Characterizations of Minimal Elements in a Non-commutative $L_p$-Space

Abstract

For $1\le p<\infty$, let $L_p({\mathcal {M}},\tau )$ be the non-commutative $L_p$-space associated with a von Neumann algebra ${\mathcal {M}}$, where ${\mathcal {M}}$ admits a normal semifinite faithful trace $\tau$. Using the trace $\tau$, Banach duality formula and Gâteaux derivative, this paper characterizes an element $a\in L_p({\mathcal {M}},\tau )$ such that

$$\begin{aligned} \Vert a\Vert _p=\inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}, \end{aligned}$$

where ${\mathcal {B}}_p$ is a closed linear subspace of $L_p({\mathcal {M}},\tau )$ and $\Vert \cdot \Vert _p$ is the norm on $L_p({\mathcal {M}},\tau )$. Such an a is called ${\mathcal {B}}_p$-minimal. In particular, minimal elements related to the finite-diagonal-block type closed linear subspaces

$$\begin{aligned} {\mathcal {B}}_p=\bigoplus \limits _{i=1}^{\infty } e_i {\mathcal {S}} e_i \end{aligned}$$

(converging with respect to $\Vert \cdot \Vert _p$) are considered, where $\{e_i\}_{i=1}^{\infty }$ is a sequence of mutually orthogonal and $\tau$-finite projections in a $\sigma$-finite von Neumann algebra ${\mathcal {M}}$, and ${\mathcal {S}}$ is the set of elements in ${\mathcal {M}}$ with $\tau$-finite supports.

1 Introduction

The non-commutative $L_p$-space theory was laid out in the 1950s by Segal [26] and Dixmier [12]. It is the intersection of operator theory and classical $L^p$-space theory, as well has been widely studied, extended and applied. In this paper, we explore the minimal elements in a non-commutative $L_p$-space, which are closely related to the orthogonality in metric geometry.

Suppose ${\mathcal {H}}$ is a Hilbert space and ${\mathcal {K}}$ is its closed linear subspace. For $h\in {\mathcal {H}}$, there exists a unique $k_0\in {\mathcal {K}}$ such that

$$\begin{aligned} \Vert h-k_0\Vert =\textrm{dist} (h,{\mathcal {K}})=\inf \{\Vert h-k\Vert : k\in {\mathcal {K}}\}, \end{aligned}$$

where $\Vert \cdot \Vert$ is the norm induced by the inner product of ${\mathcal {H}}$. Replacing $h-k_0$ with $h_0$, one has

$$\begin{aligned} \Vert h_0\Vert =\inf \{\Vert h_0+k\Vert : k\in {\mathcal {K}}\}. \end{aligned}$$

Such an $h_0$ is called ${\mathcal {K}}$-minimal [25, Definition 5.2].

In the absence of inner product, Birkhoff [6] and James [15] study the orthogonality in a normed linear space, firstly. Suppose ${\mathcal {X}}$ is a normed linear space over $\mathbb C$ and $x,y\in {\mathcal {X}}$, then x is said to be Birkhoff–James orthogonal to y if

$$\begin{aligned} \Vert x\Vert \le \Vert x+\lambda y\Vert \mathrm{\ for\ all\ \lambda \in \mathbb C}. \end{aligned}$$

Thereafter, with the help of Hahn–Banach Theorem, Lumer [19] and Giles [14] carry over the notion of inner product on a Hilbert space to the semi-inner-product on a normed linear space, put forward that x and y in a continuous semi-inner-product space are Birkhoff–James orthogonal if and only if their semi-inner-product is 0 [14, Theorem 2]. Let ${\mathcal {Y}}$ be a closed linear subspace of ${\mathcal {X}}$. Then $x_0\in {\mathcal {X}}$ is said to be ${\mathcal {Y}}$-minimal if it is Birkhoff–James orthogonal to each $y\in {\mathcal {Y}}$, or equivalently, if

$$\begin{aligned} \Vert x_0\Vert =\inf \{\Vert x_0+y\Vert : y\in {\mathcal {Y}} \}. \end{aligned}$$

The existence of minimal elements allows the description of minimal length curves (curves with minimal length joining fixed endpoints) of metric geometry in homogeneous spaces, and the characterization of minimal elements in various Banach spaces has attracted the attention of many scholars. For instance, [13] studies minimal elements and the corresponding minimal length curves of a homogeneous space ${\mathcal {P}}$ in a C*-algebra context. [3, 4, 18, 22, 32] are devoted to characterizing and constructing $D_n(\mathbb R)$-minimal hermitian matrices in $M_n(\mathbb C)$, in the sense of operator norm, where $M_n(\mathbb C)$ is the algebra of complex $n\times n$ matrices and $D_n(\mathbb R)$ is the algebra of real diagonal $n\times n$ matrices. For the study of minimal length curves in an infinite dimensional manifold, as well as the corresponding works on D(K(H))-minimal compact operators, one can refer to [2, 9, 10, 21, 31], where ${\mathcal {H}}$ is a complex separable Hilbert space with an orthonormal basis $\{\xi _i\}_{i=1}^\infty$, $K({\mathcal {H}})$ is the algebra of compact operators on ${\mathcal {H}}$, and

$$\begin{aligned} D(K({\mathcal {H}}))=\{D\in K({\mathcal {H}}):\langle D \xi _i,\xi _j\rangle =0 \mathrm{\ when\ }i\ne j\} \end{aligned}$$

is the set of diagonal compact operators. Moreover, [5, 28] study the best approximation and orthogonality in Hilbert C*-modules, which are closely related to minimal elements. Recently, with the help of semi-inner-product, minimal elements in p-Schatten ideals are explored in [7, 8]. With a view to the geometric property of orthogonality in a non-commutative $L_p$-space, illuminated by the idea of [7, 8], this paper devotes to characterizing an element $a\in L_p({\mathcal {M}},\tau )$ such that

$$\begin{aligned} \Vert a\Vert _p=\inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}, \end{aligned}$$

where $\Vert \cdot \Vert _p$ is the norm on $L_p({\mathcal {M}},\tau )$ and ${\mathcal {B}}_p$ is a closed linear subspace of $L_p({\mathcal {M}},\tau )$.

We briefly describe the contents of this paper. Section 2 lists some basic notions and prevalent results we will use throughout this paper. Section 3 provides the semi-inner-product on $L_p({\mathcal {M}},\tau )$ specifically and characterizes ${\mathcal {B}}_p$-minimal elements in terms of disjoint supports and the trace $\tau$. Section 4 describes ${\mathcal {B}}_p$-minimal elements through the Gâteaux derivative of norm $\Vert \cdot \Vert _p$ and the Banach duality formula, respectively. In Sect. 5, minimal elements related to the finite-diagonal-block type closed linear subspaces

$$\begin{aligned} {\mathcal {B}}_p=\bigoplus \limits _{i=1}^{\infty } e_i {\mathcal {S}} e_i \end{aligned}$$

(converging with respect to $\Vert \cdot \Vert _p$) of $L_p({\mathcal {M}},\tau )$ are taken into account, where $\{e_i\}_{i=1}^{\infty }$ is a sequence of mutually orthogonal and $\tau$-finite projections in a $\sigma$-finite von Neumann algebra ${\mathcal {M}}$, and ${\mathcal {S}}$ is the set of elements in ${\mathcal {M}}$ with $\tau$-finite supports.

2 Preliminaries

In this section we give some basic concepts and prevalent results on non-commutative $L_p$-spaces. One can refer to [23, Chapter 34] and [29] for more details.

• Denote by ${\mathcal {M}}$ a von Neumann algebra acting on a Hilbert space ${\mathcal {H}}$ and by ${\mathcal {M}}_+$ its positive part. A trace on ${\mathcal {M}}$ is a map $\tau :{\mathcal {M}}_+\rightarrow [0,\infty ]$ satisfying

  • (1) $\tau (x+\lambda y)=\tau (x)+\lambda \tau (y)$, for $x,y\in {\mathcal {M}}_+$ and $\lambda \in \mathbb R_+$;

  • (2) $\tau (x^*x)=\tau (xx^*)$, for $x\in {\mathcal {M}}$.

  Moreover, $\tau$ is said to be normal if $\sup _{i}\tau (x_i)=\tau (\sup _{i}x_i)$ for each bounded increasing net $\{x_i\}_{i\in \Lambda }$ in ${\mathcal {M}}_+$; to be semifinite if for any non-zero $x\in M_+$ there is a non-zero $y\in M_+$ such that $y\le x$ and $\tau (y)<\infty$; and to be faithful if $x\in {\mathcal {M}}_+$ with $\tau (x)=0$ implies that $x=0$. In the rest of this paper, the von Neumann algebra ${\mathcal {M}}$ always admits a normal semifinite faithful trace $\tau$. Denote by $P({\mathcal {M}})$ the set of projections in ${\mathcal {M}}$, namely, $e\in P({\mathcal {M}})$ if $e=e^2=e^*$. There always exists an increasing net $\{e_i\}_{i\in \Lambda }\subset P({\mathcal {M}})$ such that $\tau (e_i)<\infty$ for each $i\in \Lambda$ and $e_i\rightarrow I$ with respect to the strong operator topology, where I is the identity of ${\mathcal {M}}$.

• For $x\in {\mathcal {M}}$, let $x=u|x|$ be its polar decomposition, where u is a partial isometry from $(\ker x)^{\bot }$ onto $\overline{\textrm{ran}x}$ and $|x|=(x^*x)^{\frac{1}{2}}$ is the absolute value of x. Denote by $l(x)=uu^*$ and $r(x)=u^*u$ the left and right support for x, respectively. If $x\in {\mathcal {M}}_+$, then $l(x)=r(x)$ and we write the support as s(x). Set

  $$\begin{aligned} {\mathcal {S}}_+=\{x\in {\mathcal {M}}_+:\tau (s(x))<\infty \}, \end{aligned}$$

  and let ${\mathcal {S}}$ be the linear span of ${\mathcal {S}}_+$, namely, the set of elements in ${\mathcal {M}}$ with $\tau$-finite support. If $x\in {\mathcal {S}}$ and $0<p<\infty$, then $|x|^p\in {\mathcal {S}}$. Moreover, define $\Vert x\Vert _p=\tau (|x|^p)^{\frac{1}{p}}$, then $\Vert \cdot \Vert _p$ is a norm on ${\mathcal {S}}$ when $1\le p<\infty$ and is a quasi-norm on ${\mathcal {S}}$ when $0<p<1$. The completion of $({\mathcal {S}},\Vert \cdot \Vert _p)$, denoted by $L_p({\mathcal {M}},\tau )$, is called the non-commutative $L_p$-space associated with $({\mathcal {M}}, \tau )$. In this paper we focus on the case $1\le p<\infty$, for which $L_p({\mathcal {M}},\tau )$ forms a Banach space. For the sake of convenience, we set $L_{\infty }({\mathcal {M}},\tau )={\mathcal {M}}$ equipped with the operator norm.

• Let $1\le p<\infty$ and take $x\in L_p({\mathcal {M}},\tau )$. Then x is a closed densely defined operator on ${\mathcal {H}}$. More specifically, its domain D(x) is dense in ${\mathcal {H}}$ and its graph $G(x)=\{(\xi ,x\xi ):\xi \in D(x)\}$ is closed in ${\mathcal {H}}\oplus {\mathcal {H}}$. The adjoint $x^*$ of x is defined such that $\langle xf, g\rangle =\langle f, x^*g\rangle$ for all $f\in D(x)$ and $g\in D(x^*)$, where $D(x^*)=\{g\in {\mathcal {H}}: f\rightarrow \langle xf,g\rangle \mathrm{\ is\ continuous\ on\ } D(x)\}$. If $x=x^*$, then x is said to be self-adjoint. Similar to bounded linear operators, x has a unique polar decomposition $x=u|x|$, where u is a partial isometry from $(\ker x)^{\bot }$ onto $\overline{x(D(x))}$. In addition, the left and right supports for x can be defined. For more details on closed densely defined operators one can refer to [11, Chapter X] and [24, Chapter 13].

The following Lemma 2.1 is crucial to this paper.

Lemma 2.1

[23, 29] The following statements hold:

• (1) ${\mathcal {S}}$ is a strongly dense involutive ideal of ${\mathcal {M}}$. Moreover, for $x\in {\mathcal {M}}$, $x\in {\mathcal {S}}$ if and only if there is an $e\in P({\mathcal {M}})$ with $\tau (e)<\infty$ such that $exe=x$.

• (2) $|\tau (x)|\le \Vert x\Vert _1$ for $x\in {\mathcal {S}}$. Moreover, $\tau$ can be extended to a continuous linear functional on $L_1({\mathcal {M}},\tau )$.

• (3) For $x\in L_p({\mathcal {M}},\tau )$ and $a,b\in {\mathcal {M}}$,

  $$\begin{aligned} \Vert x\Vert _p=\Vert x^*\Vert _p=\Vert |x|\Vert _p,\ \Vert axb\Vert _p\le \Vert a\Vert \Vert x\Vert _p\Vert b\Vert . \end{aligned}$$

  (1)

• (4) (Hölder inequality) Suppose $1\le p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. Then

  $$\begin{aligned} |\tau (xy)|\le \Vert x\Vert _p\Vert y\Vert _q \end{aligned}$$

  for $x\in L_p({\mathcal {M}},\tau )$ and $y\in L_q({\mathcal {M}},\tau )$.

• (5) Let $\{a_i\}_{i\in \Lambda }$ be a bounded net in ${\mathcal {M}}$ such that $a_i\rightarrow a$ with respect to the strong operator topology, then $xa_i\rightarrow xa$ in $L_p({\mathcal {M}},\tau )$ for any $x\in L_p({\mathcal {M}},\tau )$.

3 Characterizations of ${\mathcal {B}}_p$-Minimal Elements

The aim of this section is to characterize ${\mathcal {B}}_p$-minimal elements in terms of disjoint supports and the normal semifinite faithful trace $\tau$.

Definition 3.1

Let $1\le p<\infty$ and ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$. We say that $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if

$$\begin{aligned} \Vert a\Vert _p=\inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}. \end{aligned}$$

Remark 3.2

With ${\mathcal {B}}_p$ as above, suppose $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal.

• (1) $\lambda a$ is ${\mathcal {B}}_p$-minimal for all $\lambda \in \mathbb C$, since

  $$\begin{aligned} \begin{array}{rl} \Vert \lambda a\Vert _p=|\lambda | \Vert a\Vert _p &{}=|\lambda | \inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\} \\ {} &{} =\inf \{\Vert \lambda a+b\Vert _p: b\in {\mathcal {B}}_p\}. \end{array} \end{aligned}$$

• (2) $a^*$ is ${\mathcal {B}}_p$-minimal provided that ${\mathcal {B}}_p$ is $*$-closed. Indeed, since a is ${\mathcal {B}}_p$-minimal, $\Vert a\Vert _p=\Vert a^*\Vert _p$ and ${\mathcal {B}}_p={\mathcal {B}}_p^*$, then

  $$\begin{aligned} \begin{array}{rl} \Vert a^*\Vert _p=\Vert a\Vert _p &{}=\inf \{\Vert (a+b)\Vert _p: b\in {\mathcal {B}}_p\} \\ {} &{}=\inf \{\Vert (a+b)^*\Vert _p: b\in {\mathcal {B}}_p^*\} \\ {} &{}=\inf \{\Vert a^*+b^*\Vert _p: b^*\in {\mathcal {B}}_p\} \\ {} &{}=\inf \{\Vert a^*+b\Vert _p: b\in {\mathcal {B}}_p\}. \end{array} \end{aligned}$$

• (3) Suppose u and v are two unitary operators in ${\mathcal {M}}$ and ${\mathcal {B}}_p$ is (u, v)-invariant (namely, $u{\mathcal {B}}_p v={\mathcal {B}}_p$), then uav is ${\mathcal {B}}_p$-minimal. Indeed, according to Lemma 2.1 (3), one has

  $$\begin{aligned} \Vert x\Vert _p =\Vert u^*uxvv^*\Vert _p\le \Vert uxv\Vert _p\le \Vert x\Vert _p,\ \ \forall x\in L_p({\mathcal {M}},\tau ) \end{aligned}$$

  so the norm $\Vert \cdot \Vert _p$ is unitary invariant. Therefore, if $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal, then

  $$\begin{aligned} \begin{array}{rl} \Vert uav\Vert _p=\Vert a\Vert _p&{}=\inf \{\Vert u(a+b)v\Vert _p: b\in {\mathcal {B}}_p\} \\ {} &{}=\inf \{\Vert uav+b\Vert _p: b\in {\mathcal {B}}_p\}, \end{array} \end{aligned}$$

  which implies that uav is ${\mathcal {B}}_p$-minimal as well.

In recent works, Li et al. [20] and Bottazzi et al. [8] point out that operators x and y in a p-Schatten ideal have disjoint supports if and only if $\Vert x+y\Vert _p^p=\Vert x\Vert _p^p+\Vert y\Vert _p^p$ $(0<p<\infty )$. Following this idea, we characterize ${\mathcal {B}}_p$-minimal elements in $L_p({\mathcal {M}},\tau )$ through disjoint supports. Let $L_p^+({\mathcal {M}},\tau )$ be the set of positive elements in $L_p({\mathcal {M}},\tau )$. For $x, y\in L_p^+({\mathcal {M}},\tau )$, $x\ge y$ means that $x-y\in L_p^+({\mathcal {M}},\tau )$.

Definition 3.3

[29] For $a\in L_p({\mathcal {M}},\tau )$, let $a=u|a|$ be its polar decomposition, where u is a partial isometry from $(\ker a)^{\bot }$ onto $\overline{a D(a)}$, D(a) is the domain of a, and $|a|=(a^*a)^{\frac{1}{2}}$ is the absolute value of a. We say that $l(a)=uu^*$ is the left support for a and $r(a)=u^*u$ is the right support for a.

Definition 3.4

Let $1\le p<\infty$ and ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$. We say that $a\in L_p({\mathcal {M}},\tau )$ and ${\mathcal {B}}_p$ have disjoint left supports if

$$\begin{aligned} l(a)l(b)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p, \end{aligned}$$

and have disjoint right supports if

$$\begin{aligned} r(a)r(b)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p, \end{aligned}$$

where $l(\cdot )$ and $r(\cdot )$ mean the left and right supports, respectively.

Theorem 3.5

Let $1\le p<\infty$ and ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$. If $a\in L_p({\mathcal {M}},\tau )$ and ${\mathcal {B}}_p$ have disjoint left (or right) supports, then a is ${\mathcal {B}}_p$-minimal.

Proof

Recall that for $x\in L_p({\mathcal {M}},\tau )$, l(x) is the projection from H onto $\overline{\mathrm{ran\,}x}$ and r(x) is the projection from H onto $(\ker x)^{\bot }$. Since each x in $L_p({\mathcal {M}},\tau )$ is closed and densely defined, then $(\mathrm{ran\,}x)^{\bot }=\ker x^*$, $(\mathrm{ran\,}x^*)^{\bot }=\ker x$ and $x=x^{**}$ [11, Proposition X.1.6 and X.1.13].

Suppose that a and ${\mathcal {B}}_p$ have disjoint left supports first. For $b\in {\mathcal {B}}_p$, $l(a)l(b)=0$ implies that

$$\begin{aligned} \mathrm{ran\,}a\subset (\mathrm{ran\,}b)^{\bot }=\ker b^*, \ \ \mathrm{ran\,}b\subset (\mathrm{ran\,}a)^{\bot }=\ker a^*. \end{aligned}$$

Thus, $b^*a=a^*b=0$ and

$$\begin{aligned} |a+b|^2=a^*a+a^*b+b^*a+b^*b=|a|^2+|b|^2\ge |a|^2. \end{aligned}$$

According to [16, Lemma 3.2], one has

$$\begin{aligned} \Vert a+b\Vert _p^p=\tau (|a+b|^{2\cdot \frac{p}{2}})\ge \tau (|a|^{2\cdot \frac{p}{2}})=\Vert a\Vert _p^p, \end{aligned}$$

so a is ${\mathcal {B}}_p$-minimal.

Using similar techniques, if a and ${\mathcal {B}}_p$ have disjoint right supports, then

$$\begin{aligned} \overline{\mathrm{ran\,}a^*}=(\ker a)^{\bot }\subset (\ker b)^{\bot \bot }=\ker b, \ \ \overline{\mathrm{ran\,}b^*}\subset \ker a \end{aligned}$$

and so $ba^*=ab^*=0$ for each $b\in {\mathcal {B}}_p$. Hence

$$\begin{aligned}{} & {} |a^*+b^*|^2=|a^*|^2+|b^*|^2\ge |a^*|^2, \\{} & {} \Vert a+b\Vert _p=\Vert a^*+b^*\Vert _p\ge \Vert a^*\Vert _p=\Vert a\Vert _p. \end{aligned}$$

The desired result follows. $\square$

In 1961, Lumer [19] carried over the concept of inner product on Hilbert spaces to semi-inner-product on normed linear spaces, excepting the conjugate linear property. Later, G.R. Giles pointed out that every normed linear space can be represented as a semi-inner-product space with the homogeneity property (see [14, Theorem 1]). Before moving forward, let us recall revalent notions.

Definition 3.6

[14, Page 437], [19, Definition 1]

• (1) Let ${\mathcal {X}}$ be a normed linear space. A mapping $[\cdot ,\cdot ]: {\mathcal {X}}\times {\mathcal {X}}\rightarrow \mathbb C$ satisfying

  • (a) $[\alpha x+\beta z, y]=\alpha [x, y]+\beta [z, y]$;

  • (b) $\Vert x\Vert =[x,x]^{\frac{1}{2}}$;

  • (c) $|[x,y]|^2\le [x,x][y,y];$

  for all $x,y,z\in {\mathcal {X}}$ and $\alpha ,\beta \in \mathbb C$ is called a semi-inner-product on ${\mathcal {X}}$, and then $({\mathcal {X}},[\cdot ,\cdot ]$) is called a semi-inner-product space.

• (2) A semi-inner-product space $({\mathcal {X}},[\cdot ,\cdot ]$) is said to have the homogeneity property if $[\cdot ,\cdot ]$ also satisfies

  • (d) $[x, \alpha y]=\overline{\alpha } [x,y]$

  for all $x,y\in {\mathcal {X}}$ and $\alpha \in \mathbb C$.

• (3) A semi-inner-product space $({\mathcal {X}},[\cdot ,\cdot ]$) is said to be continuous if

  • (e) $\textrm{Re}([x,y+\lambda x])\rightarrow \textrm{Re}([x,y]) \mathrm{\ for\ real\ }\lambda \rightarrow 0,$

  for every x, y in the unit sphere $S({\mathcal {X}})=\{x\in {\mathcal {X}}:\Vert x\Vert =1\}$, where $\textrm{Re}([x,y])$ is the real part of [x, y].

Draw on the experience of [8, 27], we show the semi-inner-product on $L_p({\mathcal {M}},\tau )$ specifically, where ${\mathcal {M}}$ admits a normal semifinite faithful trace $\tau$ and $1< p<\infty$.

Proposition 3.7

Suppose $1< p<\infty$. For $x, y\in L_p({\mathcal {M}},\tau )$, define

$$\begin{aligned}{}[x,y]=\Vert y\Vert _p^{2-p}\tau (|y|^{p-1}u^*x), \end{aligned}$$

(2)

where $y=u|y|$ is the polar decomposition of y. Then

$$\begin{aligned}{}[\cdot , \cdot ]: L_p({\mathcal {M}},\tau )\times L_p({\mathcal {M}},\tau )\rightarrow \mathbb C \end{aligned}$$

is a semi-inner-product on $L_p({\mathcal {M}},\tau )$ having the homogeneity property.

Proof

Suppose $\frac{1}{p}+\frac{1}{q}=1$. Take $x,y,z\in L_p({\mathcal {M}},\tau )$ and $\alpha ,\beta \in \mathbb C$. Let us check that the mapping $[\cdot ,\cdot ]$ defined in (2) satisfies (a–d) in Definition 3.6.

(a) For $y=u|y|\in L_p({\mathcal {M}},\tau )$ one has $|y|^{p-1}\in L_q({\mathcal {M}},\tau )$, since

$$\begin{aligned} \tau \Big ((|y|^{p-1})^q\Big )=\tau (|y|^p)<\infty . \end{aligned}$$

Moreover, using the Hölder inequality,

$$\begin{aligned} \left\| |y|^{p-1}u^*x\right\| _1\le \left\| |y|^{p-1} \right\| _q \Vert u^*x\Vert _p\le \left\| |y|^{p-1} \right\| _q \Vert u^*\Vert \Vert x\Vert _p<\infty , \end{aligned}$$

$|y|^{p-1}u^*x$ is in $L_1({\mathcal {M}},\tau )$. Recall that $\tau$ is linear on $L_1({\mathcal {M}},\tau )$, one has

$$\begin{aligned} \begin{array}{rl} [\alpha x+\beta z, y]&{}=\Vert y\Vert _p^{2-p}\tau (|y|^{p-1}u^*(\alpha x+\beta z)) \\ {} &{}=\alpha \Vert y\Vert _p^{2-p}\tau (|y|^{p-1}u^*x)+\beta \Vert y\Vert _p^{2-p}\tau (|y|^{p-1}u^*z) \\ {} &{}=\alpha [x, y]+\beta [z, y]. \end{array} \end{aligned}$$

(b) First we claim that $u^*u|y|=|y|$. Indeed, suppose $\xi \in \mathrm{ran\,}y^*$, then $\xi =y^*\eta \mathrm{\ for\ some\ }\eta \in (\ker y^*)^{\bot }\cap D(y^*)=\overline{\mathrm{ran\,}y}\cap D(y^*)$ and so

$$\begin{aligned} \overline{\mathrm{ran\,}y^*y}=\overline{\mathrm{ran\,}y^*}=(\ker y)^{\bot }. \end{aligned}$$

(3)

Applying (3) to |y|, one has $\overline{\mathrm{ran\,}|y|}=\overline{\mathrm{ran\,}|y|^2}=\overline{\mathrm{ran\,}y^*y}$ and thus $\mathrm{ran\,}|y|$ is dense in $(\ker y)^{\bot }$. Recall that $u^*u$ is the projection onto $(\ker y)^{\bot }$, $u^*u|y|=|y|$ as asserted. Therefore,

$$\begin{aligned} \begin{array}{rl} [y, y]&{}=\Vert y\Vert _p^{2-p}\tau (|y|^{p-1}u^*y)=\Vert y\Vert _p^{2-p}\tau (|y|^{p-1}u^*u|y|) \\ {} &{}=\Vert y\Vert _p^{2-p}\tau (|y|^{p})=\Vert y\Vert _p^2. \end{array} \end{aligned}$$

(c) Since $|y|^{p-1}\in L_q({\mathcal {M}},\tau )$, it follows from the Hölder inequality that

$$\begin{aligned} \begin{array}{rl} |[x, y]|^2&{}=\Vert y\Vert _p^{4-2p}\left| \tau (|y|^{p-1}u^*x)\right| ^2 \\ {} &{}\le \Vert y\Vert _p^{4-2p} \Big (\left\| |y|^{p-1}u^*\right\| _q^2\Big ) \Vert x\Vert _p^2, \end{array} \end{aligned}$$

meanwhile, by Lemma 2.1 (2),

$$\begin{aligned} \begin{array}{rl} \left\| |y|^{p-1}u^*\right\| _q&{}\le \left\| |y|^{p-1}\Vert _q\Vert u^*\right\| \\ {} &{}=\left\| |y|^{p-1}\right\| _q =\tau \Big ((|y|^{p-1})^q\Big )^{\frac{1}{q}} \\ {} &{}=\tau (|y|^p)^{\frac{1}{q}}=\Big (\Vert y\Vert _p\Big )^{\frac{p}{q}}=\Vert y\Vert _p^{p-1}. \end{array} \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{array}{rl} |[x, y]|^2&{}\le \Vert y\Vert _p^{4-2p} \Vert y\Vert _p^{2p-2} \Vert x\Vert _p^2 \\ {} &{}=\Vert y\Vert _p^2 \Vert x\Vert _p^2=[x,x][y,y]. \end{array} \end{aligned}$$

(d) Observe that $\alpha y=(\frac{\alpha }{|\alpha |}u)(|\alpha y|)$ is the polar decomposition of $\alpha y$,

$$\begin{aligned} \begin{array}{rl} [x, \alpha y]&{}=\Vert \alpha y\Vert _p^{2-p}\tau (|\alpha y|^{p-1}(\frac{\alpha }{|\alpha |}u)^*x) \\ {} &{}=(|\alpha |^{(2-p)+(p-1)-1})(\overline{\alpha })\Vert y\Vert _p^{2-p}\tau (|y|^{p-1}u^*x) \\ {} &{}=\overline{\alpha } [x,y]. \end{array} \end{aligned}$$

The proof is completed. $\square$

To characterize ${\mathcal {B}}_p$-minimal elements, it is necessary to review some basic definitions and known results on geometric theory of Banach space.

Let $({\mathcal {X}},\Vert \cdot \Vert )$ be a Banach space. We call $x_0\in S({\mathcal {X}})$ a smooth point of the unit ball $B({\mathcal {X}})=\{x\in {\mathcal {X}}:\Vert x\Vert \le 1\}$, if there is a unique $f\in {\mathcal {X}}^*$ such that $\Vert f\Vert =1$ and $f(x_0)=1$; and call the norm $\Vert \cdot \Vert$ is Gâteaux differentiable at $x_0\in S({\mathcal {X}})$, if for any $y\in S({\mathcal {X}})$ and $\lambda \in \mathbb R$

$$\begin{aligned} D_{x_0}(y)=\lim \limits _{\lambda \rightarrow 0}\frac{\Vert x_0+\lambda y\Vert -\Vert x_0\Vert }{\lambda } \end{aligned}$$

exists. Accordingly, the Banach space ${\mathcal {X}}$ is said to be smooth if each $x\in S({\mathcal {X}})$ is a smooth point of $B({\mathcal {X}})$, and is said to be Gâteaux differentiable if $\Vert \cdot \Vert$ is Gâteaux differentiable at each $x\in S({\mathcal {X}})$. It is well known that ${\mathcal {X}}$ is smooth if and only if ${\mathcal {X}}$ is Gâteaux differentiable [1, Theorem 2.1].

Lemma 3.8

[23, Corollary 5.2] For $1<p<\infty$, $L_p({\mathcal {M}},\tau )$ is uniformly convex and smooth.

Let $1< p<\infty$ and ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$. Since the Banach space $L_p({\mathcal {M}},\tau )$ is smooth, in other words, it is Gâteaux differentiable, then the semi-inner product defined in (2) is continuous [14, Theorem 3]. Moreover, by [14, Theorem 2], $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if and only if

$$\begin{aligned}{}[b,a]=0 \mathrm{\ for\ all} \ b \ {\in {\mathcal {B}}_p}, \end{aligned}$$

equivalently,

$$\begin{aligned} \tau (|a|^{p-1}u^*b)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p, \end{aligned}$$

where $a=u|a|$ is the polar decomposition of a. We obtain the following theorem.

Theorem 3.9

Let $1< p<\infty$ and ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$. Then $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if and only if

$$\begin{aligned} \tau (|a|^{p-1}u^*b)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p, \end{aligned}$$

where $a=u|a|$ is the polar decomposition of a.

One can simplify Theorem 3.9 when the ${\mathcal {B}}_p$-minimal element is self-adjoint. Notice that $a=u|a|=|a|u^*$ when a is self-adjoint, moreover, $|a|^{p-2}=a^{p-2}$ when p is an even integer, one has the following corollary.

Corollary 3.10

Let $2\le p<\infty$ and ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$.

(1) A self-adjoint element $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if and only if

$$\begin{aligned} \tau (|a|^{p-2}ab)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p. \end{aligned}$$

In particular, when p is an even integer, a self-adjoint element $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if and only if

$$\begin{aligned} \tau (a^{p-1}b)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p. \end{aligned}$$

(2) A positive element $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if and only if

$$\begin{aligned} \tau (a^{p-1} b)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p. \end{aligned}$$

Example 3.11

Let $M_n(\mathbb C)$ be the algebra of complex $n\times n$ matrices. For $A\in M_n(\mathbb C)$, denote by $\lambda (A)=(\lambda _1(A),\lambda _2(A),\ldots ,\lambda _n(A))$ the set of eigenvalues of A, in counting multiplicity. Then $\Vert A\Vert _p=\Big ( \sum \nolimits _{i=1}^n |\lambda _i(A)|^p \Big )^{\frac{1}{p}}.$ A class of positive minimal matrices in $M_2(\mathbb C)$ will be provided below.

Let $p=3$, denote ${\mathcal {B}}_3=\mathbb C\oplus 0$ and take $E=\left( \begin{array}{cc}1&{}0\\ 0&{}0\end{array}\right) \in {\mathcal {B}}_3$. The positive ${\mathcal {B}}_3$-minimal matrix in $M_2(\mathbb C)$ must have the form $\left( \begin{array}{cc}0&{}0\\ 0&{}a_{22}\end{array}\right)$, where $a_{22}\ge 0$. Indeed, suppose $A=\left( \begin{array}{cc}a_{11}&{}a_{12}\\ \overline{a_{12}}&{}a_{22}\end{array}\right)$ is a positive ${\mathcal {B}}_3$-minimal matrix, where $a_{11}, a_{22}\ge 0$. By Corollary 3.10 (2) one has

$$\begin{aligned} \textrm{tr}(A^2E)=\textrm{tr}(A^2E^2)=\textrm{tr}(EA^2E)=\textrm{tr}\left( \begin{array}{cc}a_{11}^2+|a_{12}|^2&{}0\\ 0&{}0\end{array}\right) =0, \end{aligned}$$

and then $a_{11}=a_{12}=0$. Moreover, since that

$$\begin{aligned} \left\| \left( \begin{array}{cc}0&{}0\\ 0&{}a_{22}\end{array}\right) +\left( \begin{array}{cc}x&{}0\\ 0&{}0\end{array}\right) \right\| _3=\root 3 \of {|x|^3+a_{22}^3}\ge a_{22}= \left\| \left( \begin{array}{cc}0&{}0\\ 0&{}a_{22}\end{array}\right) \right\| _3 \end{aligned}$$

for all $x\in \mathbb C$, $A=\left( \begin{array}{cc}0&{}0\\ 0&{}a_{22}\end{array}\right)$ is ${\mathcal {B}}_3$-minimal.

Remark 3.12

• (1) The ${\mathcal {B}}_p$-minimal element must exists (considering 0).

• (2) From Example 3.11 one can see the ${\mathcal {B}}_p$-minimal element may not be unique.

4 Banach Duality Formula and Minimal Elements

As an application of the Hahn–Banach Theorem, [9, Proposition 4] and [21, Lemma 4] put forward the Banach duality formula between sets of compact operators and trace class operators. Then [7, Proposition 3.3] generalizes this result to a p-Schatten ideal and its dual, i.e. q-Schatten ideal, where $\frac{1}{p}+\frac{1}{q}=1$. The Banach duality formula connects a Banach space ${\mathcal {X}}$ and its dual ${\mathcal {X}}^*$, which is also a tool to characterize minimal elements. In this section, we characterize ${\mathcal {B}}_p$-minimal elements by the Banach duality formula between $L_p({\mathcal {M}},\tau )$ and its dual given below.

Lemma 4.1

(Banach duality formula for non-commutative $L_p$-space) Let $1\le p<\infty$ and ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$. Denote

$$\begin{aligned} {\mathcal {B}}_p^{\bot (\tau )}=\{y\in L_q({\mathcal {M}},\tau ): \tau (by)=0 \mathrm{\ for\ all\ } b\in {\mathcal {B}}_p\}, \end{aligned}$$

where $\frac{1}{p}+\frac{1}{q}=1$. Then for $a\in L_p({\mathcal {M}},\tau )$,

$$\begin{aligned} \inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}=\sup \{|\tau (ay)|: y\in {\mathcal {B}}_p^{\bot (\tau )}, \Vert y\Vert _q=1\}. \end{aligned}$$

(4)

Proof

Take $y\in {\mathcal {B}}_p^{\bot (\tau )}$ with $\Vert y\Vert _q=1$. With the help of Hölder inequality,

$$\begin{aligned} |\tau (ay)|=|\tau (ay+by)|\le \Vert a+b\Vert _p\Vert y\Vert _q=\Vert a+b\Vert _p \end{aligned}$$

for all $b\in {\mathcal {B}}_p$, so

$$\begin{aligned} \sup \{|\tau (ay)|: y\in {\mathcal {B}}_p^{\bot (\tau )}, \Vert y\Vert _q=1\}\le \inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}. \end{aligned}$$

Without loss of generality, suppose $a\notin {\mathcal {B}}_p$, then $\inf \{\Vert a+b\Vert _p: B\in {\mathcal {B}}_p\}>0$ as ${\mathcal {B}}_p$ is closed. According to the Hahn–Banach Theorem [11, Corollary III.6.8], there is a linear functional $f\in (L_p({\mathcal {M}},\tau ))^*$ such that $\Vert f\Vert =1$, $f|_{{\mathcal {B}}_p}=0$ and

$$\begin{aligned} f(a)=\textrm{dist}(a,{\mathcal {B}}_p) =\inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}. \end{aligned}$$

Since $(L_p({\mathcal {M}},\tau ))^*=L_q({\mathcal {M}},\tau )$ [23, Page 1464], there exists a unique $y_0$ in $L_q({\mathcal {M}},\tau )$ such that

$$\begin{aligned} \Vert y_0\Vert _q=\Vert f\Vert =1 \mathrm{\ and\ } f(\cdot )=\tau (\cdot y_0). \end{aligned}$$

In addition, $f|_{{\mathcal {B}}_p}=0$ implies that $y_0\in {\mathcal {B}}_p^{\bot (\tau )}$. One gets

$$\begin{aligned} \inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}=\tau (ay_0) \le \sup \{|\tau (ay)|: y\in {\mathcal {B}}_p^{\bot (\tau )}, \Vert y\Vert _q=1\}. \end{aligned}$$

The equation (4) holds. $\square$

Theorem 4.2

Let $1<p<\infty$, ${\mathcal {B}}_p$ be a closed linear subspace of $L_p({\mathcal {M}},\tau )$ and $a(\ne 0)\in L_p({\mathcal {M}},\tau )$. The following two statements hold:

• (1) a is ${\mathcal {B}}_p$-minimal if and only if

  $$\begin{aligned} \Vert a\Vert _p=\sup \{|\tau (ay)|: y\in {\mathcal {B}}_p^{\bot (\tau )}, \Vert y\Vert _q=1\}. \end{aligned}$$

  Moreover, the supremum of the right side can be obtained at $y_a=\frac{|a|^{p-1} u^{*}}{\Vert a\Vert _p^{p-1}},$ where $a=u|a|$ is the polar decomposition of a;

• (2) If a is ${\mathcal {B}}_p$-minimal, then $D_{a}(b)=0 \mathrm{\ for\ all\ }b\in {\mathcal {B}}_p,$ where $D_{a}(b)$ is the Gâteaux derivative of $\Vert \cdot \Vert _p$ at a in the b direction.

Proof

(1) It is obvious from Lemma 4.1 that $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if and only if

$$\begin{aligned} \Vert a\Vert _p=\sup \{|\tau (ay)|: y\in {\mathcal {B}}_p^{\bot (\tau )}, \Vert y\Vert _q=1\}. \end{aligned}$$

(5)

Let us prove that the supremum of right side of (5) can be obtained at

$$\begin{aligned} y_a=\frac{|a|^{p-1} u^{*}}{\Vert a\Vert _p^{p-1}} \end{aligned}$$

when a is ${\mathcal {B}}_p$-minimal.

According to Theorem 3.9, if a is ${\mathcal {B}}_p$-minimal, then

$$\begin{aligned} \tau (by_a)=\tau (y_ab)=\frac{1}{\Vert a\Vert _p^{p-1}}\tau (|a|^{p-1}u^*b)=0 \end{aligned}$$

for all $b\in {\mathcal {B}}_p$, hence $y_a\in {\mathcal {B}}_p^{\bot (\tau )}$. In addition, by the proof of Proposition 3.7 (b) we know $u^*u|a|=|a|$, then

$$\begin{aligned} y_ay_a^*=\frac{|a|^{p-1}u^*u|a|^{p-1}}{\Vert a\Vert _p^{2p-2}}=\frac{|a|^{2p-2}}{\Vert a\Vert _p^{2p-2}}, \end{aligned}$$

$|y_a^*|=\frac{|a|^{p-1}}{\Vert a\Vert _p^{p-1}}$ and

$$\begin{aligned} \Vert y_a\Vert _q=\Vert y_a^*\Vert _q=\left( \frac{\tau (|a|^{(p-1)q})}{\Vert a\Vert _p^{(p-1)q}}\right) ^{\frac{1}{q}}=\left( \frac{\tau (|a|^p)}{\Vert a\Vert _p^p}\right) ^{\frac{1}{q}}=1. \end{aligned}$$

Moreover,

$$\begin{aligned} \tau (ay_a)=\frac{\tau (u|a|^pu^*)}{\Vert a\Vert _p^{p-1}}= \frac{\tau (|a|^{p-1}u^*u|a|)}{\Vert a\Vert _p^{p-1}}= \frac{\tau (|a|^p)}{\Vert a\Vert _p^{p-1}}=\Vert a\Vert _p. \end{aligned}$$

The desired result is obtained.

(2) By Lemma 3.8 and its previous statements, the norm $\Vert \cdot \Vert _p$ is Gâteaux differentiable at each $a\in L_p({\mathcal {M}},\tau )$. Moreover, according to [1, Theorem 1.1] and [17, Proposition 1.3],

$$\begin{aligned} D_{a}(x)=\lim \limits _{\lambda \rightarrow 0}\frac{\Vert a+\lambda x\Vert _p-\Vert a\Vert _p}{\lambda }=\textrm{Re} f_a(x) \end{aligned}$$

for $x\in L_p({\mathcal {M}},\tau )$, where $f_a(\cdot )$ is the unique linear functional on $L_p({\mathcal {M}},\tau )$ such that $\Vert f_a\Vert =1$ and $f_a(a)=\Vert a\Vert _p$. Combined with the first part proof, one gets $f_a(\cdot )=\tau (\cdot y_a)$ and

$$\begin{aligned} D_{a}(x)=\mathrm{Re\ }\tau (xy_a)=\frac{1}{\Vert a\Vert _p^{p-1}}\mathrm{Re\ }[x,a]. \end{aligned}$$

Thus, if $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal, then $D_{a}(b)=0$ for all $b\in {\mathcal {B}}_p$. $\square$

Remark 4.3

Let ${\mathcal {A}}$ be a Banach space, ${\mathcal {B}}$ be a proper closed linear subspace of ${\mathcal {A}}$ and $a(\ne 0)\in {\mathcal {A}}$ be ${\mathcal {B}}$-minimal. A witness to the ${\mathcal {B}}$-minimality of a is a linear functional f on ${\mathcal {A}}$ such that $\Vert f\Vert =1$, $f|_{{\mathcal {B}}}=0$ and $f(a)=\Vert a\Vert$ [25, Page 2267–2268]. By the proof of Theorem 4.2, we know that in the non-commutative $L_p$-space context ($1<p<\infty$), $\tau (\cdot y_a)$ is a witness to the ${\mathcal {B}}_p$-minimality of a, where $y_a=\frac{|a|^{p-1} u^{*}}{\Vert a\Vert _p^{p-1}}$ and $a=u|a|$ is the polar decomposition of a.

5 Minimal Elements Related to Finite-Diagonal-Block Type ${\mathcal {B}}_p$

Let ${\mathcal {H}}$ be a complex separable Hilbert space with an orthonormal basis $\{\xi _i\}_{i=1}^\infty$. Make a partition of $\mathbb Z^+$ by $\{1,2,\ldots ,\lambda _1\},$ $\{\lambda _1+1,\lambda _1+2,\ldots ,\lambda _1+\lambda _2\},$ $\{\lambda _1+\lambda _2+1,\lambda _1+\lambda _2+2,\ldots ,\lambda _1+\lambda _2+\lambda _3\},$ $\ldots ,$ such that each set is finite. Set $\Lambda _0=0$, $\Lambda _k=\sum \nolimits _{i=1}^k\lambda _i$ and

$$\begin{aligned} {\mathcal {H}}_{\Lambda _k}=\hbox {span} \{\xi _i: \Lambda _{k-1}+1\le i\le \Lambda _{k}\}. \end{aligned}$$

Denote by $P_{\Lambda _k}$ the orthogonal projection from ${\mathcal {H}}$ onto ${\mathcal {H}}_{\Lambda _k}$, and let

$$\begin{aligned} W({\mathcal {H}})=\sum \limits _{k=1}^{\infty }P_{\Lambda _k}F({\mathcal {H}})P_{\Lambda _k} \end{aligned}$$

(6)

(converging with respect to the operator norm), where $F({\mathcal {H}})$ is the algebra of finite rank operators on ${\mathcal {H}}$. The elements of $W({\mathcal {H}})$ are compact operators with finite square matrices of a fixed type in their diagonals, and $W({\mathcal {H}})$ is a C*-subalgebra of $K({\mathcal {H}})$ [31]. Such a $W({\mathcal {H}})$ is said to be finite-diagonal-block type. As a continuation and improvement of previous works on $D(K({\mathcal {H}}))$-minimal compact operators [2, 9, 10, 21, 31], we characterize minimal elements related to a finite-diagonal-block type C*-subalgebra of $K({\mathcal {H}})$ in [31].

Similar to the $W({\mathcal {H}})$ in (6), we can construct a closed linear subspace ${\mathcal {B}}_p$ of $L_p({\mathcal {M}}, \tau )$ with finite-diagonal-block type. Some interesting results about minimal elements related to such a type ${\mathcal {B}}_p$ can be drawn. Two projections $e_1$ and $e_2$ in $P({\mathcal {M}})$ are said to be orthogonal if $e_1e_2=0$. It is easy to check that if $e_1$ and $e_2$ are two orthogonal projections in $P({\mathcal {M}})$, then $e_1 {\mathcal {S}} e_1\cap e_2 {\mathcal {S}} e_2=\{0\}.$

Theorem 5.1

Let ${\mathcal {M}}$ be a $\sigma$-finite von Neumann algebra. Take $\{e_i\}_{i=1}^{\infty }\subset P({\mathcal {M}})$ such that

• (a) $\tau (e_i)<\infty$;

• (b) $e_i e_j=0$ when $i\ne j$;

• (c) $\sum \nolimits _{i=1}^{\infty } e_i=I$, with respect to the strong operator topology.

Set

$$\begin{aligned} {\mathcal {B}}_p=\bigoplus \limits _{i=1}^{\infty } e_i {\mathcal {S}} e_i \end{aligned}$$

(7)

(converging with respect to $\Vert \cdot \Vert _p$). The following statements hold:

• (1) For $1<p<\infty$, $a\in L_p({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal if and only if

  $$\begin{aligned} \tau (|a|^{p-1} u^*e_i x e_i)=0 \mathrm{\ for\ all\ }x\in {\mathcal {S}} \mathrm{\ and\ } i\in \mathbb Z_+, \end{aligned}$$

  where $a=u|a|$ is the polar decomposition of a.

• (2) For $2\le p<\infty$, if $a\in L_p^+({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal, then $e_ia^{p-1}e_i=0$ for all $i\in \mathbb Z_+$.

• (3) For $2\le p<\infty$, $a\in {\mathcal {S}}_+$ is ${\mathcal {B}}_p$-minimal implies that $a=0$.

We say ${\mathcal {B}}_p$ with the form (7) a finite-diagonal-block type closed linear subspace of $L_p({\mathcal {M}},\tau )$.

Proof

It is well known that for a $\sigma$-finite von Neumann algebra ${\mathcal {M}}$, the sequence of projections $\{e_i\}_{i=1}^{\infty }$ satisfying (a-c) must exist.

(1) Suppose that $\tau (|a|^{p-1} u^*e_i x e_i)=0$ for all $x\in {\mathcal {S}}$ and $i\in \mathbb Z_+$. Take any $b=\sum \nolimits _{i=1}^{\infty } e_i x_i e_i\in {\mathcal {B}}_p$, where $x_i\in {\mathcal {S}}$. Since $|a|^{p-1}u^*\in L_q({\mathcal {M}},\tau )$, one has

$$\begin{aligned} \begin{aligned}&\left\| |a|^{p-1}u^*\sum \limits _{i=1}^n e_i x_i e_i-|a|^{p-1}u^*b\right\| _1 \\ {}&\quad \le \left\| |a|^{p-1}u^*\right\| _q \left\| \sum \limits _{i=1}^n e_i x_i e_i-b\right\| _p\rightarrow 0 \end{aligned} \end{aligned}$$

when $n\rightarrow \infty$. Moreover, as $\tau$ is continuous on $L_1({\mathcal {M}},\tau )$,

$$\begin{aligned} \begin{aligned} \tau (|a|^{p-1}u^*b)&=\lim \limits _{n\rightarrow \infty }\tau (|a|^{p-1}u^*\sum \limits _{i=1}^n e_i x_i e_i) \\ {}&=\lim \limits _{n\rightarrow \infty }\sum \limits _{i=1}^n\tau (|a|^{p-1}u^*e_i x_i e_i) \\ {}&=0. \end{aligned} \end{aligned}$$

Using Theorem 3.9 and by the arbitrariness of b, a is ${\mathcal {B}}_p$-minimal. The necessity is obvious.

(2) Suppose $a\in L_p^+({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal. From Corollary 3.10 we know $\tau (a^{p-1}x)=0$ for all $x\in {\mathcal {B}}_p.$ Take $x=e_{i}$, then

$$\begin{aligned} \tau (a^{p-1}e_{i})=\tau (a^{p-1}e_{i}^2)=\tau (e_{i} a^{p-1} e_{i})=0. \end{aligned}$$

One can get that $e_{i} a^{p-1}e_{i}=0$, since $\tau$ is faithful and $e_{i} a^{p-1}e_{i}$ is positive.

(3) Suppose $a\in {\mathcal {S}}_+$ is ${\mathcal {B}}_p$-minimal, then $\tau \Big (a^{p-1}(\sum \nolimits _{i=1}^ne_{i})\Big )=0$ for each $n\in \mathbb Z_+$ (Corollary 3.10). Note that $a^{p-1}$ is in ${\mathcal {S}}_+$ and then in $L_1({\mathcal {M}},\tau )$, according to Lemma 2.1 (5) one has $\lim \limits _{n\rightarrow \infty } a^{p-1}\sum \nolimits _{i=1}^{n} e_i=a^{p-1}$ in $L_1({\mathcal {M}},\tau )$. Since $\tau$ is continuous and faithful on $L_1({\mathcal {M}},\tau )$,

$$\begin{aligned} \tau (a^{p-1})=\lim \limits _{n\rightarrow \infty }\tau \Big (a^{p-1}\left( \sum \limits _{i=1}^ne_{i}\right) \Big )=0, \end{aligned}$$

further, $a^{p-1}=a=0$. $\square$

Corollary 5.2

Let ${\mathcal {M}}$ be a finite von Neumann algebra, $\{e_i\}_{i=1}^n$ be mutually orthogonal projections in $P({\mathcal {M}})$ such that $\tau (e_i)<\infty$ and $\sum \nolimits _{i=1}^n e_i=I$. Set

$$\begin{aligned} {\mathcal {B}}_p=\bigoplus \limits _{i=1}^n e_i{\mathcal {S}}e_i, \end{aligned}$$

(8)

where $2\le p<\infty$. Then ${\mathcal {B}}_p$ is a closed linear subspace of $L_p({\mathcal {M}},\tau )$, and $a\in L_p^+({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal implies that $a=0$.

Proof

Obviously, ${\mathcal {B}}_p$ is a linear subspace of $L_p({\mathcal {M}},\tau )$. Recall that a finite direct sum of closed sets is also closed, it is enough to prove that $e{\mathcal {S}}e$ is closed for any $e\in P({\mathcal {M}})$ with $\tau (e)<\infty$.

Take z from the closure of $e{\mathcal {S}}e$ and suppose $e x_{i} e\rightarrow z$ in $L_p({\mathcal {M}},\tau )$ when $i\rightarrow \infty$, where $x_{i}\in S$. Note that

$$\begin{aligned} \Vert ex_{i}e-eze\Vert _p=\left\| e (ex_{i}e)e-eze\right\| _p\le \Vert e\Vert \Vert ex_{i}e-z\Vert _p \Vert e\Vert , \end{aligned}$$

so $ex_{i}e\rightarrow eze$ in $L_p({\mathcal {M}},\tau )$ when $i\rightarrow \infty$. By the uniqueness of the limit, one has $z=eze=e(eze)e$. On the other hand, since $\tau (e)<\infty$, it follows from Lemma 2.1 (1) that $eze\in {\mathcal {S}}$, then $z\in e{\mathcal {S}}e$ and $e{\mathcal {S}}e$ is closed in $L_p({\mathcal {M}},\tau )$. Obviously, the ${\mathcal {B}}_p$ in (8) contains identity I. Thus if $a\in L_p^+({\mathcal {M}},\tau )$ is ${\mathcal {B}}_p$-minimal, then $\tau (a^{p-1})=0$ and $a^{p-1}=a=0$. $\square$

Example 5.3

Let $2\le p<\infty$. Consider $M_n(\mathbb C)$ and its closed linear subspace

$$\begin{aligned} {\mathcal {B}}_p=M_{\mu _1}(\mathbb C)\oplus M_{\mu _2}(\mathbb C)\oplus \cdots \oplus M_{\mu _k}(\mathbb C), \end{aligned}$$

where $k\ge 1$, $1\le \mu _i\le n$ and $\sum \nolimits _{i=1}^k\mu _i=n$. Different from the ${\mathcal {B}}_3$ in Example 3.11, such a ${\mathcal {B}}_p$ contains the identity matrix $I_n$. By Corollary 5.2, the only positive ${\mathcal {B}}_p$-minimal matrix in $M_n(\mathbb C)$ is the zero matrix.

Example 5.4

This example is obtained from [7]. Let ${\mathcal {H}}$ be a complex separable Hilbert space with an orthonormal basis $\{\xi _i\}_{i=1}^\infty$ and $S_p({\mathcal {H}})$ be the set of p-Schatten class operators on ${\mathcal {H}}$, namely,

$$\begin{aligned} S_p({\mathcal {H}})=\{x\in K({\mathcal {H}}):\Vert x\Vert _p<\infty \}, \end{aligned}$$

where $\Vert x\Vert _p^p=\textrm{tr}(|x|^p)=\sum \nolimits _{i=1}^{\infty }\langle |x|^p \xi _i, \xi _i\rangle$. Set

$$\begin{aligned} {\mathcal {B}}_p=\bigoplus \limits _{i=1}^{\infty } e_i F({\mathcal {H}}) e_i, \end{aligned}$$

(converging with respect to $\Vert \cdot \Vert _p$), where $e_i$ is the orthogonal projection from ${\mathcal {H}}$ onto $\textrm{span}\{\xi _i\}$. That is, ${\mathcal {B}}_p$ consists of all diagonal p-Schatten operators. From Theorem 5.1 (1) we know if $a\in S_p^+({\mathcal {H}})$ is ${\mathcal {B}}_p-$minimal $(p\ge 2)$, then $e_i a^{p-1} e_i=0$ for each $i\in \mathbb Z_+$. Further,

$$\begin{aligned} \begin{aligned} \textrm{tr}(a^{p-1})&=\sum \limits _{i=1}^{\infty }\langle a^{p-1} \xi _i, \xi _i\rangle =\sum \limits _{i=1}^{\infty }\langle a^{p-1} e_i\xi _i, e_i\xi _i\rangle \\ {}&=\sum \limits _{i=1}^{\infty }\langle e_i a^{p-1} e_i\xi _i, \xi _i\rangle =0, \end{aligned} \end{aligned}$$

and a must be 0.