\mathscr {B}}(H)\). Suppose that \(e\in {\mathscr {B}}(H)\) is a rankone projection. Since \(\dim H\ge 2\), it follows that there exists an equivalent projection \(e'\) such that \(e'\perp e\). Then it follows from the above claim that \(P(e)=0\). Let \(e\in {\mathscr {B}}(H)\) be a finite projection. Then there exist mutually orthogonal projections \(e_1,\ldots ,e_n\) such that \(e_1+\cdots +e_n=e\). Using the preceding observation and the orthogonal additivity of P we get \(P(e)=P(e_1)+\cdots +P(e_n)=0\). We now assume that \(e\in {\mathscr {B}}(H)\) is an infinite projection. Then there exist mutually orthogonal, equivalent projections \(e_1\) and \(e_2\) such that \(e_1+e_2=e\). By the claim, we have \(P(e)=P(e_1)+P(e_2)=0\).
We finally proceed to show that \(P=0\). By Lemma 1, it suffices to show that \(P(x)=0\) for each \(x\in {\mathscr {B}}(H)_+\). Suppose that \(x\in {\mathscr {B}}(H)_+\) can be written in the form \(x=\sum _{j=1}^k\rho _j e_j\), where \(e_1,\ldots ,e_k\in {\mathscr {B}}(H)\) are mutually orthogonal projections and \(\rho _1,\ldots ,\rho _k\in {\mathbb {R}}^+\). Then we have \(P(x)=\sum _{j=1}^k{\rho _j}^mP(e_j)=0\). Now let \(x\in {\mathscr {B}}(H)_{+}\) be an arbitrary element. From the spectral decomposition we deduce that there exists a sequence \((x_n)\) in \({\mathscr {B}}(H)_{+}\) such that each \(x_n\) is a positive linear combination of mutually orthogonal projections and \(\lim x_n=x\). On account of the preceding observation, \(P(x_n)=0\) (\(n\in {\mathbb {N}}\)), and the continuity of P implies that \(P(x)=\lim P(x_n)=0\), as required. \(\square \)
Noncommutative \(L^p\)spaces
Before giving the next results we make the following preliminary remarks.
A fundamental fact for us is the behaviour of the product of \(L^0({{\mathscr {M}}},\tau )\) when restricted to the \(L^p\)spaces. Specifically, if \(0<p,q,r\le \infty \) are such that \(\tfrac{1}{p}+\tfrac{1}{q}=\tfrac{1}{r}\), then the Hölder inequality states that
Suppose that \(x,y\in L^p({{\mathscr {M}}},\tau )_+\), \(0<p<\infty \), are mutually orthogonal and that \(\omega \in {\mathbb {C}}\) with \(\vert \omega \vert =1\). Then it is immediately seen that \(\vert x+\omega y\vert =x+y\), and it follows, by considering the spectral resolutions of x, y, and \(x+y\), that \((x+y)^p=x^p+y^p\). Hence
Each \(x\in L^{p}({{\mathscr {M}}},\tau )\) can be written in the form
Indeed, first we write \(x=\mathfrak {R}x+i\mathfrak {I}x\), where
and, since \(\Vert x^*\Vert _{p}=\Vert x\Vert _{p}\), it follows that \(\Vert \mathfrak {R}x\Vert _{p},\Vert \mathfrak {I}x\Vert _{p}\le \Vert x\Vert _{p}\). Further, we take the positive operators
Then \(x_1,x_2,x_3,x_4\in L^p({{\mathscr {M}}},\tau )\), \(\mathfrak {R}x=x_1x_2\) with \(x_1\perp x_2\), so that (9) gives
and \(\mathfrak {I}x= x_3x_4\) with \(x_3\perp x_4\), so that (9) gives
Theorem 4
Let \({{\mathscr {M}}}\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau \), let X be a topological linear space, and let \(\varPhi :L^{p/m}({{\mathscr {M}}},\tau )\rightarrow X\) be a continuous linear map with \(0<p<\infty \). Then:

(i)
the map \(P_\varPhi :L^p({{\mathscr {M}}},\tau )\rightarrow X\) defined by \(P_\varPhi (x)=\varPhi (x^m)\)\((x\in L^p({{\mathscr {M}}},\tau ))\) is a continuous mhomogeneous polynomial which is orthogonally additive on \(L^p({{\mathscr {M}}},\tau )_{\mathrm{sa}}\);

(ii)
the polynomial \(P_\varPhi \) is uniquely specified by the map \(\varPhi \).
Suppose, further, that X is a qnormed space, \(0<q\le 1\). Then:

(iii)
\(2^{1/q}\Vert \varPhi \Vert \le \Vert P_\varPhi \Vert \le \Vert \varPhi \Vert \).
Moreover, in the case where \(X={\mathbb {C}}\),

(iv)
the functional \(\varPhi \) is hermitian if and only if the polynomial \(P_\varPhi \) is hermitian, in which case \(\Vert P_\varPhi \Vert =\Vert \varPhi \Vert \).
Proof
The proof of this result is similar to that establishing Theorem 1.
(i) It follows immediately from (8) that, for each \(x_1,\ldots ,x_m\in L^p({{\mathscr {M}}},\tau )\),
On the one hand, this clearly implies that the map \(P_\varPhi \) is welldefined, on the other hand, the map \(x\mapsto x^m\) from \(L^p({{\mathscr {M}}},\tau )\) into \(L^{p/m}({{\mathscr {M}}},\tau )\) is continuous, and so \(P_\varPhi \) is continuous. Further, \(P_\varPhi \) is the mhomogeneous polynomial associated with the symmetric mlinear map \(\varphi :L^p({{\mathscr {M}}},\tau )^m\rightarrow X\) defined by
Suppose that \(x,y\in L^p({{\mathscr {M}}},\tau )_{\mathrm{sa}}\) are such that \(x\perp y\). Then \(xy=yx=0\), and so \((x+y)^m=x^m+y^m\), which gives
(ii) Suppose that \(\Psi :L^{p/m}({{\mathscr {M}}},\tau )\rightarrow X\) is a linear map such that \(P_\Psi =P_\varPhi \). For each \(x\in L^{p/m}({{\mathscr {M}}},\tau )_+\), we have \(x^{1/m}\in L^{p}({{\mathscr {M}}},\tau )\) and
By linearity we obtain \(\varPhi =\Psi \).
(iii) Next, assume that X is a qnormed space. For each \(x\in L^p({{\mathscr {M}}},\tau )\), by (11), we have
which clearly implies that \(\Vert P_\varPhi \Vert \le \Vert \varPhi \Vert \). Now take \(x\in L^{p/m}({{\mathscr {M}}},\tau )\), and take \(\omega \in {\mathbb {C}}\) with \(\omega ^m=1\). Write
as in (10) (with p / m instead of p). Since \(x_1\perp x_2\) and \(x_3\perp x_4\), it follows that \(x_1^{1/m}\perp x_2^{1/m}\) and \(x_3^{1/m}\perp x_4^{1/m}\), so that (9) gives
and
Further, we have \(x_1^{1/m},x_2^{1/m},x_3^{1/m},x_4^{1/m}\in L^p({{\mathscr {M}}},\tau )\) and
On the other hand, we have
whence
Hence, by (14),
This clearly forces \(\Vert \varPhi \Vert \le 2^{1/q}\Vert P_\varPhi \Vert \), as claimed.
(iv) It is straightforward to check that \(P_\varPhi ^*=P_{\varPhi ^*}\). From this deduce that \(\varPhi \) is hermitian if and only if \(P_\varPhi \) is hermitian as in the proof of Theorem 1(iv). Suppose that \(\varPhi \) is a hermitian functional. By direct calculation, we see that \(P_\varPhi \) is hermitian, and it remains to prove that \(\Vert P_\varPhi \Vert =\Vert \varPhi \Vert \). We only need to show that \(\Vert \varPhi \Vert \le \Vert P_\varPhi \Vert \). To this end, let \(\varepsilon \in {\mathbb {R}}^+\), and choose \(x\in L^{p/m}({{\mathscr {M}}},\tau )\) such that \(\Vert x\Vert _{p/m}=1\) and \(\Vert \varPhi \Vert \varepsilon <\vert \varPhi (x)\vert \). We take \(\alpha \in {\mathbb {C}}\) with \(\vert \alpha \vert =1\) and \(\vert \varPhi (x)\vert =\alpha \varPhi (x)\), so that
We see that \(\mathfrak {R}(\alpha x)\in L^{p/m}({{\mathscr {M}}},\tau )_{\mathrm{sa}}\), \(\Vert \mathfrak {R}(\alpha x)\Vert _{p/m}\le 1\), and \(\Vert \varPhi \Vert \varepsilon <\varPhi (\mathfrak {R}(\alpha x))\). Now we consider the decomposition \(\mathfrak {R}(\alpha x)=x_1x_2\) as in (10) (with p / m instead of p), and take \(\omega \in {\mathbb {C}}\) with \(\omega ^m=1\). As in (14), we see that \(\big \Vert x_1^{1/m}+\omega x_2^{1/m}\big \Vert =\Vert \mathfrak {R}(\alpha x)\Vert ^{1/m}\le 1\). Moreover, we have
and so \(\Vert \varPhi \Vert \varepsilon <\Vert P_\varPhi \Vert \). \(\square \)
Theorem 5
Let \({{\mathscr {M}}}\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau \), let X be a topological linear space, and let \(P:L^p({{\mathscr {M}}},\tau )\rightarrow X\) be a continuous mhomogeneous polynomial with \(0<p<\infty \). Then the following conditions are equivalent:

(i)
there exists a continuous linear map \(\varPhi :L^{p/m}({{\mathscr {M}}},\tau )\rightarrow X\) such that \(P(x)=\varPhi (x^m)\)\((x\in L^p({{\mathscr {M}}},\tau ))\);

(ii)
the polynomial P is orthogonally additive on \(L^p({{\mathscr {M}}},\tau )_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \(S({{\mathscr {M}}},\tau )_+\).
If the conditions are satisfied, then the map \(\varPhi \) is unique.
Proof
Theorem 4 shows that (i)\(\Rightarrow \)(ii), and it is obvious that (ii)\(\Rightarrow \)(iii). We proceed to prove that (iii)\(\Rightarrow \)(i).
Suppose that (iii) holds. Let \(e\in {\mathscr {M}}\) be a projection such that \(\tau (e)<\infty \), and consider the von Neumann algebra \({{\mathscr {M}}}_e=e{{\mathscr {M}}}e\). We claim that \({\mathscr {M}}_e\subset S({{\mathscr {M}}},\tau )\) and that there exists a unique continuous linear map \(\varPhi _e:{\mathscr {M}}_e\rightarrow X\) such that
Set \(x\in {\mathscr {M}}_e\), and write \(x=(x_1x_2)+i(x_3x_4)\) with \(x_1,x_2,x_3,x_4\in {\mathscr {M}}_{e \, +}\). Then \({{\,\mathrm{supp}\,}}(x_j)\le e\) and therefore \(\tau \bigl ({{\,\mathrm{supp}\,}}(x_j)\bigr )\le \tau (e)<\infty \)\((j\in \{1,2,3,4\})\). This shows that \(x_j\in S({\mathscr {M}},\tau )\)\((j\in \{1,2,3,4\})\), whence \(x\in S({\mathscr {M}},\tau )\). Our next goal is to show that the restriction \(P\mid _{{\mathscr {M}}_e}\) is continuous (with respect to the norm that \({\mathscr {M}}_e\) inherits as a closed subspace of \({\mathscr {M}}\)). Let \(x\in {\mathscr {M}}_e\), and let \(U\subset X\) be a neighbourhood of P(x). Since P is continuous, the set \(P^{1}(U)\) is a neighbourhood of x in \(L^p({\mathscr {M}},\tau )\), which implies that there exists \(r\in {\mathbb {R}}^+\) such that \(P(y)\in U\) whenever \(y\in L^p({\mathscr {M}},\tau )\) and \(\Vert y x\Vert _p<r\). If \(y\in {\mathscr {M}}_e\) is such that \(\Vert y x\Vert <r/\Vert e\Vert _p\), then, from (8), we obtain
and therefore \(P(y)\in U\). Hence \(P\mid _{{\mathscr {M}}_e}\) is continuous. Since, by hypothesis, the polynomial \(P\mid _{{\mathscr {M}}_e}\) is orthogonally additive on \({\mathscr {M}}_{e \, +}\), Corollary 1 states that there exists a unique continuous linear map \(\varPhi _e:{\mathscr {M}}_e\rightarrow X\) such that (15) holds.
For each \(x\in S({\mathscr {M}},\tau )\), define
where \(e\in {\mathscr {M}}\) is any projection such that
We will show that \(\varPhi \) is welldefined. For this purpose we first check that, if \(x\in S({\mathscr {M}},\tau )\), then there exists a projection e such that (16) holds. Indeed, we write \(x=\sum _{j=1}^k\alpha _j x_j\) with \(\alpha _1,\ldots ,\alpha _k\in {\mathbb {C}}\) and \(x_1,\ldots ,x_k\in S({\mathscr {M}},\tau )_+\), and define \(e={{\,\mathrm{supp}\,}}(x_1)\vee \cdots \vee {{\,\mathrm{supp}\,}}(x_k)\). Then \(ex=xe=x\) and \(\tau (e)\le \sum _{j=1}^k\tau \bigl ({{\,\mathrm{supp}\,}}(x_j)\bigr )<\infty \), as required. Suppose that \(x\in S({\mathscr {M}},\tau )\) and that \(e_1,e_2\in {\mathscr {M}}\) are projections satisfying (16). Then the projection \(e=e_1\vee e_2\) satisfies (16) and \({\mathscr {M}}_{e_1},{\mathscr {M}}_{e_2}\subset {\mathscr {M}}_e\). The uniqueness of the representation (15) on both \({\mathscr {M}}_{e_1}\) and \({\mathscr {M}}_{e_2}\) gives \(\varPhi _{e}\mid _{{\mathscr {M}}_{e_1}}=\varPhi _{e_1}\) and \(\varPhi _{e}\mid _{{\mathscr {M}}_{e_2}}=\varPhi _{e_2}\), which implies that \(\varPhi _{e_1}(x)=\varPhi _{e}(x)=\varPhi _{e_2}(x)\).
We now show that \(\varPhi \) is linear. Take \(x_1,x_2\in S({\mathscr {M}},\tau )\) and \(\alpha ,\beta \in {\mathbb {C}}\). Let \(e_1,e_2\in {\mathscr {M}}\) be projections such that \(e_jx_j=x_je_j=x_j\) and \(\tau (e_j)<\infty \)\((j\in \{1,2\})\). Then the projection \(e=e_1\vee e_2\) satisfies
and
Thus
and
We see from the definition of \(\varPhi \) that
Our next concern will be the continuity of \(\varPhi \) with respect to the norm \(\Vert \cdot \Vert _{p/m}\). Let U be a neighbourhood of 0 in X. Let V be a balanced neighbourhood of 0 in X with \(V+V+V+V\subset U\). The set \(P^{1}(V)\) is a neighbourhood of 0 in \(L^p({\mathscr {M}},\tau )\), which implies that there exists \(r\in {\mathbb {R}}^+\) such that \(P(x)\in V\) whenever \(x\in L^p({\mathscr {M}},\tau )\) and \(\Vert x\Vert _p<r\). Take \(x\in S({\mathscr {M}},\tau )\) with \(\Vert x\Vert _{p/m}<r^m\), and write \(x=(x_1x_2)+i(x_3x_4)\) as in (10) (with p / m instead of p). Then it is immediate to check that actually \(x_1,x_2,x_3,x_4\in S({\mathscr {M}},\tau )_+\) and, further, \(\Vert x_j\Vert _{p/m}\le \Vert x\Vert _{p/m}\)\((j\in \{1,2,3,4\})\). For each \(j\in \{1,2,3,4\}\), we have
whence
which establishes the continuity of \(\varPhi \). Since \(S({\mathscr {M}},\tau )\) is dense in \(L^{p/m}({\mathscr {M}},\tau )\), the map \(\varPhi \) extends uniquely to a continuous linear map from \(L^{p/m}({\mathscr {M}},\tau )\) into the completion of X. By abuse of notation we continue to write \(\varPhi \) for this extension. Since both P and \(\varPhi \) are continuous, (17) gives \(P(x)=\varPhi (x^m)\) for each \(x\in L^p({\mathscr {M}})\). The task is now to show that the image of \(\varPhi \) is actually contained in X. Of course, it suffices to show that \(\varPhi \) takes \(L^{p/m}({\mathscr {M}},\tau )_+\) into X. Let \(x\in L^{p/m}({\mathscr {M}},\tau )_+\). Then \(x^{1/m}\in L^{p}({\mathscr {M}},\tau )_+\) and
as required.
The uniqueness of the map \(\varPhi \) is given by Theorem 4(ii). \(\square \)
Let us note that the space of all continuous mhomogeneous polynomials from \(L^p({\mathscr {M}},\tau )\) into any topological linear space X which are orthogonally additive on \(S({\mathscr {M}},\tau )_+\) is sufficiently rich in the case where \(p/m\ge 1\), because of the existence of continuous linear functionals on \(L^{p/m}({\mathscr {M}},\tau )\). However, some restriction on the space X must be imposed when we consider the case \(p/m<1\) and the von Neumann algebra \({\mathscr {M}}\) has no minimal projections, because in this case the dual of \(L^{p/m}({\mathscr {M}},\tau )\) is trivial ([15]). In fact, there are no nonzero continuous linear maps from \(L^p({\mathscr {M}},\tau )\) into any qnormed space X with \(q>p\). We think that this property is probably wellknown, but we have not been able to find any reference, so that we next present a proof of this result for completeness.
Proposition 2
Let \({\mathscr {M}}\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau \) and with no minimal projections, let X be a qnormed space, \(0<q\le 1\), and let \(\varPhi :L^p({\mathscr {M}},\tau )\rightarrow X\) be a continuous linear map with \(0<p<q\). Then \(\varPhi =0\).
Proof
The proof will be divided in a number of steps.
Our first step is to show that for each projection \(e_0\in {\mathscr {M}}\) with \(\tau (e_0)<\infty \) and each \(0\le \rho \le \tau (e_0)\), there exists a projection \(e\in {\mathscr {M}}\) such that \(e\le e_0\) and \(\tau (e)=\rho \). Set
Note that \(e_0\in {\mathscr {P}}_1\), so that \({\mathscr {P}}_1\) is nonempty. Let \({\mathscr {C}}\) be a chain in \({\mathscr {P}}_1\), and let \(e'=\wedge _{e\in {\mathscr {C}}}e\). Then \(e'\) is a projection and \(e'\le e_0\). For each \(e\in {\mathscr {C}}\), since \(\tau (e_0)<\infty \), it follows that \(\tau (e_0)\tau (e)=\tau (e_0e)\). From the normality of \(\tau \) we now deduce that
Hence \(\tau (e')=\inf _{e\in {\mathscr {C}}}\tau (e)\ge \rho \), which shows that \(e'\) is a lower bound of \({\mathscr {C}}\), and so, by Zorn’s lemma, \({\mathscr {P}}_1\) has a minimal element, say \(e_1\). We now consider the set
Note that \(0\in {\mathscr {P}}_2\), so that \({\mathscr {P}}_2\) is nonempty. Let \({\mathscr {C}}\) be a chain in \({\mathscr {P}}_2\), and let \(e'=\vee _{e\in {\mathscr {C}}}e\). Then \(e'\le e_1\), and the normality of \(\tau \) yields
This implies that \(e'\) is an upper bound of \({\mathscr {C}}\), and so, by Zorn’s lemma, \({\mathscr {P}}_2\) has a maximal element, say \(e_2\). Assume towards a contradiction that \(e_1\ne e_2\). Since, by hypothesis, \({\mathscr {M}}\) has no minimal projections, it follows that there exists a nonzero projection \(e<e_1e_2\). Since \(e\perp e_2\), we see that \(e_2+e\) is a projection. Further, we have \(e_2<e_2+e<e_1\). The maximality of \(e_2\) implies that \(\tau (e_2+e)>\rho \), which implies that \(e_2+e\in {\mathscr {P}}_1\), contradicting the minimality of \(e_1\). Thus \(e_1=e_2\), and this clearly implies that \(\tau (e_1)=\tau (e_2)=\rho \).
Our next goal is to show that \(\varPhi (e_0)=0\) for each projection \(e_0\) with \(\tau (e_0)<\infty \). From the previous step, it follows that there exists a projection \(e\le e_0\) with \(\tau (e)=\frac{1}{2}\tau (e_0)\). Set \(e'=e_0e\). Then \(\tau (e')=\frac{1}{2}\tau (e_0)\). Further,
and therefore either \(\Vert \varPhi (e)\Vert ^q\ge \frac{1}{2}\Vert \varPhi (e_0)\Vert ^q\) or \(\Vert \varPhi (e')\Vert ^q\ge \frac{1}{2}\Vert \varPhi (e_0)\Vert ^q\). We define \(e_1\) to be any of the projections \(e,e'\) for which the inequality holds. We thus get \(e_1\le e_0\), \(\tau (e_1)=\frac{1}{2}\tau (e_0)\), and \(\Vert \varPhi (e_1)\Vert \ge 2^{1/q}\Vert \varPhi (e_0)\Vert \). By repeating the process, we get a decreasing sequence of projections \((e_n)\) such that
Then
which converges to zero, because \(p<q\). Since \(\varPhi \) is continuous and \(\Vert \varPhi (e_0)\Vert \le \big \Vert \varPhi \bigl (2^{n/q}e_n\bigr )\big \Vert _p\)\((n\in {\mathbb {N}})\), it may be concluded that \(\varPhi (e_0)=0\), as claimed.
Our next concern is to show that \(\varPhi \) vanishes on \(S({\mathscr {M}},\tau )\). Of course, it suffices to show that \(\varPhi \) vanishes on \(S({\mathscr {M}},\tau )_+\). Take \(x\in S({\mathscr {M}},\tau )_+\), and let \(e={{\,\mathrm{supp}\,}}(x)\), so that \(\tau (e)<\infty \). The spectral decomposition implies that there exists a sequence \((x_n)\) in \({\mathscr {M}}_+\) such that \(\lim x_n=x\) with respect to the operator norm and each \(x_n\) is of the form \(x_n=\sum _{j=1}^k\rho _j e_j\), where \(\rho _1,\ldots ,\rho _k\in {\mathbb {R}}^+\) and \(e_1,\ldots ,e_k\in {\mathscr {M}}\) are mutually orthogonal projections with \(e_je=ee_j=e_j\)\((j\in \{1,\ldots ,k\})\). From the previous step, we conclude that \(\varPhi (x_n)=0\)\((n\in {\mathbb {N}})\). Further, from (8) we deduce that
and the continuity of \(\varPhi \) implies that \(\varPhi (x)=0\), as required.
Finally, since \(S({\mathscr {M}},\tau )\) is dense in \(L^p({\mathscr {M}},\tau )\) and \(\varPhi \) is continuous, it may be concluded that \(\varPhi =0\). \(\square \)
Corollary 2
Let \({\mathscr {M}}\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau \) and with no minimal projections, let X be a qnormed space, \(0<q\le 1\), and let \(P:L^p({\mathscr {M}},\tau )\rightarrow X\) be a continuous mhomogeneous polynomial with \(0<p/m<q\). Suppose that P is orthogonally additive on \(S({\mathscr {M}},\tau )_+\). Then \(P=0\).
Proof
This is a straightforward consequence of Theorem 5 and Proposition 2. \(\square \)
We now turn our attention to the complexvalued polynomials. In this setting the representation given in Theorem 5 has a particularly significant integral form, because of the wellknown representation of the dual of the \(L^p\)spaces. The trace gives rise to a distinguished contractive positive linear functional on \(L^1({\mathscr {M}},\tau )\), still denoted by \(\tau \). By (8), if \(\frac{1}{p}+\frac{1}{q}=1\), for each \(\zeta \in L^q({\mathscr {M}},\tau )\), the formula
defines a continuous linear functional on \(L^p({\mathscr {M}},\tau )\). Further, in the case where \(1\le p<\infty \), the map \(\zeta \mapsto \varPhi _\zeta \) is an isometric isomorphism from \(L^q({\mathscr {M}},\tau )\) onto the dual space of \(L^p({\mathscr {M}},\tau )\). It is immediate to see that \(\varPhi _\zeta ^*=\varPhi _{\zeta ^*}\), so that \(\varPhi _\zeta \) is hermitian if and only if \(\zeta \) is selfadjoint.
Corollary 3
Let \({\mathscr {M}}\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau \), and let \(P:L^p({\mathscr {M}},\tau )\rightarrow {\mathbb {C}}\) be a continuous mhomogeneous polynomial with \(m\le p<\infty \). Then the following conditions are equivalent:

(i)
there exists \(\zeta \in L^{r}({\mathscr {M}},\tau )\) such that \(P(x)=\tau (\zeta x^m)\)\((x\in L^p({\mathscr {M}},\tau ))\), where \(r=p/(pm)\) (with the convention that \(p/0=\infty \));

(ii)
the polynomial P is orthogonally additive on \(L^p({\mathscr {M}},\tau )_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \(S({\mathscr {M}},\tau )_+\).
If the conditions are satisfied, then \(\zeta \) is unique and \(\Vert P\Vert \le \Vert \zeta \Vert _{r}\le 2\Vert P\Vert \); moreover, if P is hermitian, then \(\zeta \) is selfadjoint and \(\Vert \zeta \Vert _{r}=\Vert P\Vert \).
Proof
This follows from Theorems 4 and 5. \(\square \)
Let H be a Hilbert space. We denote by \({{\,\mathrm{Tr}\,}}\) the usual trace on the von Neumann algebra \({\mathscr {B}}(H)\). Then \(L^p({\mathscr {B}}(H),{{\,\mathrm{Tr}\,}})\), with \(0<p<\infty \), is the Schatten class \(S^p(H)\). In the case where \(0<p<q\), we have \(S^p(H)\subset S^q(H)\subset {\mathscr {K}}(H)\) and \(\Vert x\Vert \le \Vert x\Vert _q\le \Vert x\Vert _p\)\((x\in S^p(H))\). It is clear that \(S({\mathscr {B}}(H),{{\,\mathrm{Tr}\,}})={\mathscr {F}}(H)\), the twosided ideal of \({\mathscr {B}}(H)\) consisting of the finiterank operators. Thus, the following result is an immediate consequence of Corollary 3.
Corollary 4
Let H be a Hilbert space, and let \(P:S^p(H)\rightarrow {\mathbb {C}}\) be a continuous mhomogeneous polynomial with \(m< p<\infty \). Then the following conditions are equivalent:

(i)
there exists \(\zeta \in S^{r}(H)\) such that \(P(x)={{\,\mathrm{Tr}\,}}(\zeta x^m)\)\((x\in S^p(H))\), where \(r=p/(pm)\);

(ii)
the polynomial P is orthogonally additive on \(S^p(H)_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \({\mathscr {F}}(H)_+\).
If the conditions are satisfied, then \(\zeta \) is unique and \(\Vert P\Vert \le \Vert \zeta \Vert _{r}\le 2\Vert P\Vert \); moreover, if P is hermitian, then \(\zeta \) is selfadjoint and \(\Vert \zeta \Vert _{r}=\Vert P\Vert \).
Corollary 5
Let H be a Hilbert space, and let \(P:{\mathscr {K}}(H)\rightarrow {\mathbb {C}}\) be a continuous mhomogeneous polynomial. Then the following conditions are equivalent:

(i)
there exists \(\zeta \in S^1(H)\) such that \(P(x)={{\,\mathrm{Tr}\,}}(\zeta x^m)\)\((x\in {\mathscr {K}}(H))\);

(ii)
the polynomial P is orthogonally additive on \({\mathscr {K}}(H)_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \({\mathscr {F}}(H)_+\).
If the conditions are satisfied, then \(\zeta \) is unique and \(\Vert P\Vert \le \Vert \zeta \Vert _{1}\le 2\Vert P\Vert \); moreover, if P is hermitian, then \(\zeta \) is selfadjoint and \(\Vert \zeta \Vert _{1}=\Vert P\Vert \).
Proof
In order to prove the equivalence of the conditions we are reduced to prove that (iii)\(\Rightarrow \)(i). Suppose that (iii) holds. Let \(x,y\in {\mathscr {K}}(H)_+\) such that \(x\perp y\). From the spectral decomposition of both x and y we deduce that there exist sequences \((x_n)\) and \((y_n)\) in \({\mathscr {F}}(H)_+\) such that \(\lim x_n=x\), \(\lim y_n=y\), and \(x_m\perp y_n\)\((m,n\in {\mathbb {N}})\). Then
This shows that P is orthogonally additive on \({\mathscr {K}}(H)_+\). Since the \(C^*\)algebra \({\mathscr {K}}(H)\) has real rank zero and the net consisting of all finiterank projections is an increasing approximate unit, Theorem 3 applies and gives a continuous linear functional \(\varPhi \) on \({\mathscr {K}}(H)\) such that \(P(x)=\varPhi (x^m)\)\((x\in {\mathscr {K}}(H))\). It is wellknown that the map \(\zeta \mapsto \varPhi _\zeta \), as defined in (18), gives an isometric isomorphism from \(S^1(H)\) onto the dual of \({\mathscr {K}}(H)\), so that there exists \(\zeta \in S^1(H)\) such that \(\varPhi (x)={{\,\mathrm{Tr}\,}}(\zeta x)\)\((x\in {\mathscr {K}}(H))\) and \(\Vert \zeta \Vert _1=\Vert \varPhi \Vert \). Thus we obtain (i). The additional properties of the result follow from Theorem 1. \(\square \)
Corollary 6
Let H be a Hilbert space, and let \(P:S^p(H)\rightarrow {\mathbb {C}}\) be a continuous mhomogeneous polynomial with \(0<p\le m\). Then the following conditions are equivalent:

(i)
there exists \(\zeta \in {\mathscr {B}}(H)\) such that \(P(x)={{\,\mathrm{Tr}\,}}(\zeta x^m)\)\((x\in S^p(H))\);

(ii)
the polynomial P is orthogonally additive on \(S^p(H)_{\mathrm{sa}}\);

(iii)
the polynomial P is orthogonally additive on \({\mathscr {F}}(H)_+\).
If the conditions are satisfied, then \(\zeta \) is unique and \(\Vert P\Vert \le \Vert \zeta \Vert \le 2\Vert P\Vert \); moreover, if P is hermitian, then \(\zeta \) is selfadjoint and \(\Vert \zeta \Vert =\Vert P\Vert \).
Proof
By Theorems 4 and 5, it suffices to show that the map \(\zeta \mapsto \varPhi _\zeta \), as defined in (18), gives isometric isomorphism from \({\mathscr {B}}(H)\) onto the dual of \(S^{p/m}(H)\). This is probably wellknown, but we are not aware of any reference. Consequently, it may be helpful to include a proof of this fact. If \(\zeta \in {\mathscr {B}}(H)\) and \(x\in S^{p/m}(H)\), then, by (8), \(\zeta x\in S^{p/m}(H)\), so that \(\zeta x\in S^1(H)\) and
which shows that \(\varPhi _\zeta \) is a continuous linear functional on \(S^{p/m}(H)\) with \(\Vert \varPhi _\zeta \Vert \le \Vert \zeta \Vert \). Conversely, assume that \(\varPhi \) is a continuous linear functional on \(S^{p/m}(H)\). For each \(\xi ,\eta \in H\), let \(\xi \otimes \eta \in {\mathscr {F}}(H)\) defined by
and define \(\varphi :H\times H\rightarrow {\mathbb {C}}\) by
It is easily checked that \(\varphi \) is a continuous sesquilinear functional with \(\Vert \varphi \Vert \le \Vert \varPhi \Vert \). Therefore there exists \(\zeta \in {\mathscr {B}}(H)\) such that \(\langle \zeta (\xi )\vert \eta \rangle =\varphi (\xi ,\eta )\) for all \(\xi ,\eta \in H\) and \(\Vert \zeta \Vert \le \Vert \varPhi \Vert \). The former condition implies that
for all \(\xi ,\eta \in H\), which gives \(\varPhi _\zeta (x)=\varPhi (x)\) for each \(x\in {\mathscr {F}}(H)\). Since \({\mathscr {F}}(H)\) is dense in \(S^{p/m}(H)\), it follows that \(\varPhi _\zeta =\varPhi \). Further, we have \(\Vert \zeta \Vert \le \Vert \varPhi \Vert =\Vert \varPhi _\zeta \Vert \le \Vert \zeta \Vert \). Finally, it is immediate to see that \(\varPhi _\zeta ^*=\varPhi _{\zeta ^*}\), so that \(\varPhi _\zeta \) is hermitian if and only if \(\zeta \) is selfadjoint. \(\square \)
Proposition 3
Let H be a Hilbert space with \(\dim H \ge 2\), let X be a topological linear space, and let \(P:S^p(H) \rightarrow X\) be a continuous mhomogeneous polynomial with \(0<p<\infty \). Suppose that P is orthogonally additive on \(S^p(H)\). Then \(P=0\).
Proof
Since \({\mathscr {F}}(H)\) is dense in \(S^p(H)\) and P is continuous, it suffices to prove that P vanishes on \({\mathscr {F}}(H)\). On account of Lemma 1, we are also reduced to prove that P vanishes on \({\mathscr {F}}(H)_{\mathrm{sa}}\). We continue to use the notation \(\xi \otimes \eta \) which was introduced in the proof of Corollary 6.
Let \(x\in {\mathscr {F}}(H)_{\mathrm{sa}}\). Then \(x=\sum _{j=1}^k\alpha _j\xi _j\otimes \xi _j\), where \(k\ge 2\), \(\alpha _1,\ldots ,\alpha _k\in {\mathbb {R}}\), and \(\{\xi _1,\ldots ,\xi _k\}\) is an orthonormal subset of H. It is clear that the subalgebra \({\mathscr {M}}\) of \({{\mathscr {B}}}(H)\) generated by \(\bigl \{\xi _i\otimes \xi _j : i,j\in \{1,\ldots ,k\}\bigr \}\) is contained in \({{\mathscr {F}}}(H)\) and it is \(*\)isomorphic to the von Neumann algebra \({{\mathscr {B}}}(K)\), where K is the linear span of the set \(\{\xi _1,\ldots ,\xi _k\}\). By Proposition 1, \(P\mid _{{\mathscr {M}}}=0\), and therefore \(P(x)=0\). We thus get \(P\mid _{{{\mathscr {F}}}(H)_{\mathrm{sa}}}=0\), as required. \(\square \)
References
 1.
Alaminos, J., Extremera, J., Villena, A.R.: Orthogonally additive polynomials on Fourier algebras. J. Math. Anal. Appl. 422, 72–83 (2015)
 2.
Alaminos, J., Extremera, J., Godoy, M.L.C., Villena, A.R.: Orthogonally additive polynomials on convolution algebras associated with a compact group. J. Math. Anal. Appl. 472, 285–302 (2019)
 3.
Alaminos, J., Godoy, M.L.C., Villena, A.R.: Orthogonally additive polynomials on the algebras of approximable operators. Linear Multilinear Algebra 67, 1922–1936 (2019). https://doi.org/10.1080/03081087.2018.1476445
 4.
Benyamini, Y., Lassalle, S., Llavona, J.G.: Homogeneous orthogonally additive polynomials on Banach lattices. Bull. Lond. Math. Soc. 38, 459–469 (2006)
 5.
Blackadar, B.: Operator algebras. Theory of \(C^*\)algebras and von Neumann algebras. In: Cuntz, J., Jones, V.F.R. (eds.) Encyclopaedia of Mathematical Sciences. Operator Algebras and Noncommutative Geometry. III, vol. 122. Springer, Berlin (2006)
 6.
Carando, D., Lassalle, S., Zalduendo, I.: Orthogonally additive polynomials over \(C(K)\) are measures: a short proof. Integral Equ. Oper. Theory 56, 597–602 (2006)
 7.
Davidson, K.R.: \(C^*\)Algebras by Example. Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996)
 8.
Dineen, S.: Complex Analysis on InfiniteDimensional Spaces. Springer Monographs in Mathematics. Springer, London (1999)
 9.
Ibort, A., Linares, P., Llavona, J.G.: A representation theorem for orthogonally additive polynomials on Riesz spaces. Rev. Mat. Complut. 25, 21–30 (2012)
 10.
Nelson, E.: Notes on noncommutative integration. J. Funct. Anal. 15, 103–116 (1974)
 11.
Palazuelos, C., Peralta, A.M., Villanueva, I.: Orthogonally additive polynomials on \(C^*\)algebras. Q. J. Math. 59, 363–374 (2008)
 12.
PérezGarcía, D., Villanueva, I.: Orthogonally additive polynomials on spaces of continuous functions. J. Math. Anal. Appl. 306, 97–105 (2005)
 13.
Pisier, G., Xu, Q.: Noncommutative \(L^p\)Spaces. Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. NorthHolland, Amsterdam (2003)
 14.
Raynaud, Y., Xu, Q.: On subspaces of noncommutative \(L_p\)spaces. J. Funct. Anal. 203, 149–196 (2003)
 15.
Saito, K.S.: Noncommutative \(L^p\)spaces with \(0<p<1\). Math. Proc. Camb. Philos. Soc. 89, 405–411 (1981)
 16.
Sundaresan, K.: Geometry of spaces of homogeneous polynomials on Banach lattices. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 571–586. American Mathematical Society, Providence (1991)
 17.
Takesaki, M.: Theory of Operator Algebras. I. Springer, New York (1979)
 18.
Terp, M.: \(L^p\) spaces associated with von Neumann algebras. Notes, Mathematical Institute, Copenhagen University (1981)
 19.
Villena, A.R.: Orthogonally additive polynomials on Banach function algebras. J. Math. Anal. Appl. 448, 447–472 (2017)
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The authors were supported by MINECO Grant PGC2018–093794–B–I00. and Junta de Andalucía Grant FQM–185. The second named author was supported by Contrato Predoctoral FPU, Plan propio de Investigación y Transferencia 2018, University of Granada.
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Alaminos, J., Godoy, M.L.C. & Villena, A.R. Orthogonally additive polynomials on noncommutative \(L^p\)spaces. Rev Mat Complut 33, 835–858 (2020). https://doi.org/10.1007/s13163019003301
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Keywords
 Noncommutative \(L^p\)space
 Schatten classes
 Orthogonally additive polynomial
Mathematics Subject Classification
 46L10
 46L52
 47H60