\section{Introduction}

In this paper, $\Gamma$ is a discrete group, $K$ is a (complex) Hilbert space and $\pi$ is a unitary representation of $\Gamma$ on $K$.
We denote by $1_\Gamma$ the one dimensional trivial representation of $\Gamma$, and $K^\pi$ the set of $\pi$-invariant vectors in $K$.
Recall that (see \cite{BC}) $\pi$ is said to have a \emph{spectral gap} if the restriction of $\pi$ on the orthogonal complement $(K^\pi)^\bot$ does not weakly contain $1_\Gamma$ (in the sense of Fell), i.e.\ there does not exist a net of unit vectors $\xi_i\in (K^\pi)^\bot$ satisfying $\|\pi_t\xi_i -\xi_i\| \to 0$ for every $t\in \Gamma$.

The following result on group actions, which relates the existence of spectral gap of certain representation and the uniqueness of the invariant states (means), is crucial to the solution of the Banach-Ruziewicz problem (see \cite{Marg},\cite{Sull},\cite{Drin}).

\begin{thm2}\label{motivation}
Let $\Gamma$ be a countable group with a measure preserving ergodic action on a standard non-atomic probability space $(X, \mu)$, and $\rho$ be the associated unitary representation of $\Gamma$ on $L^2(X, \mu)$.

\smallskip\noindent
(a) (\cite{dR}, \cite{Sch}, \cite{LH}) If $\rho$ has a spectral gap, then the integration with respect to $\mu$ is the unique $\Gamma$-invariant state on $L^\infty(X,\mu)$.

\smallskip\noindent
(b) (\cite{Sch}) If there is a unique $\Gamma$-invariant state on $L^\infty(X,\mu)$, then $\rho$ has a spectral gap.
\end{thm2}

Motivated by this result we will define spectral gap actions of discrete groups on von Neumann algebras, and study the corresponding invariant states on the von Neumann algebras.

From now on, $N$ is a von Neumann algebra  with standard form $(N, \mathfrak{H}, \mathfrak{J}, \mathfrak{P})$, i.e.\ $\mathfrak{H}$ is a Hilbert space, $\mathfrak{J}:\mathfrak{H} \to \mathfrak{H}$ is a conjugate linear bijective isometry, and $\mathfrak{P}$ is a self-dual cone in $\mathfrak{H}$ such that there is a (forgettable) faithful representation of $N$ on $\mathfrak{H}$ satisfying some compatibility conditions with $\mathfrak{J}$ and $\mathfrak{P}$.
The readers may find in \cite{Haag-st-form} more information on this topic.
As proved in \cite[Theorem 3.2]{Haag-st-form}, there exists a unique unitary representation $\mathcal{U}$ of the $^*$-automorphism group ${\rm Aut}(N)$ on $\mathfrak{H}$ such that
$$g(x) = \mathcal{U}_{g} x \mathcal{U}_{g}^*, \quad \mathcal{U}_g\mathfrak{J}=\mathfrak{J}\mathcal{U}_g \quad \text{and} \quad \mathcal{U}_g(\mathfrak{P})\subseteq \mathfrak{P}\qquad (x\in N; g\in{\rm Aut}(N)).$$
Hence, an action $\alpha$ of $\Gamma$ on $N$ gives rise to a unitary representation $\mathcal{U}_{\alpha}=\mathcal{U}\circ\alpha$ of $\Gamma$ on $\mathfrak{H}$.

\begin{defn2}
Let $\alpha$ be an action of $\Gamma$ on $N$.

\smallskip\noindent
(a) $\alpha$ is said to have a \emph{spectral gap} if the representation $\mathcal{U}_{\alpha}$ has a spectral gap.

\smallskip\noindent
(b) $\alpha$ is said to be \emph{standard} if every $\alpha$-invariant state on $N$ is a weak-$^*$-limit of a net of normal $\alpha$-invariant states.
\end{defn2}

We will study the relationship between spectral gap actions and standard actions of discrete groups on von Neumann algebras. This is the main topic of Section 2. Our first main result generalizes Theorem \ref{motivation}(a) to the situation of discrete group actions on von Neumann algebras.

\begin{thm2}\label{thm:main}Let $\alpha$ be an action of a (possibly uncountable) discrete group $\Gamma$ on a von Neumann algebra $N$. If $\alpha$ has a spectral gap, then $\alpha$ is standard.
\end{thm2}

In contrast to Theorem \ref{motivation}(b), the converse statement of Theorem \ref{thm:main} is not true in general (see Example \ref{eg:4-not-2}(b)).
Nevertheless, we will give several situations in which the converse  does hold (see e.g.\ Proposition \ref{prop:inv-mean-sp-gap-act}(b)).
Let us also note that Theorem \ref{motivation} (and hence Theorem \ref{thm:main}) does not extend to locally compact groups.
Indeed, if $G$ is the circle group (equipped with the canonical compact topology) and $N := L^\infty(G)$ with the left translation action $\alpha$ by $G$, then there is a unique normal $\alpha$-invariant state on $N$, but there exist more than one $\alpha$-invariant states (see e.g.\ \cite[Proposition 2.2.11]{Lub}).

We will give two applications of Theorem \ref{thm:main}.
The first one concerns inner amenability and will be considered in Section 3.
The notion of inner amenability was first introduced by
Effros, aiming to give a group theoretic description of property Gamma of the group von Neumann algebra $L(\Gamma)$ of an ICC group.
Recall that  that
$\Gamma$ is an \emph{ICC group} if $\Gamma \neq \{e\}$ and all the non-trivial conjugacy classes of $\Gamma$ are infinite.
Moreover, $\Gamma$ is \emph{inner amenable} if there exist more than one inner invariant states on $\ell^\infty(\Gamma)$.
Effros showed in \cite{Eff} that a countable ICC group $\Gamma$ is inner amenable if $L(\Gamma)$ has property Gamma.
However, Vaes recently gave, in \cite{Vaes}, a counter example to the converse.
From an opposite angle, it is natural to ask whether one can express inner amenability of $\Gamma$ in terms of certain property of $L(\Gamma)$.
One application of Theorem \ref{thm:main} is  the following result.

\begin{thm2}\label{thm:inner}
Let $\Gamma$ be a finitely generated ICC group. Then $\Gamma$ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra $L(\Gamma)$.
\end{thm2}

We will also study an alternative generalization of inner amenability (which is called ``strongly inner amenability'') of  (not necessarily ICC) discrete groups, which is of independent interest.

Section 4 is concerned with our second application, which is related to property $(T)$.
Recall that $\Gamma$ is said to have \emph{property $(T)$} if every unitary representation of $\Gamma$ has a spectral gap.
It follows from Theorem \ref{thm:main} that if $\Gamma$ has property $(T)$, then every action of $\Gamma$ on a von Neumann algebra is standard, and in particular, the absence of normal invariant state implies the absence of invariant state for this action.
We will show that this property actually characterizes property $(T)$.

\begin{thm2}\label{thm:property-T} Let $\Gamma$ be a countable discrete group.
The following statements are equivalent.
\begin{enumerate}
\item $\Gamma$ has property $(T)$.
\item All actions of $\Gamma$ on von Neumann algebras are standard.
\item For any action $\alpha$ of $\Gamma$ on a von Neumann algebra with only one normal $\alpha$-invariant state, there is only one $\alpha$-invariant state.
\item For every action $\alpha$ of $\Gamma$ on a von Neumann algebra without normal $\alpha$-invariant state, there is no $\alpha$-invariant state.
\end{enumerate}
\end{thm2}

We will also consider the implication $(4)\Rightarrow (1)$ for general discrete group  $\Gamma$ with property $(T, FD)$ in the sense of \cite{LZ89} (Proposition \ref{prop:no-inv-state}). Consequently, a minimally almost period discrete group satisfying (4) is finitely generated.

We end this introduction by introducing more notation.
Throughout this article, $\overline{K}$ and $\mathcal{L}(K)$ denote the conjugate Hilbert space of $K$ and the space of bounded linear maps on $K$ respectively.
For any $\xi,\eta\in K$, we define $\omega_{\xi,\eta}\in \mathcal{L}(K)^*$ by $\omega_{\xi,\eta}(x) := \langle x\xi, \eta\rangle$ ($x\in \mathcal{L}(K)$) and set $\omega_\xi := \omega_{\xi,\xi}$.
We consider $p_\pi\in \mathcal{L}(K)$ to be the orthogonal projection onto $K^\pi$.
Furthermore, we denote by $C^*(\pi)$ the $C^*$-subalgebra generated by $\{\pi_t\}_{t\in \Gamma}$ and by $vN(\pi)$ the bicommutant of $C^*(\pi)$.
We define a functional $\varepsilon_\pi: C^*(\pi) \to \mathbb{C}$ formally by $\varepsilon_\pi\big(\sum_{k=1}^n c_k \pi_{t_k}\big) = \sum_{k=1}^n c_k$ (warning: $\varepsilon_\pi$ is not necessarily well-defined.)

\section{Invariant States and Spectral Gap Actions (Theorem \ref{thm:main})}

Let us start with the following lemma, which may be a known result.
Note that part (d) of this lemma is motivated by \cite{Pasch} (and one can also obtain this part using the argument in \cite{Pasch}).

\begin{lem}\label{lem:p_pi}
(a) $\varepsilon_\pi$ is well-defined if and only if $1_\Gamma$ is weakly contained in $\pi$.
This is equivalent to the existence of $\psi\in \mathcal{L}(K)^*_+$ satisfying
\begin{equation}\label{eqt:m-pi}
\psi(\pi_t)\ =\ 1 \qquad (t\in \Gamma)
\end{equation}
(in this case, $\varepsilon_\pi = \psi|_{C^*(\pi)}$).

\smallskip\noindent
(b) $\pi$ does not have a spectral gap if and only if there is $\psi\in \mathcal{L}(K)_+^*$ satisfying \eqref{eqt:m-pi} and $\psi(p_{\pi}) = 0$.

\smallskip\noindent
(c) If $\psi\in \mathcal{L}(K)_+^*$ satisfies \eqref{eqt:m-pi}, then $\psi(\pi_t x) = \psi(x)$ ($t\in \Gamma, x\in \mathcal{L}(K)$) and $\psi$ is ${\rm Ad}\ \!\pi$-invariant.

\smallskip\noindent
(d) $p_\pi\notin C^*(\pi)$ if and only if $p_\pi \neq 0$ and $\pi$ does not have a spectral gap.
\end{lem}
\begin{prf}
Let $\pi^0$ be the restriction of $\pi$ on $(K^\pi)^\bot$.
One has $\pi_t = 1_{K^\pi}\oplus \pi^0_t$ ($t\in \Gamma$) and $p_\pi = 1_{K^\pi}\oplus 0$.

\smallskip\noindent
(a) For any unitary representation $\rho$ of $\Gamma$, we denote by $\tilde \rho$ the induced $^*$-representation of the full group $C^*$-algebra of $\Gamma$.
Then $\varepsilon_\pi$ is well-defined if and only if $\ker \tilde\pi \subseteq \ker \tilde 1_\Gamma$, which in turn is equivalent to $1_\Gamma$ being weakly contained in $\pi$ (see e.g.\ \cite[Theorem F.4.4]{BHV}).
The second statement is trivial.

\smallskip\noindent
(b) Part (a) implies that $\pi$ does not have a spectral gap if and only if there is $\phi\in \mathcal{L}\big((K^\pi)^\bot\big)^*_+$ satisfying $\phi(\pi_t^0) = 1$ for each $t\in \Gamma$.
This, in turn, is equivalent to the existence of $\varphi\in \big(\mathcal{L}(K^\pi)\oplus \mathcal{L}((K^\pi)^\bot)\big)^*_+$ with $\varphi(\pi_t) = 1$ ($t\in \Gamma$) and $\varphi(p_\pi) = 0$.

\smallskip\noindent
(c) This can be obtained by applying the Cauchy-Schwarz inequality to $\psi\big((1-\pi_t)x\big)$.

\smallskip\noindent
(d) Notice that any of the two statements implies that $p_\pi \neq 0$, and so $\varepsilon_\pi$ is well-defined (because of part (a)).

\smallskip\noindent
$\Rightarrow)$.
Since $p_\pi\notin C^*(\pi)$, the map $\rho:C^*(\pi) \to \mathcal{L}(K)$ defined by  $\rho(x) := x - \varepsilon_\pi(x) p_\pi$ is injective with its image being $0\oplus C^*(\pi^0)$.
Consequently, $\varepsilon_{\pi^0} = \varepsilon_\pi\circ \rho^{-1}$ (we identify $0\oplus C^*(\pi^0)$ with $C^*(\pi^0)$) and is well-defined.
This shows that $1_\Gamma$ is weakly contained in $\pi^0$ (by part (a)).

\smallskip\noindent
$\Leftarrow)$.
By parts (a) and (b), there is $\psi\in \mathcal{L}(K)^*_+$ with $\psi|_{C^*(\pi)} = \varepsilon_\pi$ and $\psi(p_\pi) = 0$.
Take any unit vector $\xi\in K^\pi$.
If $p_\pi \in C^*(\pi)$, then $1 = \omega_\xi(p_\pi) = \varepsilon_\pi(p_\pi) = 0$ (as $\varepsilon_\pi = \omega_\xi$ on $C^*(\pi)$), which is absurd.
\end{prf}

Let $N$ be a von Neumann algebra.
We denote by $N_*$ the pre-dual space of $N$, which is naturally identified with the subspace of all weak-$^*$-continuous linear functionals in the dual space $N^*$.
Let
$$N^*_\alpha:= \{f\in N^*: f\circ \alpha_t =f \text{ for any } t\in \Gamma\}, \quad \mathfrak{M}_{N,\alpha} := \{f\in N^*_\alpha: f\geq 0 \text{ and } f(1) =1\}$$
and $\mathfrak{M}^{N,\alpha} := \overline{\mathfrak{M}_{N,\alpha}\cap N_*}^{\sigma(N^*,N)}$ (where $\overline{E}^{\sigma(N^*,N)}$ means the weak-$^*$-closure of a subset $E\subseteq N^*$).
Clearly, $\mathfrak{M}^{N,\alpha}\subseteq \mathfrak{M}_{N,\alpha}$.

The following theorem is a refined version of Theorem \ref{thm:main}.
This result can also be regarded as generalizations of the main results in \cite{Pasch} and in \cite{LH} (see also \cite[Proposition 3.4.1]{Lub}).

\begin{thm}\label{thm:sp-gap-act-inv-mean}
Let $\alpha$ be an action of a group $\Gamma$ on a von Neumann algebra $N$  with standard form $(N, \mathfrak{H}, \mathfrak{J}, \mathfrak{P})$.
Consider the following statements.

\begin{enumerate}[(G1)]
\item If $\psi\in \mathcal{L}(\mathfrak{H})_+^*$ satisfies $\psi(\mathcal{U}_{\alpha_t}) = 1$ ($t\in \Gamma$), then $\psi(p_{\mathcal{U}_{\alpha}}) \neq 0$.
\item $\alpha$ has a spectral gap.
\item $p_{\mathcal{U}_{\alpha}}\in C^*(\mathcal{U}_{\alpha})$
\item $\mathfrak{M}_{N,\alpha} = \mathfrak{M}^{N,\alpha}$.
\end{enumerate}

\noindent
One has $(G1) \Leftrightarrow (G2) \Rightarrow (G3)$ and $(G2)\Rightarrow (G4)$.
\end{thm}
\begin{prf}
$(G1) \Leftrightarrow (G2)$.
This follows from Lemma \ref{lem:p_pi}(b).

\smallskip\noindent
$(G2) \Rightarrow (G3)$.
This follows from Lemma \ref{lem:p_pi}(d).

\smallskip\noindent
$(G2) \Rightarrow (G4)$.
Let $m\in N^*_+$ be an $\alpha$-invariant state.
If $\{f_i\}_{i\in I}$ is a net of states in $N_*$ that $\sigma(N^*, N)$-converges to $m$, the ``convergence to invariance'' type argument (see e.g.\ \cite[p.33-34]{Green}) will produce a net $\{g_j\}_{j\in J}$ in the convex hull of $\{f_i\}_{i\in I}$ that $\sigma(N^*, N)$-converges to $m$ and satisfies $\|\alpha_t^*(g_j) - g_j\|_{N_*} \to 0$ for every $t\in \Gamma$.

For each $j\in J$, there is a unique unit vector $\zeta_j\in \mathfrak{P}$ with $g_j = \omega_{\zeta_j}$.
As $\alpha_{t^{-1}}^*(g_j) = \omega_{\mathcal{U}_{\alpha_{t}}(\zeta_j)}$,
we have, by \cite[Lemma 2.10]{Haag-st-form} (and the fact that $\mathcal{U}_{\alpha_{t}}(\mathfrak{P})\subseteq \mathfrak{P}$),
\begin{equation}\label{eqt:Pow-Stor-ineq}
\|\mathcal{U}_{\alpha_{t}}(\zeta_j) - \zeta_j\|^2
\ \leq\  \|\alpha_{t^{-1}}^*(g_j) - g_j\|_{N_*}
\qquad (t\in \Gamma).
\end{equation}

Let $\zeta^0_j := \zeta_j - p_{\mathcal{U}_{\alpha}}(\zeta_j)$.
If $\|\zeta_j^0\| \nrightarrow 0$, a subnet of $\{\zeta_j^0\}_{j\in J}$ will produce an almost $\mathcal{U}_{\alpha}$-invariant unit vector in  $(\mathfrak{H}^{\mathcal{U}_{\alpha}})^\bot$, which contradicts the spectral gap assumption of $\mathcal{U}_{\alpha}$.
Consequently, if we set $\zeta_j^1:= \frac{p_{\mathcal{U}_{\alpha}}(\zeta_j)}{\|p_{\mathcal{U}_{\alpha}}(\zeta_j)\|}$, then $\{\omega_{\zeta_j^1}\}_{j\in J}$ is a net of $\alpha$-invariant states that  $\sigma(N^*, N)$-converges to $m$.
\end{prf}

It is easy to see that $\mathfrak{M}_{N,\alpha}$ spans $N^*_\alpha$ (see e.g.\ \cite[Proposition 2.2]{Pat}).
Moreover, if we denote $N_*^\alpha :=N^*_\alpha\cap N_*$, it is not hard to check that $\mathfrak{M}^{N,\alpha}$ spans $\overline{N_*^\alpha}^{\sigma(N^*, N)}$.
Thus, Statement (G4) is equivalent to the fact that $N^*_\alpha = \overline{N_*^\alpha}^{\sigma(N^*, N)}$.

\begin{lem}\label{lem:H-N_*}
(a) The map sending $\xi \in \mathfrak{H}$ to $\omega_\xi \in N_*$ restricts to a bijection from $\mathfrak{P} \cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ onto the positive part of
$N_*^\alpha$.

\smallskip\noindent
(b) If $\mathfrak{H}^{\mathcal{U}_{\alpha}}$ is finite dimensional, then $N_*^\alpha$ is finite dimensional.

\smallskip\noindent
(c) If $\dim \mathfrak{H}^{\mathcal{U}_{\alpha}} =1$, then $\dim N_*^\alpha = 1$.
\end{lem}
\begin{prf}
(a) This follows from the fact that for any $g\in (N_*^\alpha)_+$,  if $\zeta\in \mathfrak{P}$ is the unique element satisfying $g = \omega_\zeta$, we have $\zeta\in \mathfrak{H}^{\mathcal{U}_{\alpha}}$, by the inequality in \eqref{eqt:Pow-Stor-ineq}.

\smallskip\noindent
(b) For any $\phi\in N_*^\alpha$, there exist $\alpha$-invariant elements $\phi_1,\phi_2,\phi_3,\phi_4\in (N_*^\alpha)_+$ with $\phi = (\phi_1 - \phi_2) + \mathrm{i} (\phi_3 - \phi_4)$.
By part (a), one can find $\zeta_k\in \mathfrak{P}\cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ such that $\phi_k = \omega_{\zeta_k}$ ($k=1,2,3,4$).
Therefore, the linear map from $(\mathfrak{H}^{\mathcal{U}_{\alpha}} \otimes \overline{\mathfrak{H}^{\mathcal{U}_{\alpha}}}) \oplus (\mathfrak{H}^{\mathcal{U}_{\alpha}} \otimes \overline{\mathfrak{H}^{\mathcal{U}_{\alpha}}})$ to $N_*^\alpha$ given by $(\xi_1\otimes \overline{\eta_1}, \xi_2\otimes \overline{\eta_2}) \mapsto \omega_{\xi_1,\eta_1} + \mathrm{i} \omega_{\xi_2,\eta_2}$ is surjective and $N_*^\alpha$ is finite dimensional.

\smallskip\noindent
(c) Clearly, $N_*^\alpha \neq (0)$.
If $\dim N_*^\alpha > 1$, there will be two different norm one elements in $(N_*^\alpha)_+$.
Thus, part (a) gives two norm one elements in $\mathfrak{P} \cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$, which is not possible.
\end{prf}

The following proposition concerns some converses of the implications in Theorem \ref{thm:sp-gap-act-inv-mean}.
Note that the idea of the argument for $p_{\mathcal{U}_{\alpha}}\in vN(\mathcal{U}_{\alpha})$ in part (b) comes from \cite{Pasch}.

\begin{prop}\label{prop:inv-mean-sp-gap-act}
Let $\alpha$ be an action of a group $\Gamma$ on a von Neumann algebra $N$.

\smallskip\noindent
(a) If $p_{\mathcal{U}_{\alpha}}\neq 0$, then Statement (G3) implies Statement (G1).

\smallskip\noindent
(b) If $p_{\mathcal{U}_{\alpha}}\in N$, then Statement (G4) implies Statement (G1) and $p_{\mathcal{U}_{\alpha}}$ belongs to the center $Z\big(vN(\mathcal{U}_{\alpha})\big)$ of $vN(\mathcal{U}_{\alpha})$.
\end{prop}
\begin{prf}
(a) This follow from Lemma \ref{lem:p_pi}(d).

\smallskip\noindent
(b) Suppose that Statement (G4) holds, but there exists $\psi\in \mathcal{L}(\mathfrak{H})^*_+$ with $\psi(\mathcal{U}_{\alpha_t}) = 1$ ($t\in \Gamma$) and $\psi(p_{\mathcal{U}_{\alpha}}) = 0$.
Then Lemma \ref{lem:p_pi}(c) implies that $\psi|_{N}$ is $\alpha$-invariant, and Statement (G4) gives a net $\{g_j\}_{j\in J}$ of states in $N_*^\alpha$ that $\sigma(N^*,N)$-converges to $\psi|_N$.
By Lemma \ref{lem:H-N_*}(a), one can find $\zeta_j\in \mathfrak{P}\cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ with $g_j = \omega_{\zeta_j}$, which means that $g_j(p_{\mathcal{U}_{\alpha}}) = 1$ ($j\in J$).
This contradicts $\psi(p_{\mathcal{U}_{\alpha}}) = 0$.

For the second conclusion, it is clear that $p_{\mathcal{U}_{\alpha}}\in vN(\mathcal{U}_{\alpha})'$ because $\mathfrak{H}^{\mathcal{U}_{\alpha}}$ is $\mathcal{U}_{\alpha}$-invariant.
Suppose on the contrary that $p_{\mathcal{U}_{\alpha}}\notin vN(\mathcal{U}_{\alpha})$.
Since $\mathbb{C} p_{\mathcal{U}_{\alpha}}$ is one dimensional and
\begin{equation}\label{eqt:rel-p}
\mathcal{U}_{\alpha_t} p_{\mathcal{U}_{\alpha}}\ =\ p_{\mathcal{U}_{\alpha}} \qquad (t\in \Gamma),
\end{equation}
one sees that $vN(\mathcal{U}_{\alpha}) + \mathbb{C} p_{\mathcal{U}_{\alpha}}$ is a von Neumann algebra.
Moreover, if $\xi\in \mathfrak{H}^{\mathcal{U}_{\alpha}}$ is a norm one vector (note that we assume $p_{\mathcal{U}_{\alpha}}\notin vN(\mathcal{U}_{\alpha})$), we define a functional on $vN(\mathcal{U}_{\alpha}) + \mathbb{C} p_{\mathcal{U}_{\alpha}}$ by
$$\phi(x+ c p_{\mathcal{U}_{\alpha}})\ :=\ \omega_{\xi}(x)
\qquad (x\in vN(\mathcal{U}_{\alpha}); c\in \mathbb{C}).$$
Clearly, $\phi$ is weak-$^*$-continuous and it is a $^*$-homomorphism since $xp_{\mathcal{U}_{\alpha}}\in \mathbb{C} p_{\mathcal{U}_{\alpha}}$ (because of \eqref{eqt:rel-p}).
If $\psi$ is a normal state extension of $\phi$ on $\mathcal{L}(\mathfrak{H})$, then $\psi(p_{\mathcal{U}_{\alpha}}) = 0$ and Lemma \ref{lem:p_pi}(c) implies that $\psi|_N$ is an $\alpha$-invariant normal state.
Now, Lemma \ref{lem:H-N_*}(a) produces $\zeta\in \mathfrak{P}\cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ with $\psi|_N = \omega_{\zeta}$ and we have the contradiction that $\psi(p_{\mathcal{U}_{\alpha}}) = 1$.
\end{prf}

Our next example shows that Statement (G4) does not imply Statement (G2) in general.
We first set some notation and recall some facts.
Let $\mathcal{I}_N$ be the inner automorphism group of $N$ and $\beta$ be the canonical action of $\mathcal{I}_N$ on $N$.
Then $\beta$-invariant states on $N$ are precisely tracial states.
Suppose, in addition, that $N$ admits a normal faithful tracial state $\tau$.
If $(H_\tau, \Psi_\tau)$ is the GNS construction with respect to $\tau$ and $\Lambda_\tau: N \to H_\tau$ is the canonical map, then $\mathfrak{H} \cong H_\tau$ and the canonical action of $N$ on $\mathfrak{H}$ can be identified with $\Psi_\tau$.
For any $g\in {\rm Aut}(N)$, one has $\mathcal{U}_{\beta_g}(\Lambda_\tau (x)) = \Lambda_\tau(g(x))$ ($x\in N$).

\begin{eg}\label{eg:4-not-2}
Suppose that $\lambda$ is the left regular representation of $\Gamma$ and put $L(\Gamma):= vN(\lambda)$.
Then $\mathfrak{H} = \ell^2(\Gamma)$ and we let $\mathcal{I}_{L(\Gamma)}$ and $\beta$ be as in the above.

\smallskip\noindent
(a) The representation $\mathcal{U}_{\beta}\circ {\rm Ad}\circ \lambda: \Gamma\to \mathcal{L}(\ell^2(\Gamma))$ coincides with the ``conjugate representation'' $\gamma$ defined by  $\gamma_t(\xi)(s) := \xi (t^{-1}st)$ ($s,t\in \Gamma; \xi\in \ell^2(\Gamma)$).

\smallskip\noindent
(b) Suppose that $\Gamma$ is an amenable countable ICC group.
Since $L(\Gamma)$ is a type $I\!I_1$-factor, it has only one tracial state and this state is normal.
Consequently, $\mathfrak{M}_{L(\Gamma), \beta} = \mathfrak{M}^{L(\Gamma), \beta}$ (because $\beta$-invariant states are precisely tracial states).

On the other hand, we have $\mathfrak{H}^{\mathcal{U}_\beta} = \mathbb{C}\delta_e$.
Moreover, as $L(\Gamma)$ is semi-discrete, it has property Gamma (see \cite[Corollary 2.2]{Con}), and the restriction of $\mathcal{U}_{\beta}$ on $(\mathfrak{H}^{\mathcal{U}_\beta})^\bot$ weakly contains the trivial representation (by an equivalent form of property Gamma in \cite[Theorem 2.1(c)]{Con}).
Thus, the representation $\mathcal{U}_\beta$ (and hence the action $\beta$) does not have a spectral gap.
\end{eg}

\section{Invariant States and Inner Amenability (Theorem \ref{thm:inner})}

In this section, we will give an application of Theorem \ref{thm:sp-gap-act-inv-mean} to inner amenability.
We recall (from the main theorem in \cite{Eff}) that an ICC group $\Gamma$ is inner amenable if and only if there is a net $\{\xi_i\}_{i\in I}$ of unit vectors in $\ell^2(\Gamma\setminus \{e\})$ such that $\|\gamma_t(\xi_i) - \xi_i\| \to 0$ for any $t\in \Gamma$ (where $\gamma$ is as in Example \ref{eg:4-not-2}(a)).
Notice that this inner amenability is slightly different from the one in \cite{LP} and \cite{Pat}, and it is called ``non-trivially inner amenable'' in \cite[p.84]{Pat}.

Let us consider another extension of inner amenability to general (not necessarily ICC) discrete groups.

\begin{defn}
$\Gamma$ is said to be \emph{strongly inner amenable} if the conjugate representation $\gamma$ does not have a spectral gap.
\end{defn}

It is obvious that strong inner amenability implies inner amenability, and the converse holds when $\Gamma$ is an ICC group.
On the other hand, all abelian groups and all property $(T)$ groups are not strongly inner amenable.

\begin{prop}\label{prop:st-inner}
Let $\alpha$ be the action of $\Gamma$ on $\mathcal{L}(\ell^2(\Gamma))$ given by $\alpha_t(x) := \gamma_t x \gamma_{t^{-1}}$, $\Gamma_{\rm fin}$ be the normal subgroup consisting of elements in $\Gamma$ with finite conjugacy classes and $A(\Gamma)$ be the predual of $L(\Gamma)$.

\smallskip\noindent
(a) Consider the following statements.

\begin{enumerate}[(S1)]
\item $\Gamma/\Gamma_{\rm fin}$ is inner amenable.
\item $\ell^\infty(\Gamma)^*_\alpha$ is not the $\sigma(\ell^\infty(\Gamma)^*, \ell^\infty(\Gamma))$-closure of $\ell^1(\Gamma)^\alpha$.
\item $\Gamma$ is strongly inner amenable.
\item There is $\psi\in \mathcal{L}(\ell^2(\Gamma))_+^*$ satisfying $\psi(\gamma_t) = 1$ ($t\in \Gamma$) and $\psi(p_\gamma) = 0$.
\item $p_\gamma\notin C^*(\gamma)$.
\item $L(\Gamma)^*_\alpha$ is not the $\sigma(L(\Gamma)^*, L(\Gamma))$-closure of $A(\Gamma)^\alpha$.
\item $\mathfrak{M}_{L(\Gamma),\alpha}$ does not coincide with the set of tracial states on $L(\Gamma)$.
\end{enumerate}

\noindent
Then $(S1) \Rightarrow (S2) \Rightarrow (S3) \Leftrightarrow (S4) \Leftrightarrow (S5)$ and  $(S6)\Leftrightarrow (S7) \Rightarrow (S3)$.

\smallskip\noindent
(b) If $\Gamma_{\rm fin}$ coincides with the center of $\Gamma$, then $(S5) \Rightarrow (S1)$ and $p_\gamma\in Z(vN(\gamma))$.

\smallskip\noindent
(c) If $\Gamma_{\rm fin}$ is finite, we have $(S5) \Rightarrow (S1)$.
\end{prop}
\begin{prf}
Notice that if $x\in L(\Gamma)$, then $\alpha_t(x) = \lambda_t x\lambda_t^*\in L(\Gamma)$.
On the other hand, if $y\in \ell^\infty(\Gamma)$ (considered as a subalgebra of $\mathcal{L}(\ell^2(\Gamma))$ in the canonical way), then $\alpha_t(y)\in \ell^\infty(\Gamma)$ and $\alpha_t(y)(s) = y(t^{-1}st)$.
Moreover, if we either set
$$N = \ell^\infty(\Gamma) \quad \text{or} \quad N = L(\Gamma)$$
(and we denote by $\alpha$ the restriction of $\alpha$ on $N$ by abuse of notation), then $\mathfrak{H} = \ell^2(\Gamma)$ and $\mathcal{U}_{\alpha}$ coincides with $\gamma$ (even though $\mathfrak{J}$ and $\mathfrak{P}$ are different in these two cases).

\smallskip\noindent
(a) By Theorem \ref{thm:sp-gap-act-inv-mean} and Proposition \ref{prop:inv-mean-sp-gap-act}(a) (for both $N=\ell^\infty(\Gamma)$ and $N= L(\Gamma)$), it remains to show that $(S1)\Rightarrow (S2)$ and $(S6)\Leftrightarrow (S7)$.

\noindent
$(S1)\Rightarrow (S2)$.
Since $\Gamma/\Gamma_{\rm fin}$ is inner amenable, we know from \cite[Corollary 1.4]{Kan-Mark} as well as the argument of Lemma \ref{lem:p_pi}(c) that there is an $\alpha$-invariant mean $m$ on $\Gamma$ such that $m(\chi_{\Gamma_{\rm fin}}) \neq 1$.
Obviously, $\ell^2(\Gamma)^\gamma \subseteq \ell^2(\Gamma_{\rm fin})$ and Lemma \ref{lem:H-N_*}(a) implies that $g(\chi_{\Gamma_{\rm fin}}) = \|g\|$ if $g\in \ell^1(\Gamma)^\alpha_+$.
Consequently, $m\notin \mathfrak{M}^{\ell^\infty(\Gamma),\alpha}$.

\noindent
$(S6)\Rightarrow (S7)$.
Assume that $\mathfrak{M}_{L(\Gamma),\alpha}$ coincides with the set of all tracial states on $L(\Gamma)$.
It is well-known that every tracial state on $L(\Gamma)$ is a weak-$^*$-limit of a net of normal tracial states (see e.g.\ \cite[Proposition 8.3.10]{KRII}).
Thus, $\mathfrak{M}_{L(\Gamma),\alpha} = \mathfrak{M}^{L(\Gamma),\alpha}$, which contradicts Statement (S6).

\noindent
$(S7)\Rightarrow (S6)$.
Clearly, any tracial state on $L(\Gamma)$ is $\alpha$-invariant.
Moreover, $\omega\in A(\Gamma)$ is $\alpha$-invariant if and only if $$\omega(\lambda_t x)\ =\ \omega(x\lambda_{t}) \qquad (x\in L(\Gamma); t\in \Gamma),$$
which is equivalent to $\omega(yx) = \omega(xy)$ ($x,y\in L(\Gamma)$).
Hence, $\mathfrak{M}_{L(\Gamma),\alpha}\cap A(\Gamma)$ is the set of all normal tracial states on $L(\Gamma)$.
Consequently, if $L(\Gamma)^*_\alpha$ is the weak-$^*$-closure of $A(\Gamma)^\alpha$, then $\mathfrak{M}_{L(\Gamma),\alpha}$ coincides with the set of tracial states, which contradicts Statement (S7).

\smallskip\noindent
(b) The assumption implies that every finite conjugacy class of $\Gamma$ is a singleton set.
Consequently, one has $\ell^2(\Gamma)^\gamma = \ell^2(\Gamma_{\rm fin})$ and $p_\gamma = \chi_{\Gamma_{\rm fin}}\in \ell^\infty(\Gamma)$.
Thus, $p_\gamma\in Z(vN(\gamma))$ by Proposition \ref{prop:inv-mean-sp-gap-act}(b).
Furthermore, Statement (S4) produces an $\alpha$-invariant mean $m$ on $\Gamma$ (see Lemma \ref{lem:p_pi}(c)) such that $m(p_\gamma) =0$.
Now, by \cite[Corollary 1.4]{Kan-Mark}, $\Gamma/\Gamma_{\rm fin}$ is inner amenable.

\smallskip\noindent
(c) Let $C_1,...,C_n$ be all the finite conjugacy classes of $\Gamma$.
Let $p_k\in \mathcal{L}(\ell^2(\Gamma))$ be the orthogonal projection onto $\ell^2(C_k)$ ($k=1,...,n$) and $p_0\in \mathcal{L}(\ell^2(\Gamma))$ be the orthogonal projection onto $\ell^2(\Gamma\setminus \Gamma_{\rm fin})$.
Then
$$\ell^2(\Gamma)^\gamma = \{\xi\in \ell^2(\Gamma): p_0\xi = 0 \text{ and } p_k\xi \text{ is a constant function for every } k=1,...,n\}.$$
Suppose that $\{\xi_i\}_{i\in I}$ is an almost $\gamma$-invariant unit vector in $(\ell^2(\Gamma)^\gamma)^\bot$.
One can find a subnet $\{\xi_{i_j}\}_{j\in J}$ such that $\{p_k\xi_{i_j}\}_{j\in J}$ is norm-converging to some $\xi^{(k)} \in \ell^2(C_k)$ with $\sum_{s\in C_k}\xi^{(k)}(s) = 0$ (for $k =1,...,n$).
As $\|\gamma_t(p_k \xi_{i_j}) - p_k \xi_{i_j}\|\to 0$, we see that $\gamma_t \xi^{(k)} = \xi^{(k)}$ ($t\in \Gamma$), which means that $\xi^{(k)} = 0$.
Consequently, $\|p_0\xi_{i_j}\| \to 1$ and a subnet of $\{\omega_{p_0\xi_{i_j}}\}_{j\in J}$ will produce a $\gamma$-invariant mean $m$ satisfying $m(\chi_{\Gamma_{\rm fin}}) =0$.
Now, Statement (S1) follows from \cite[Corollary 1.4]{Kan-Mark}.
\end{prf}

The following theorem concerns the case when there is a unique inner invariant state on $L(\Gamma)$.
Part (b) of which gives Theorem \ref{thm:inner}.

\begin{thm}\label{thm:ICC}
Let $\alpha$ be as in Proposition \ref{prop:st-inner}.

\smallskip\noindent
(a) If $\Gamma$ is amenable, then there are more than one $\alpha$-invariant states on $L(\Gamma)$.

\smallskip\noindent
(b) Suppose that $\Gamma$ is an ICC group.
If $\Gamma$ is not inner amenable, there is a unique $\alpha$-invariant state on $L(\Gamma)$.
The converse holds when $\Gamma$ is finitely generated.
\end{thm}
\begin{prf}
(a) Suppose that $n\in \ell^\infty(\Gamma)^*_+$ is an invariant mean.
Using the ``convergence to invariance'' argument, there is a net $\{g_j\}_{j\in J}$ of norm one elements in $\ell^1(\Gamma)_+$ such that $\|t\cdot g_j - g_j\|_{\ell^1(\Gamma)} \to 0$ and $\|g_j\cdot t- g_j\|_{\ell^1(\Gamma)} \to 0$, for every $t\in \Gamma$ (where $(t\cdot g_i)(s) = g_i(t^{-1}s)$ and  $(g_i\cdot t)(s) = g_i(st)$).
Set $\eta_j := g_j^{1/2}\in \ell^2(\Gamma)$ ($j\in J$).
Then
\begin{equation*}\label{eqt:left-and-adj-inv}
\|\lambda_t(\eta_j) - \eta_j\|_{\ell^2(\Gamma)} \to 0
\qquad \text{and}\qquad
\|\gamma_t(\eta_j) - \eta_j\|_{\ell^2(\Gamma)} \to 0,
\end{equation*}
If $\phi$ is the $\sigma(L(\Gamma)^*, L(\Gamma))$-limit of a subnet of $\{\omega_{\eta_j}\}_{j\in J}$, then $\phi$ is an $\alpha$-invariant state on $L(\Gamma)$ but $\phi\neq \omega_{\delta_e}$ because $\phi(\lambda_t) = 1$ for all $t\in \Gamma$.

\smallskip\noindent
(b) Since $\Gamma$ is an ICC group, one has $\mathfrak{H}^{\mathcal{U}_{\alpha}} = \mathbb{C} \delta_e$ and $A(\Gamma)^\alpha = \mathbb{C} \omega_{\delta_e}$.
The first statement follows from Proposition \ref{prop:st-inner}(a).

To show the second statement, we suppose, on the contrary, that $\Gamma$ has a finite generating subset $\{t_1,...,t_n\}$, $\omega_{\delta_e}$ is the only $\alpha$-invariant state on $N:=L(\Gamma)$, but $\gamma$ does not have a spectral gap.
Then there is a sequence $\{\xi_k\}_{k\in \mathbb{N}}$ of unit vectors in $\ell^2(\Gamma)$ satisfying
$$\langle \xi_k, \delta_e\rangle\ =\ 0 \qquad \text{and} \qquad  \|\lambda_{t_j}(\rho_{t_j}(\xi_k)) -\xi_k\| \to 0 \quad (j=1,...,n)$$
(notice that $\gamma = \lambda\circ \rho$, where $\rho$ is the right regular representation of $\Gamma$).
Set $u_j := \lambda_{t_j}\in N$ ($j=1,...,n$) and consider $\mathfrak{F}$ to be a free ultrafilter on $\mathbb{N}$.
Since the relative commutant of $F:=\{u_1,...,u_n\}$ in $N$ is the same as the relative commutant of $F$ in the ultrapower $N^\mathfrak{F}$ (by the argument of \cite[Lemma 2.6]{Con}; see lines 3 and 4 of \cite[p.86]{Con}), the relative commutant of $F$ in $N^\mathfrak{F}$ is $\mathbb{C}$.
Now, one may employ the argument for Case (1) of ``(c) $\Rightarrow$ (b)'' in \cite[p.87]{Con} to conclude that there exists a subsequence of $\{\omega_{\xi_k}\}_{k\in \mathbb{N}}$ which is not relatively $\sigma(N_*,N)$-compact in $N_*$.
Hence, this subsequence has a subnet that $\sigma(N^*,N)$-converges to a non-normal state $\varphi\in N^*$.
As $\varphi$ is invariant under $\{{\rm Ad}\ \! \gamma_{t_1}, ..., {\rm Ad}\ \! \gamma_{t_n}\}$, it is $\alpha$-invariant and we obtain a contradiction.
\end{prf}

\begin{rem}\label{rem:icc}
(a) It was shown in \cite{Pasch} that inner amenability of an ICC group $\Gamma$ is equivalent to $p_\gamma\notin C^*(\gamma)$.
However, Proposition \ref{prop:st-inner}(b) tells us that $p_\gamma\in vN(\gamma)$ for any ICC group $\Gamma$, whether or not it is inner amenable.

\smallskip\noindent
(b) A functional $\varphi\in L(\Gamma)^*$ is $\alpha$-invariant if and only if $\varphi(ax) = \varphi(xa)$ ($x\in L(\Gamma);a\in C^*(\lambda) = C^*_r(\Gamma)$).
Hence, the difference between $\alpha$-invariant states and tracial states on $L(\Gamma)$ is similar to the different between inner amenability of $\Gamma$ and property Gamma of $L(\Gamma)$ (see e.g.\ \cite[p.394]{Vaes}).
It is shown in Proposition \ref{prop:st-inner}(a) that if $\Gamma$ is not strongly inner amenable, all $\alpha$-invariant states on $L(\Gamma)$ are tracial states.
On the other hand, if $\Gamma$ is an amenable ICC group, then $L(\Gamma)$ has only one tracial state, but there exist more than one $\alpha$-invariant states on $L(\Gamma)$ (by Theorem \ref{thm:ICC}(a)).
\end{rem}

\section{Invariant States and Property ($T$) (Theorem \ref{thm:property-T})}

Although Statement (G4) does not imply Statement (G2) for individual action, if Statement (G4) holds for every action, then so does Statement (G2).
This can be seen in the following result, which is a more elaborated version of Theorem \ref{thm:property-T}.

\begin{thm}\label{thm:equiv-prop-T}
Let $\Gamma$ be a countable discrete group.
The following statements are equivalent.

\begin{enumerate}[(T1)]
\item $\Gamma$ has property $(T)$.
\item If $\alpha$ is an action of $\Gamma$ on a von Neumann algebra $N$ and $\psi\in \mathcal{L}(\mathfrak{H})_+^*$ satisfying $\psi(\mathcal{U}_{\alpha_t}) = 1$ ($t\in \Gamma$), one has $\psi(p_{\mathcal{U}_{\alpha}}) \neq 0$.
\item $\mathfrak{M}_{N, \alpha} = \mathfrak{M}^{N, \alpha}$ for every action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$.
\item For each action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$ with $\dim N_*^\alpha  =1$, there is only one $\alpha$-invariant state on $N$.
\item For any Hilbert space $K$ and any action $\alpha$ of $\Gamma$ with $\mathcal{L}(K)_*^\alpha  = (0)$, there is no $\alpha$-invariant state on $\mathcal{L}(K)$.
\end{enumerate}
\end{thm}
\begin{prf}
Theorem \ref{thm:sp-gap-act-inv-mean} gives $(T1) \Rightarrow (T2) \Rightarrow (T3)$.
Moreover, it is clearly that $(T3) \Rightarrow (T4)$ and $(T3) \Rightarrow (T5)$. We need $(T4) \Rightarrow (T1)$ and $(T5) \Rightarrow (T1)$.

\smallskip\noindent
$(T4) \Rightarrow (T1)$.
This follows from \cite[Theorem 2.5]{Sch}  (by considering the case when $N= L^\infty(X,\mu)$, where $(X,\mu)$ is a fixed non-atomic standard probability space, and $\alpha$ runs through all ergodic actions of $\Gamma$ on $(X,\mu)$).

\smallskip\noindent
$(T5) \Rightarrow (T1)$.
Suppose that $\Gamma$ does not have property $(T)$.
By \cite[Theorem 1]{BV}, there is a unitary representation $\mu: \Gamma\to \mathcal{L}(K)$ such that $\mu$ does not have a nonzero finite dimensional subrepresentation and $\pi:=\mu\otimes \overline{\mu}$ weakly contains $1_\Gamma$.
It is easy to see, by using \cite[Proposition A.1.12]{BHV}, that $\pi$ does not have a nonzero finite dimensional subrepresentation as well.

Let $N := \mathcal{L}(K\otimes \overline{K})$ and $\alpha := {\rm Ad}\ \!\pi$.
Then $\mathfrak{H} = K\otimes \overline{K}\otimes K\otimes \overline{K} = \mathcal{H}\mathcal{S}(K\otimes \overline{K})$ (where $\mathcal{H}\mathcal{S}$ denote the space of all Hilbert-Schmidt operators), $\mathcal{U}_{\alpha} = \pi \otimes \overline{\pi}$ and the $^*$-representation of $N$ on $\mathfrak{H}$ is given by compositions.
As $\pi$ does not have any nonzero finite dimensional subrepresentation, we see that $\mathfrak{H}^{\mathcal{U}_{\alpha}} = (0)$ (again by \cite[Proposition A.1.12]{BHV}).
On the other hand, if $\{\xi_i\}_{i\in I}$ is an almost $\pi$-invariant unit vector in $K\otimes \overline{K}$, the $\sigma(N^*,N)$-limit of a subnet of $\{\omega_{\xi_i}\}_{i\in I}$ will produce an $\alpha$-invariant state on $N$.
This contradicts Statement (T5).
\end{prf}

\begin{rem}
One may regard Theorem \ref{thm:equiv-prop-T} as a sort of non-commutative analogue of \cite[Theorem 2.5]{Sch}.
There is also a non-commutative analogue of \cite[Theorem 2.4]{Sch}, which is basically a reformulation of \cite[Theorem 2.2]{B}.
More precisely, the following are equivalence for a discrete group $\Gamma$:
\begin{enumerate}[({A}1)]
\item $\Gamma$ is amenable.
\item $\mathfrak{M}_{N, \alpha} \neq \emptyset$ for every action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$.
\item For any action $\alpha$ of $\Gamma$ on $\mathcal{L}(K)$ (where $K$ is a Hilbert space) with $\mathcal{L}(K)_*^\alpha = (0)$, there exists an $\alpha$-invariant state on $\mathcal{L}(K)$.
\end{enumerate}
In fact, if $\Gamma$ is amenable, then \cite[Theorem 5.1]{B} implies that $\mathcal{U}_\alpha\otimes \overline{\mathcal{U}_\alpha}$ weakly contains $1_\Gamma$.
Thus, if we identify $N$ with the subalgebra $N\otimes 1$ of $\mathcal{L}(\mathfrak{H}\otimes \mathfrak{H})$, an almost $\mathcal{U}_\alpha\otimes \overline{\mathcal{U}_\alpha}$-invariant unit vector will produce an $\alpha$-invariant state on $N$.
To show (A3) $\Rightarrow$ (A1), one may consider the action $\alpha := {\rm Ad}\ \! \lambda$ on $\mathcal{L}(\ell^2(\Gamma))$.
It is easy to see that $\mathcal{L}(\ell^2(\Gamma))^\alpha_* = (0)$ (e.g.\ by Lemma \ref{lem:H-N_*}(a)) and an $\alpha$-invariant state will restricts to a left invariant state on $\ell^\infty(\Gamma)$, which implies that $\Gamma$ is amenable.
\end{rem}

Recall from \cite{LZ89} that $\Gamma$ has property $(T,FD)$ if $1_\Gamma$ is not in the closure of the subset of $\hat \Gamma$ consisting of non-trivial finite dimensional irreducible representations.
\begin{prop}\label{prop:no-inv-state}
Let $\Gamma$ be an infinite discrete group with property $(T,FD)$.
Then (T1) is equivalent to the following statement.
\begin{enumerate}[(T5')]
\item For any action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$ with $N_*^\alpha = (0)$, there is no $\alpha$-invariant state on $N$.
\end{enumerate}
\end{prop}
\begin{prf}
By Theorem \ref{thm:sp-gap-act-inv-mean}, we have (T1)$\Rightarrow$(T5').
Now, suppose that $\Gamma$ does not have property $(T)$.
As $\Gamma$ has property $(T,FD)$, there is a net $\{(\pi^i,K^i)\}_{i\in I}$ in $\hat \Gamma\setminus\{1_\Gamma\}$ with each $K^i$ being infinite dimensional and there is a unit vector $\xi_i\in K^i$ ($i\in I$) such that
$$\|\pi_t^i\xi_i - \xi_i\|\ \to\ 0
\qquad (t\in \Gamma).$$
Let $N$ be the von Neumann algebra $\bigoplus_{i\in I} \mathcal{L}(K^i)$ and set $\pi:=\bigoplus_{i\in I}\pi^i$ as well as $\alpha := {\rm Ad}\ \!\pi$.
Then $\mathfrak{H} = \bigoplus_{i\in I} \mathcal{H}\mathcal{S}(K^i)\cong \bigoplus_{i\in I} K^i\otimes \overline{K^i}$ and the representation of $N$ on $\mathfrak{H}$ is given by compositions.
In this case, one has
$$\mathcal{U}_{{\rm Ad}w}\big((\zeta^i \otimes \overline{\eta^i}\big)_{i\in I}\big)\ =\ \big(w_i\zeta^i \otimes \overline{w_i\eta^i}\big)_{i\in I}
\qquad \Big(w = (w_i)_{i\in I}\in {\bigoplus}_{i\in I}U(K^i); (\zeta^i \otimes \overline{\eta^i})_{i\in I}\in \mathfrak{H} \Big),$$
where $U(K^i)$ is the group of unitaries  on $K^i$.
Thus, $\mathcal{U}_{\alpha_t}(y) = \pi_t \circ y \circ \pi_{t^{-1}}$ ($y\in \mathfrak{H}; t\in \Gamma$), which implies that
$$\mathfrak{H}^{\mathcal{U}_{\alpha}}\ \subseteq\ {\bigoplus}_{i\in I}\{S_i\in \mathcal{H}\mathcal{S}(K^i): S_i\pi^i_t = \pi^i_t S_i,\ \! \forall t\in \Gamma\}$$
and hence $\mathfrak{H}^{\mathcal{U}_{\alpha}} = (0)$.

On the other hand, the $\sigma(N^*,N)$-limit of a subnet of  $\{\omega_{\xi_i}\}_{i\in I}$ (we consider $\mathcal{L}(K^i)\subseteq N$ for all $i\in I$) will give an $\alpha$-invariant state on $N$.
\end{prf}

A similar statement as the above for the strong property $(T)$ of locally compact groups can be found in \cite{LN}.

\begin{cor}
Let $\Gamma$ be a minimally almost periodic group in the sense of \cite{vNW40} (i.e.\ there is no non-trivial finite dimensional irreducible representation of $\Gamma$).
If $\Gamma$ satisfies (T5'), then it has property (T) and hence is finitely generated.
\end{cor}

\section*{Acknowledgement}
We would like to thank the referee for helpful comments and suggestions that lead to a better presentation of the article, and for informing us that one can use the materials in \cite{BV} to obtain (T5) $\Rightarrow$ (T1) for general countable groups (we only had Proposition \ref{prop:no-inv-state} in the original version.)
We would also like to thank Prof. Roger Howe for helpful discussion leading to a simpler argument for Theorem \ref{thm:equiv-prop-T}.

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\section{Introduction}
The physics of $F=2$ spinor Bose-Einstein condensates (BECs) started to gain the attention of both theorists and experimentalists during the last decade. The interest was motivated by the structure of $F=2$ condensates: being more complex than that of $F=1$ condensates, it made  possible properties and phenomena which are not present in an $F=1$ system.
One example of this can been seen in the structure of the ground states.
The energy functional of an $F=2$ condensate is characterized by one additional degree of freedom compared to the $F=1$ case.
This leads to a rich ground state manifold
as now there are two free parameters parametrizing the ground states \cite{Ciobanu00,Zheng10}.
This should be contrasted with an $F=1$ condensate, where the ground state is determined by the sign
of the spin-dependent interaction term \cite{Ho98,Ohmi98}.
Another difference can be seen in the structure of topological defects. It has been shown that non-commuting vortices can exist in an $F=2$ condensate \cite{Makela03}, while these are not possible in an $F=1$ BEC \cite{Ho98,Makela03}. The topological defects of $F=2$ condensates have been studied further by the authors of Refs. \cite{Makela06,Huhtamaki09,Kobayashi09}.
Experimental studies of $F=2$ BECs have been advancing in the past ten years.
Experiments on $F=2$ ${}^{87}$Rb atoms
cover topics such as spin dynamics \cite{Schmaljohann04,Chang04,Kuwamoto04,Kronjager06,Kronjager10}, creation of skyrmions \cite{Leslie09a}, spin-dependent inelastic collisions \cite{Tojo09}, amplification of fluctuations \cite{Klempt09,Klempt10}, spontaneous breaking of spatial and spin symmetry \cite{Scherer10}, and atomic homodyne detection \cite{Gross11}. An $F=2$ spinor condensate of ${}^{23}$Na atoms has been obtained experimentally \cite{Gorlitz03}, but it has a much shorter lifetime than $F=2$ rubidium condensates.

In this work, we study the dynamical stability of nonstationary states of homogeneous
$F=2$ spinor condensates. The stability of stationary states has been examined
both experimentally \cite{Klempt09,Klempt10,Scherer10} and theoretically \cite{Martikainen01,Ueda02}.
Interestingly, the experimental studies show that the observed instability of the $|m_F=0\rangle$ state can be used to amplify vacuum fluctuations \cite{Klempt10} and to analyze symmetry breaking \cite{Scherer10}
(see Refs. \cite{Lamacraft07,Leslie09b} for related studies in an $F=1$ system).
The stability of nonstationary states of spinor condensates, on the other hand, has received only little attention. Previous studies on the topic  concentrate on $F=1$ condensates  \cite{Matuszewski08,Matuszewski09,Matuszewski10,Zhang05,Makela11}.
Here we extend the analysis of the authors of Ref. \cite{Makela11} to an $F=2$ rubidium condensate and present results
concerning the magnetic field dependence of the excitation spectrum and stability.
Although we concentrate on the stability of ${}^{87}$Rb condensates, many of the excitation spectra and stability conditions given in this article are not specific to rubidium condensates but have a wider applicability.
We show that, in comparison with an $F=1$ system, the stability analysis of an $F=2$ condensate is considerably
more complicated. This is partly due to the presence of a spin-singlet term in the energy functional of the latter system, but the main reason for the increased complexity is seen to be the much larger number of states available in an $F=2$ condensate.

This article is organized as follows. Section \ref{sec:overview} introduces the system and presents the Hamiltonian and equations of motion. In Sec. \ref{sec:stability} the Bogoliubov analysis of nonstationary states is introduced. This method is applied to study the stability both in the presence and absence of a magnetic field.
In this section it is also described how Floquet theory can be used in the stability analysis.
In Sec. \ref{sec:g2not0} the stability is studied under the (physically motivated) assumption that one of the interaction coefficients vanishes. Finally, Sec. \ref{sec:conclusions} contains the concluding remarks.

\section{Theory of a spin-2 condensate}\label{sec:overview}

The order parameter of a spin-$2$ Bose-Einstein condensate can be written as $\psi=(\psi_2,\psi_{1},\psi_{0},\psi_{-1},\psi_{-2})^T$, where $T$ denotes the transpose. The normalization is $\sum_{m=-2}^2|\psi_{m}|^2=n$, where $n$ is the total particle density. We assume that the trap confining the condensate is such that all the
components of the hyperfine spin can be trapped simultaneously and are degenerate
in the absence of magnetic field. This can be readily achieved in experiments \cite{Stamper-Kurn98}.
If the system is exposed to an external magnetic field which
is parallel to the $z$ axis, the energy functional reads
\begin{align}
\label{energy}
&E[\psi] =\!\! \int d{\bm r}
\left[ \langle \hat{h}\rangle  +\frac{1}{2}\left(g_0 n^2  + g_1 \langle\hat{\mathbf{F}}\rangle^2
 + g_2 |\Theta|^2\right)\right],
\end{align}
where $\hat{\mathbf{F}}=(\hat{F}_x,\hat{F}_y,\hat{F}_z)$ is the (dimensionless) spin operator of a spin-2 particle.
 $\Theta$ describes singlet pairs and is given by $\Theta=2\psi_2\psi_{-2}-2\psi_1\psi_{-1}+\psi_0^2$. It can also be written as $\Theta=\psi^T e^{-i\pi \hat{F}_y}\psi$. The single-particle Hamiltonian $\hat{h}$ reads
\begin{align}
\label{h}
\hat{h}= -\frac{\hbar^2 \nabla^2}{2m} + U(\mathbf{r})  -\mu-p\hat{F}_z+q\hat{F}_z^2.
\end{align}
Here $U$ is the external trapping potential, $\mu$ is the chemical potential, and
$p=-g\mu_{\rm B}B$ is the linear Zeeman term. In the last of these  $g$ is the Land\'e hyperfine $g$-factor, $\mu_{\rm B}$ is the Bohr magneton, and $B$ is the external magnetic field.
The last term in Eq. (\ref{h}) is  the quadratic Zeeman term, $q=-(g\mu_{\rm B}B)^2/E_{\rm hf}$,   where $E_{\rm hf}$ is the hyperfine splitting. The sign of $q$ can be controlled experimentally   by using a linearly polarized microwave field \cite{Gerbier06}. In this article we consider both
 positive and negative values of $q$.

The strength of the spin-independent interaction is characterized by $g_0=4\pi \hbar^2(4a_2+3a_4)/7m$, whereas $g_1=4\pi \hbar^2(a_4-a_2)/7m$ and $g_2=4\pi\hbar^2[(a_0-a_4)/5-2(a_2-a_4)/7]$ describe spin-dependent scattering.  Here $a_F$ is the $s$-wave scattering length for two atoms colliding with total angular momentum $F$.
In the case of $^{87}$Rb, we calculate $g_0$ using the scattering lengths given in Ref.\ \cite{Ciobanu00}, and $g_2$ and $g_4$ are calculated using the experimentally measured scattering length differences from Ref.\ \cite{Widera06}.

Two important quantities characterizing the state $\psi$ are the spin vector
\begin{align}
\mathbf{f}(\mathbf{r})=  \frac{\psi^\dag(\mathbf{r}) \hat{\mathbf{F}} \psi(\mathbf{r})}{n(\mathbf{r})},
\end{align}
and the magnetization in the direction of the magnetic field
\begin{align}
\label{Mz}
M_z= \frac{\int d\mathbf{r}\,n(\mathbf{r}) f_z (\mathbf{r})}{\int d\mathbf{r}\,n(\mathbf{r})}.
\end{align}
The length of $\mathbf{f}$ is denoted by $f$.
For rubidium the magnetic dipole-dipole interaction is weak and consequently the magnetization
is a conserved quantity. The Lagrange multiplier related to the conservation of magnetization
can be included into $p$.
The time evolution equation obtained from Eq. (\ref{energy}) is
\begin{align}
i\hbar \frac{\partial }{\partial t}\psi =\hat{H}[\psi] \psi,
\end{align}
where
\begin{align}
\label{H}
\hat{H}[\psi]= \hat{h}+ g_0 \psi^\dag\psi +g_1 \langle\hat{\mathbf{F}}\rangle\cdot\hat{\mathbf{F}} + g_2 \Theta \hat{{\mathcal{T}}}.
\end{align}
Here $\hat{{\mathcal{T}}}=e^{-i\pi \hat{F}_y}\hat{C}$ is the time-reversal operator, where $\hat{C}$ is the
complex conjugation operator.

\section{Stability of nonstationary states when $g_2\not=0$}\label{sec:stability}
The stability analysis is performed in a basis where the state in question
is time independent. This requires that the time evolution operator of the state
is known. As we are interested in analytical calculations, an analytical expression for  this operator has to be known. To calculate the time evolution operator analytically, the Hamiltonian has to be time independent. In particular, the singlet term $\Theta$ should not depend on time. This is clearly the case if the time evolution of the state is such that $\Theta$ vanishes at all times, and we now study this case.
We define a state
\begin{align}
\label{eq:psiparallel}
\psi_{2;-1}=
\sqrt{\frac{n}{3}}
\begin{pmatrix}
\sqrt{1+f_z}\\
0\\
0\\
\sqrt{2- f_z}\\
0
\end{pmatrix},\quad -1\leq f_z\leq 2.
\end{align}
For this state $\Theta=0$, $\langle \hat{F}_x\rangle=\langle \hat{F}_y\rangle=0$, and $\langle\hat{F}_z\rangle =f_z$. Furthermore, the populations of the state $\psi_{2;-1}$ remain unchanged during the time evolution
determined by the Hamiltonian (\ref{H}). Consequently, $\Theta=0$ throughout the time evolution.
The state $\psi_{2;-1}$ with $f_z=0$, called the cyclic state, is a ground state at zero magnetic field \cite{Ciobanu00}. The creation of vortices with fractional winding number in states of the form $\psi_{2;-1}$ has been discussed by the authors of Ref. \cite{Huhtamaki09}. The stability properties of the state
$\psi_{1;-2}=\sqrt{n}(0,\sqrt{2+f_z},0,0,\sqrt{1-f_z})^T/\sqrt{3}$ are similar to those of $\psi_{2;-1}$ and will therefore  not be studied separately.

The Hamiltonian giving the time evolution of $\psi_{2;-1}$ is
\begin{align}
\label{eq:Hparallel}
\hat{H}[\psi_{2;-1}]=g_0 n-\mu +(g_1 n f_z - p)\hat{F}_z +q\hat{F}_z^2,
\end{align}
where we have set $U=0$ as the system is assumed to be homogeneous.
This is of the same form as the Hamiltonian of an $F=1$ system discussed  by the authors of Ref. \cite{Makela11}.
The time evolution operator of $\psi_{2;-1}$ is given by
\begin{align}
\label{eq:Uparallel}
\hat{U}_{2;-1}(t)= e^{-i t \hat{H}[\psi_{2;-1}]/\hbar}.
\end{align}

We analyze the stability in a basis where the state $\psi_{2;-1}$ is time independent.
In the new basis, the energy of an arbitrary state $\phi$ is given by \cite{Makela11}
\begin{align}
\label{Erot}
E^{\textrm{new}}[\phi]&=E[\hat{U}_{2;-1}\phi]+i\hbar\langle\phi|\left(\frac{\partial}{\partial t}\hat{U}^{-1}_{2;-1}\right)\hat{U}_{2;-1}\phi\rangle,
\end{align}
and the time evolution of the components of $\phi$ can be obtained from the equation
\begin{align}\label{variE}
i\hbar\frac{\partial\phi_\nu}{\partial t}=\frac{\delta E^{\textrm{new}}[\phi]}{\delta\phi_\nu^*},\quad \nu =-2,-1,0,1,2.
\end{align}
We replace $\phi\rightarrow \psi_{2;-1} +\delta\psi$ in the time evolution equation (\ref{variE}) and expand the resulting equations to first order in $\delta\psi$. The perturbation $\delta\psi=(\delta\psi_2,\delta\psi_1,\delta\psi_0,\delta\psi_{-1},\delta\psi_{-2})^T$ is written as
\begin{align*}
\delta\psi_j=\sum_{\mathbf{k}} \left[ u_{j;\mathbf{k}}(t)\,e^{i\mathbf{k}\cdot\mathbf{r}}
-v_{j;\mathbf{k}}^{*}(t)\, e^{-i\mathbf{k}\cdot\mathbf{r}}\right],
\end{align*}
where $j=-2,-1,0,1,2$.
Straightforward calculation gives the differential equation for the time evolution of the  perturbations as
\begin{align}
i\hbar\frac{\partial}{\partial t}
\begin{pmatrix}
\mathbf{u}_\mathbf{k}\\
\mathbf{v}_\mathbf{k}
\end{pmatrix}
&=\hat{B}_{2;-1}
\begin{pmatrix}
\mathbf{u}_\mathbf{k}\\
\mathbf{v}_\mathbf{k}
\end{pmatrix},\\
\label{HBG}
\hat{B}_{2;-1} &=\begin{pmatrix}
\hat{X} &- \hat{Y}\\
\hat{Y}^* & -\hat{X}^*
\end{pmatrix},
\end{align}
where
$\mathbf{u}_\mathbf{k}=(u_{2;\mathbf{k}},u_{1;\mathbf{k}},u_{0;\mathbf{k}},u_{-1;\mathbf{k}},
u_{-2;\mathbf{k}})^T$, $\mathbf{v}_\mathbf{k}$ is defined similarly, and the $5\times 5$ matrices $\hat{X}$ and $\hat{Y}$ are
\begin{align}
\nonumber
\hat{X} =&\,\, \epsilon_k + g_0 |\psi_{2;-1}\rangle\langle\psi_{2;-1}|
+g_1 \!\!\!\!\sum_{j=x,y,z} |\psi_{2;-1}^j (t)\rangle\langle\psi_{2;-1}^j (t)| \\
\label{X}
&+2g_2  |\psi_{2;-1}^\textrm{s} (t)\rangle\langle\psi_{2;-1}^\textrm{s} (t)|, \\
\label{Y}
\hat{Y} =&\,\, g_0 |\psi_{2;-1}\rangle\langle\psi^*_{2;-1}|
+g_1 \!\!\!\!\sum_{j=x,y,z} |\psi_{2;-1}^j (t)\rangle\langle (\psi_{2;-1}^{j})^*(t)|.
\end{align}
Here we have defined
\begin{align}
\label{epsilonk}
\epsilon_k &= \, \frac{\hbar^2 k^2}{2m},\\
\psi_{2;-1}^j (t) &= \, \hat{U}_{2;-1}^\dag(t)\hat{F}_j U_{2;-1} (t)\psi_{2;-1},\quad j=x,y,z,\\
\psi_{2;-1}^\textrm{s} (t) &= \, \hat{U}_{2;-1}^T(t)e^{-i\pi \hat{F}_y} U_{2;-1} (t)\psi_{2;-1}.
\end{align}
In the rest of the article we call the operator determining the time evolution of the perturbations the Bogoliubov matrix. In the present case, $\hat{B}_{2;-1}$ is the Bogolibov matrix of $\psi_{2;-1}$.
It is possible to write $\hat{B}_{2;-1}$ as a direct sum of three operators
\begin{align}
\hat{B}_{2;-1} (t) &=\hat{B}_{2;-1}^{4}\oplus \hat{B}_{2;-1}^3(t) \oplus \hat{B}_{2;-1}^{3'}(t),\\
\hat{B}_{2;-1}^{3'} &=-(\hat{B}_{2;-1}^3)^*.
\end{align}
Here $\hat{B}_{2;-1}^{4}$ is a time independent $4\times 4$ matrix and $\hat{B}_{2;-1}^3$
is a time-dependent $3\times 3$ matrix. The bases in which these operators are defined are given in Appendix \ref{sec:appendixa}.
The time-dependent terms of $\hat{B}_{2;-1}^3$ are proportional to
$e^{\pm i k q t/\hbar}$, where $k=2,4$, or $6$, and consequently the system is periodic with minimum period $T=\pi\hbar/q$. Hence it is possible to use Floquet theory to analyze the stability of the system \cite{Makela11}. In the following we first calculate the eigenvalues of $\hat{B}_{2;-1}^{4}$, then those of $\hat{B}_{2;-1}^3$ and $ \hat{B}_{2;-1}^{3'}$ in the case $q=0$, and finally we discuss the general case $q\not =0$ using  Floquet theory.

\subsection{Eigenvalues of $\hat{B}_{2;-1}^{4}$}
First we calculate the eigenvalues and eigenvectors of $\hat{B}_{2;-1}^{4}$. This
operator is independent of $q$. The eigenvalues are
\begin{align}
\nonumber
\label{psi2m1omega1234}
\hbar\omega_{1,2,3,4} &=\pm\Big[\epsilon_k\Big(\epsilon_k +g_0 n+g_1 n(2+f_z)\\
&\pm n\sqrt{[g_0-g_1 (2+f_z)]^2+4 g_0 g_1 f_z^2}\Big)\Big]^{1/2}.
\end{align}
Here we use a labeling such that $++,-+,+-$, and $--$ correspond to $\omega_1,\omega_2,\omega_3$, and $\omega_4$, respectively. Now $\omega_{1,2}$ have a non-vanishing
imaginary part only if $g_0$ and $g_1$ are both negative, while $\omega_{3,4}$ have an imaginary component if $g_0$ and $g_1$ are not both  positive. Consequently,  these modes are stable for rubidium for which $g_0,g_1>0$.

The eigenvectors can be calculated straightforwardly, see Appendix \ref{sec:appendixa}.
The eigenvectors, like the eigenvalues, are independent of $g_2$.
The perturbations corresponding to the eigenvectors of $\hat{B}_{2;-1}^{4}$ can be written as
\begin{align}
\delta\psi^{1,2,3,4}(\mathbf{r},\mathbf{k};t) = C_{1,2,3,4}(\mathbf{r},\mathbf{k};t)\,\psi_{2;-1},
\end{align}
where the $C_j$'s include all position, momentum, and time dependence.
These change the total density of the condensate and are therefore called  density modes.

\subsection{Eigenvalues of $\hat{B}_{2;-1}^3$ and $\hat{B}_{2;-1}^{3'}$ at $q=0$}
In the absence of an external magnetic field $\hat{B}_{2;-1}^3$ is time independent.
The eigenvalues of $\hat{B}_{2;-1}^{3'}$ can be obtained from
those of $\hat{B}_{2;-1}^3$ by complex conjugating and changing the sign. For this
reason we give only the eigenvalues of $\hat{B}_{2;-1}^3$:
\begin{align}
\label{psi2m1omega5}
\hbar\omega_{5} &= \epsilon_k +2 g_2 n,\\
\label{psi2m1omega67}
\hbar\omega_{6,7} &=\frac{1}{2}\Big[g_1 n f_z
\pm \sqrt{(2\epsilon_k-g_1 n f_z)^2+ 16 g_1 n\epsilon_k}\Big].
\end{align}
The eigenvalues $\hbar\omega_6$ and $\hbar\omega_7$ have a non-vanishing complex
part if $g_1<0$. For rubidium all eigenvalues are real.
There are two gapped excitations: at $\epsilon_k=0$ we get $\hbar\omega_5=2g_2 n$ and $\hbar\omega_{6}
\,(\hbar\omega_{7})=g_1 nf_z$ if $g_1 f_z>0$ $(g_1 f_z<0)$.
The eigenvectors are given in Appendix A.
The corresponding perturbations become
\begin{align}
\label{omega3}
\delta\psi^{5}(\mathbf{r},\mathbf{k};t) &= C_5\,
\begin{pmatrix}
0\\
\sqrt{2-f_z}\\
0\\
0\\
-\sqrt{1+f_z}
\end{pmatrix},\\
\delta\psi^{6,7}(\mathbf{r},\mathbf{k};t) &= \sum_\mathbf{k} C_{6,7} \,
\begin{pmatrix}
0\\
e^{i \mathbf{k}\cdot \mathbf{r}} g_1 n\sqrt{(2-f_z)(1+f_z)}\\
e^{-i \mathbf{k}\cdot \mathbf{r}}\sqrt{\frac{3}{2}}(\epsilon_k+2 g_1 n-\hbar\omega_{6,7})\\
0\\
e^{i \mathbf{k}\cdot \mathbf{r}} g_1 n (2-f_z),
\end{pmatrix}
\end{align}
where $C_{5,6,7}$ are functions of $\mathbf{r},\mathbf{k}$, and $t$.
These modes change both the direction of the spin and magnetization and are therefore called spin-magnetization modes.

\subsection{Non-vanishing magnetic field}
If $q\not =0$, the stability can be analyzed using Floquet theory due to the
periodicity of  $\hat{B}_{2;-1}^3$ \cite{Makela11}. We denote the time evolution operator
determined by $\hat{B}_{2;-1}^3$ by $\hat{U}_{2;-1}^3$.
According to the Floquet theorem (see, e.g., Ref. \cite{Chicone}), $\hat{U}_{2;-1}^3$ can be written as
\begin{align}
\hat{U}_{2;-1}^3(t)=\hat{M}(t) e^{-i t \hat{K}},
\end{align}
where $\hat{M}$ is a periodic matrix with minimum period $T$ and $\hat{M}(0)=\hat{\textrm{I}}$,
and $\hat{K}$ is some time-independent matrix.
At times $t=nT$, where $n$ is an integer, we get $\hat{U}_{2;-1}^3(nT)=e^{-i n T \hat{K}}$. The eigenvalues of $\hat{K}$ determine the stability of the system.
We say that the system is unstable if at least one of the eigenvalues of $\hat{K}$ has a positive imaginary part. We calculate the eigenvalues $\{\hbar\omega\}$ of $\hat{K}$ from the equation
\begin{align}
\hbar\omega=\hbar\omega^\textrm{r}+i \hbar\omega^\textrm{i}=i\frac{\ln\lambda}{T},
\end{align}
 where $\{\lambda\}$ are the eigenvalues of $\hat{U}_{2;-1}^3(T)$.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=.7]{psi2m1fig_rev.eps}
\end{center}
\caption{(Color online) The positive imaginary part $\omega^\textrm{i}$ related to  $\hat{U}_{2;-1}(T)$ for different values of the magnetic field parameter $q$.
The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$. Note that the scales of $\epsilon_k$ and $\omega^\textrm{i}$ are not equal in the top and bottom rows. Note also that the scale of the $q=5|g_1|n$ figure
is shifted with respect to the scale of the $q=|g_1|n$ case. The solid white line gives the approximate location of the fastest-growing instability, and the dashed white line corresponds to the largest possible size of a
stable condensate, see Eq. (\ref{lambda2m1}) and Table \ref{table}.
\label{fig:psi2m1}}
\end{figure}

We plot $\omega^\textrm{i}$ for several values of the magnetic field in Fig. \ref{fig:psi2m1}.
By comparing this to the case of a rubidium condensate with $g_2=0$,
we found that the instabilities are essentially determined by $g_1$, the effect of $g_2$ is negligible.
The eigenvectors of $\hat{U}_{2;-1}^3(T)$ correspond to perturbations which affect both
spin direction and magnetization.
With the help of numerical results we find that a good fitting formula is given by
\begin{align}
\hbar\omega^\textrm{i}
\approx &\textrm{Im}\Big\{
\sqrt{(\epsilon_k+q)[\epsilon_k+q+\frac{5}{3}(2-f_z) g_1 n]}\Big\}, q<0,\\
\nonumber
\approx &\textrm{Im}\Big\{\sqrt{(\epsilon_k-2q+ g_1 n)^2-\frac{4}{9}|(f_z-2)(f_z+1)g_1n|}\Big\},\\
& q>0.
\end{align}
We see that for $q>0$ the fastest-growing instability is located approximately at $\epsilon_k=\max\{0,2q-g_1 n\}$ regardless of the value of $f_z$. For $q<0$ the location
of this instability becomes magnetization dependent and is approximately given
by $\epsilon_k=\max \{0, |q|-5(2-f_z)|g_1|n/6\}$.
The values of $\epsilon_k$ corresponding to unstable wavelengths are bounded above approximately  by the inequality $\epsilon_k\leq (3|q|+q)/2$. Therefore, the state $\psi_{2;-1}$
is stable if the condensate is smaller than the shortest unstable wavelength
\begin{align}
\lambda_{2;-1} =\frac{2\pi\hbar}{\sqrt{m(3|q|+q)}}.
\label{lambda2m1}
\end{align}
At $q=0$ the system is stable regardless of its size.

Figure \ref{fig:psi2m1} shows that the shape of the unstable region
depends strongly on the sign of $q$. This can be understood qualitatively
with the help of the energy functional of Eq. (\ref{energy}).
We choose $\psi_{\textrm{ini}}=\sqrt{n}|m_F=-1\rangle$ to be the initial state of the system and assume that the
final state is of the form
\begin{align}
\psi_{\textrm{fin}}(x,y,z)=
\begin{cases}
\sqrt{n}|0\rangle, & x\, \textrm{mod}\, 2L \in [0,L),\\
\sqrt{n}|-2\rangle, & x\, \textrm{mod}\, 2L \in [L,2L).
\end{cases}
\end{align}
Then the energy of the initial state is $E_{\textrm{ini}}=g_1 n/2+q$ (dropping constant terms),
while the energy of the final configuration
is $E_{\textrm{fin}}=g_1 n+ 2q$. If $g_1,q>0$, $E_{\textrm{ini}}<E_{\textrm{fin}}$ and domain formation is suppressed for energetic reasons.
If, on the other hand, $g_1>0$ and $q<0$, the energy of the final state is smaller than the energy of the initial state if $q<-g_1 n/2$ and domain formation is possible.

\section{Stability of nonstationary states when $g_2=0$}\label{sec:g2not0}
For rubidium the value of $g_2$ is small in comparison with $g_0$ and $g_1$.
Consequently, it can be assumed that this term has only a minor effect on the stability of the system. This assumption is supported by the results of the previous section.
In the following we will therefore study the stability in the limit $g_2=0$.
This makes it possible to obtain an analytical expression for the time evolution
operator also for states other than $\psi_{2;-1}$. First we discuss a state that has three nonzero components,
and then two states that have two nonzero components.

\subsection{Nonzero $\psi_2$, $\psi_0$, and $\psi_{-2}$}
We consider a state of the form
\begin{align}
\psi_{2;0;-2}=
\frac{\sqrt{n}}{2}\begin{pmatrix}
\sqrt{2-2 \rho_0+f_z}\\
0\\
2e^{i\theta} \sqrt{\rho_0}\\
0\\
\sqrt{2-2 \rho_0-f_z}
\end{pmatrix},\quad |f_z|\leq 2-2\rho_0
\end{align}
For this $\langle\hat{F}_x\rangle=\langle\hat{F}_y\rangle=0$ and
$\langle\hat{F}_z\rangle= nf_z$. The Hamiltonian and time evolution operator of this state
are given by Eqs. (\ref{eq:Hparallel}) and (\ref{eq:Uparallel}), respectively.
The equations determining the time evolution of the perturbations can be obtained
from Eqs. (\ref{X}) and (\ref{Y}) by replacing $\psi_{2;-1}$ with $\psi_{2;0;-2}$
and setting $g_2=0$. In this way, we obtain a time dependent Bogoliubov matrix
$\hat{B}_{2;0;-2}$, which is a function of the population of the zero component $\rho_0$.
The Bogoliubov matrix can now be written as
\begin{align}
\label{eq:HB20m2}
\hat{B}_{2;0;-2} (t) =\hat{B}_{2;0;-2}^6\oplus \hat{B}_{2;0;-2}^4(t),
\end{align}
where $\hat{B}_{2;0;-2}^6$ is time independent and
$\hat{B}_{2;0;-2}^4$ is periodic in time with period $T=\pi/q$.
The bases in which these operators are defined are given in Appendix B.
The eigenvalues of $ \hat{B}_{2;0;-2}^6$ are
\begin{align}
\hbar\omega_{1,2} =& \pm\epsilon_k,\\
\nonumber
\hbar\omega_{3,4,5,6} =& \pm\Big[\epsilon_k^2 +\epsilon_k [g_0+4g_1(1-\rho_0)]n \\
&\pm \epsilon_k n\sqrt{[g_0-4g_1(1-\rho_0)]^2+4 g_0 g_1 f_z^2}\Big]^{1/2}.
\end{align}
Here $++,-+,+-$, and $--$ correspond to $\omega_3,\omega_4,\omega_5$,
and $\omega_6$, respectively.
These eigenvalues are always real if $g_0$ and $g_1$ are positive.
From the eigenvectors given in Appendix \ref{sec:appendixb} we see that $\omega_{3,4}$ are density modes and $\omega_{1,2}$ and $\omega_{5,6}$ are magnetization modes. All these are gapless excitations.
Note that the eigenvalues are independent of $\theta$.

We discuss next the stability properties determined by $\hat{B}_{2;0;-2}^4$. We consider first the special case $\rho_0=0$  and proceed then to the case $\rho_0>0$.

\subsubsection{Stability at $\rho_0=0$}
In the case $\rho_0=0$ a complete analytical solution of the excitation spectrum can be obtained.
In Appendix \ref{sec:appendixb} we show that by a suitable choice of basis the time dependence of the  Bogoliubov matrix can be eliminated. The eigenvalues are
\begin{align}
\nonumber
\label{psi20m2omega78910}
&\hbar\omega_{7,8,9,10} = \frac{1}{2}\Big[\pm g_1 n f_z + 6q \\
&\pm\sqrt{4(\epsilon_k+g_1 n-3 q)^2
-(4-f_z^2)(g_1 n)^2} \Big].
\end{align}
These are gapped excitations and correspond to spin-magnetization modes (see Appendix \ref{sec:appendixb}).
If $g_1>0$, these eigenvalues have a non-vanishing complex part when $3q-2g_1 n\leq \epsilon_k \leq 3q$. This is possible only if $q$ is positive.
The location of the fastest-growing unstable mode, determined by $\epsilon_k=\max\{0,3q - g_1 n\}$, is independent of $f_z$. The maximal width of the unstable region in the $\epsilon_k$ direction, obtained at $f_z=0$,
 is $2|g_1|n$. The state is stable if the system is smaller than the size given by
\begin{align}
\label{lambda20m2}
\lambda_{2;0;-2} =\frac{2\pi\hbar}{\sqrt{6mq}},\quad q>0.
\end{align}
If $q<0$, the state is stable regardless of the size of the condensate.
In Fig. \ref{fig:psi20m2} we plot the positive imaginary part of the eigenvalues (\ref{psi20m2omega78910}) for various values of $q$.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=.86]{psi20m2fig_rev.eps}
\end{center}
\caption{The positive imaginary part $\omega^\textrm{i}$ related to the eigenvalues (\ref{psi20m2omega78910}) and
to  $\hat{U}_{2;0;-2}(T)$
for various values of the quadratic Zeeman term $q$ and population $\rho_0$.
The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$.
We have chosen $f_z=0$ as this choice gives the fastest-growing instabilities and the smallest size of a stable condensate. In the top row the dashed, dotted, and solid lines correspond to $q=1,2,3$, respectively, while in the bottom row they correspond to $q=-1,-2,-3$, respectively. We have set $\theta=0$ in $\psi_{2;0;-2}$ as
the stability was found to be independent of $\theta$.
 \label{fig:psi20m2}}
\end{figure}

\subsubsection{Stability when $\rho_0>0$}
In the case $\rho_0>0$ the stability can be studied using Floquet theory.
The stability properties can be shown to be independent of the sign of $f_z$.
At $q=0$ the operator $\hat{B}_{2;0;-2}^4$ is time independent. The eigenvalues
can be obtained analytically but are not given here.
The eigenvalues show that in the absence of magnetic field
the state is stable in a rubidium condensate regardless of the value of $\rho_0$.
Figure \ref{fig:psi20m2} illustrates how the stability depends on the value of $q$ and the population $\rho_0$. We plot only the case $f_z=0$ as it gives the fastest-growing
instabilities and the smallest size of a stable condensate.
We found numerically that the stability properties are independent of the value of $\theta$. We have set $\theta=0$ in the calculations described here.
If $q>0$, the amplitude $\omega^\textrm{i}$ of the short-wavelength instabilities is suppressed as $\rho_0$ increases.
This can be understood with the help of the energy functional
\begin{align}
E_{2;0;-2} = \frac{1}{2} g_1 n f_z^2 + 8q (1-\rho_0).
\end{align}
If $q>0$, the energy decreases as $\rho_0$ increases.
Therefore there is less energy available to be converted into the kinetic energy
of the domain structure. From the top row of Fig. \ref{fig:psi20m2} it can be seen that Eq. (\ref{lambda20m2}) gives an upper bound for the size of a stable condensate also when the value of $\rho_0$ is larger than zero.
If $q<0$, the state is stable at $\rho_0=0$.  The bottom row of Fig. \ref{fig:psi20m2} shows that now  $\psi_{2;0;-2}$ becomes more unstable as $\rho_0$ grows. This is natural because the energy $E_{2;0;-2}$ grows as $\rho_0$ increases, the energy surplus can be converted into kinetic energy of the domains.
Figure \ref{fig:psi20m2} shows that Eq. (\ref{lambda20m2}) gives an upper bound for the size of a stable condensate also in the case $q<0$.

\subsection{Nonzero $\psi_1$ and $\psi_{-1}$}
As the next example we consider a state of the form
\begin{align}
\psi_{1;-1}=
\sqrt{\frac{n}{2}}\begin{pmatrix}
0\\
\sqrt{1+f_z}\\
0\\
\sqrt{1-f_z}\\
0
\end{pmatrix},\quad |f_z|\leq 1.
\end{align}
Also for this state $\langle\hat{F}_x\rangle=\langle\hat{F}_y\rangle=0$ and
$\langle\hat{F}_z\rangle= nf_z$ and therefore
the Hamiltonian and time evolution operator are given by Eqs. (\ref{eq:Hparallel}) and (\ref{eq:Uparallel}), respectively.
The Bogoliubov matrix reads
\begin{align}
\label{eq:HB1m1}
\hat{B}_{1;-1} (t) =\hat{B}_{1;-1}^6(t)\oplus \hat{B}_{1;-1}^4.
\end{align}
Here $\hat{B}_{1;-1}^6(t)$ is time dependent with period $T=\pi/q$ and
$\hat{B}_{1;-1}^4$ is time independent. The eigenvalues of $\hat{B}_{1;-1}^4$ are
\begin{align}
\nonumber
\hbar\omega_{1,2,3,4} =&\pm\Big[\epsilon_k\Big(\epsilon_k+ (g_0+g_1)n\\
& \pm n\sqrt{(g_0-g_1)^2+4 g_0 g_1 f_z^2}\Big)\Big]^{1/2}.
\end{align}
Now $++,-+,+-$, and $--$ correspond to $\omega_1,\omega_2,\omega_3$,
and $\omega_4$, respectively.
These are all gapless modes. For rubidium the eigenvalues are real.
In Appendix C we show that $\omega_{1,2}$ are density modes
and $\omega_{3,4}$ are magnetization modes.

We now turn to the eigenvalues of $\hat{B}_{1;-1}^6$.
At $q=0$ $\hat{B}_{1;-1}^6$ becomes time independent and the eigenvalues are
\begin{align}
\hbar\omega_{5,6} &= \pm\epsilon_k,\\
\nonumber
\hbar\omega_{7,8,9,10} &= \pm\frac{1}{\sqrt{2}}
\Big[2\epsilon_k^2+10\epsilon_k g_1 n +(g_1 n f_z)^2 \\
&\pm g_1 n \sqrt{(6\epsilon_k+g_1 n f_z^2)^2-8\epsilon_k f_z^2(4\epsilon_k-g_1 n)}\Big]^{1/2}.
\end{align}
For rubidium these are all real.
One of the eigenvalues $\hbar\omega_{4,5}$ has an energy gap $|g_1 n f_z|$.
These eigenvalues describe spin-magnetization modes.

For non-zero $q$ the stability can be analyzed using Floquet theory.
As in the previous section, the fastest growing instabilities are obtained
at $f_z=0$. This case can be studied analytically by changing basis as described in Appendix \ref{sec:appendixc}. The eigenvalues for the case $f_z=0$ are
\begin{align}
\label{omega1m1a}
&\hbar\omega_{5,6} = -3 q\pm \sqrt{(\epsilon_k+3 q)(\epsilon_k+ 2 g_1 n+3 q)},\\
\nonumber
\label{omega1m1b}
&\hbar\omega_{7,8,9,10} = 3q \pm \Big[(\epsilon_k+q)^2+4(\epsilon_k g_1 n+q^2)\\
&\pm 4\sqrt{[q^2+\epsilon_k (g_1 n+q)]^2-3 g_1 n q (\epsilon_k^2-q^2)}\Big]^{1/2}.
\end{align}
These are gapped excitations with a magnetic-field-dependent gap. In more detail,
at $\epsilon_k=0$ we get $\hbar\omega_{5,6}=-3q\pm\sqrt{3q(2 g_1 n+3q)}$ and
$\hbar\omega_{7,8,9,10}=3q\pm\sqrt{5q^2\pm 4q\sqrt{q(3 g_1 n+q)}}$.
For positive $q$, the fastest-growing instability
is determined by $\omega_8^{\textrm{i}}$ and is located approximately at $\epsilon_k =\textrm{max}\{0,-3+q\}.$ For negative $q$ there are three local maxima for $\omega^{\textrm{i}}$. The one with the largest amplitude is given by
 $\omega^{\textrm{i}}_{7}$ and  $\omega^{\textrm{i}}_{10}$  and is located at $\epsilon_k\approx \textrm{max}\{0,q^2(|q|-1)/(q^2+|q|+1)\}$. The second largest is given by
 $\omega^{\textrm{i}}_5$ and is at  $\epsilon_k\approx \textrm{max}\{0,3|q|+1\}$. Finally, the instability with the smallest amplitude is related to  $\omega^{\textrm{i}}_8$ and is at
$\epsilon_k\approx \textrm{max}\{0,29 (10 |q|-1)/100\}$. In Fig. \ref{fig:psi1m1fig}
we plot the behavior of $\omega^\textrm{i}$ for $q=6$ and $q=-3$.
From Eqs. (\ref{omega1m1a}) and (\ref{omega1m1b}) it can be seen (see also Fig. \ref{fig:psi1m1fig})
that the state is stable if the size of the condensate is smaller than
\begin{align}
\label{lambda1m1}
\lambda_{1;-1} =\frac{2\pi\hbar}{\sqrt{2m(2|q|-q)}}.
\end{align}

\begin{figure}[h]
\begin{center}
\includegraphics[scale=.75]{psi1m1fig_rev.eps}
\end{center}
\caption{The positive imaginary part $\omega^\textrm{i}$ of the eigenvalues (\ref{omega1m1a})
and (\ref{omega1m1b}) related to $\psi_{1;-1}$ for $q=3$ and $q=-6$.
The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$.
We have chosen $f_z=0$ as it gives the fastest-growing instabilities and the smallest size of a stable system.
The solid and dashed lines correspond to $\omega_8^\textrm{i}$ and $\omega^{\textrm{i}}_5$, respectively, while the dotted line gives $\omega^{\textrm{i}}_7$ and $\omega^{\textrm{i}}_{10}$
[see Eqs. (\ref{omega1m1a}) and (\ref{omega1m1b})].
\label{fig:psi1m1fig}}
\end{figure}

\subsection{Nonzero $\psi_2$ and $\psi_{0}$}
As the final example we consider a state
\begin{align}
\psi_{2;0}=
\sqrt{\frac{n}{2}}\begin{pmatrix}
\sqrt{f_z}\\
0\\
\sqrt{2-f_z}\\
0\\
0
\end{pmatrix},\quad 0\leq f_z\leq 2.
\end{align}
As for other states considered in this article, now $\langle\hat{F}_x\rangle=\langle\hat{F}_y\rangle=0$ and
$\langle\hat{F}_z\rangle= nf_z$ and
the Hamiltonian and time evolution operator are given by Eqs. (\ref{eq:Hparallel}) and (\ref{eq:Uparallel}), respectively. We note that the stability properties of the states $\psi_{2;0}$ and
$\psi_{0;-2}=\sqrt{n}(0,0,\sqrt{2-f_z},0,\sqrt{f_z})/\sqrt{2}$ are similar. Therefore the latter
state will not be discussed in more detail.
The Bogoliubov matrix of $\psi_{2;0}$ reads
\begin{align}
\label{eq:HB20}
\hat{B}_{2;0} (t) =\hat{B}_{2;0}^2\oplus\hat{B}_{2;0}^4\oplus \hat{B}_{2;0}^{4'}(t),
\end{align}
where only $\hat{B}_{2;0}^{4'}$ is time dependent (with period $T=\pi/q$). The eigenvalues of $\hat{B}_{0;2}^2$ and $\hat{B}_{0;2}^4$ are
\begin{align}
\hbar\omega_{1,2} = &\pm \epsilon_k, \\
\nonumber
\hbar\omega_{3,4,5,6} =& \pm\Big[\epsilon_k^2+ \epsilon_k( g_0 n+2 g_1 n f_z)\\
&\pm \epsilon_k n\sqrt{(g_0-2 g_1 f_z)^2+4 g_0 g_1 f_z^2} \Big]^{1/2}.
\end{align}
In the lower equation, $++, -+, +-$, and $--$ correspond to $\omega_3,\omega_4,\omega_5$, and $\omega_6$, respectively.
These are gapless excitations. In Appendix \ref{sec:appendixd} we show that $\omega_{3,4}$ correspond to density modes, while $\omega_{1,2,5,6}$ are magnetization modes. For rubidium, these are all stable modes.

After a suitable change of basis the Bogoliubov matrix $\hat{B}_{2;0}^{4'}$ becomes
time independent, see Appendix \ref{sec:appendixd}. The eigenvalues of the new matrix are found to be
\begin{align}
\label{omega20}
&\hbar\omega_{7,8,9,10} = \pm\frac{1}{\sqrt{2}}\sqrt{s_1 \pm \sqrt{(2\epsilon_k+g_1 n f_z +2 q)s_2}},
\end{align}
where
\begin{align}
\nonumber
&s_1 = 2 \epsilon_k^2+(g_1 n f_z)^2 +4 \epsilon_k [(3-f_z)g_1 n +q]\\
\nonumber
&\hspace{0.7cm} - 8 f_z g_1 n q +2 q (6 g_1 n+5q),\\
\nonumber
& s_2 = f_z (g_1 n)^2[24(\epsilon_k+q)-10 \epsilon_k f_z -18 q f_z+ g_1 n f_z^2]\\
\nonumber
&\hspace{0.7cm} +32 q^2 [q+\epsilon_k +3 g_1 n (2- f_z)]  - 16 g_1 n q \epsilon_k f_z.
\end{align}
Now $++,-+,+-$, and $--$ are related to $\omega_7,\omega_8,\omega_9$,
and $\omega_{10}$, respectively. These are gapped excitations and correspond to spin-magnetization modes.
These modes can be unstable for rubidium; an example of the behavior of the positive imaginary component of
$\omega_{7,8,9,10}$ is shown in Fig. \ref{fig:psi20fig}.

An upper bound for the size of a stable condensate is the same as in the case of $\psi_{1;-1}$, see
Eq. (\ref{lambda1m1}). With the help of Eq. (\ref{omega20}) it can be seen that the fastest-growing
instability is approximately at $\epsilon_k=\textrm{max}\{0,-2+0.9 q+0.04 f_z (1+q)\}$ when $q>0$ and
at  $\epsilon_k=\textrm{max}\{0,|q|-3+1.3 f_z-0.16 f_z^2\}$ when $q<0$.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=.65]{psi20fig_rev.eps}
\end{center}
\caption{(Color online) The positive imaginary part $\omega^\textrm{i}$ of the eigenvalues
(\ref{omega20}) related to $\psi_{2;0}$ for $q=5$ and $q=-2$.
The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$. The solid white line gives the approximate location of the fastest-growing instability, and the dashed white line corresponds to the largest possible size of a
stable condensate, see Table (\ref{table}).
\label{fig:psi20fig}}
\end{figure}

\begin{table}[h]
\begin{tabular}{|c|c|c|c|}
\hline
State & $q$ & Stable size & Fastest-growing instability ($\epsilon_k$)\\
\hline
\hline
$\psi_{2;-1},$ & $>0$ & $\frac{2\pi\hbar}{\sqrt{4mq}}$ & $2q-g_1 n$\\
\cline{2-4}
$\psi_{1;-2}$ &$<0$  & $\frac{2\pi\hbar}{\sqrt{2m|q|}}$ & $|q|-\frac{5}{6}(2\mp f_z)|g_1|n$\\
\hline
$\psi_{2;0;-2}$ & $>0$ & $\frac{2\pi\hbar}{\sqrt{6mq}}$ &  $3q-g_1 n$\\
\cline{2-4}
  & $<0$ & $\infty$ & -\\
\hline
$\psi_{1;-1}$ & $>0$ & $\frac{2\pi\hbar}{\sqrt{2mq}}$ & $q-3g_1 n$\\
\cline{2-4}
 & $<0$ & $\frac{2\pi\hbar}{\sqrt{6m|q|}}$ & $\frac{q^2(|q|-g_1 n)}{q^2+|q| g_1 n+(g_1 n)^2}$\\
\hline
$\psi_{2;0},$ & $>0$ & $\frac{2\pi\hbar}{\sqrt{2mq}}$ & $-2+0.9 q+0.04 |f_z| (1+q)$\\
\cline{2-4}
$\psi_{0;-2}$ & $<0$ & $\frac{2\pi\hbar}{\sqrt{6m|q|}}$ & $|q|-3+1.3 |f_z| -0.16 f_z^2$\\
\hline
\end{tabular}
\caption{Summary of the results. Stable size gives the largest possible size of a stable
homogeneous condensate and the fastest-growing instability indicates the approximate value of $\epsilon_k$ corresponding to the fastest growing instability. If $q$ is such that the $\epsilon_k$ given in the table
is negative, the fastest-growing instability is at $\epsilon_k=0$.  On the second line of the table,
the $-$ sign holds for $\psi_{2;-1}$ and the $+$ sign for $\psi_{1;-2}$.
\label{table}}
\end{table}

\section{Conclusions} \label{sec:conclusions}
In this article, we have studied the dynamical
stability of some nonstationary states of homogeneous $F=2$ rubidium
BECs. The states were chosen to be such
that the spin vector remains parallel to the magnetic field throughout the
time evolution, making it possible to study the stability analytically.
The stability analysis was done using the Bogoliubov approach in a frame of reference where the states were stationary. The states considered had two or three spin components populated simultaneously. These types of states were found to be stable in a rubidium condensate in the absence of a magnetic field, but a finite magnetic field makes them unstable. The wavelength and the growth rate of the instabilities depends on the strength of the magnetic field. The locations of the fastest-growing instabilities and the upper bounds for the sizes of stable condensates are given in Table \ref{table}.
For positive $q$, the most unstable state, in the sense that its upper bound for the size of a stable condensate is the smallest, is $\psi_{2;0;-2}$.
However, this is the only state that is stable when $q$ is negative. For $q<0$, the states giving the smallest size of a stable condensate are $\psi_{1;-1}$ and $\psi_{2;0}$.

In comparison with $F=1$ condensates, the structure of the instabilities is much richer in an $F=2$
condensate. In an $F=1$ system, there is only one type of a state whose spin is parallel to the magnetic field.
The excitations related to this state can be classified into spin and magnetization excitations \cite{Makela11}.
In the present system, there are many types of states which are parallel to the magnetic field;
we have discussed six of these. In addition to the spin and magnetization excitations, there
exist also modes which change spin and magnetization simultaneously. The increase in the complexity can
be attributed to the number of components of the spin vector.

The stability properties of the states discussed in this article can be studied experimentally straightforwardly.
These states had two or three non-zero components, a situation which can be readily achieved by current experimental means  \cite{Ramanathan11}.
Furthermore, the stability of these states does not depend on the relative phases of the populated components,
making it unnecessary to prepare states with specific relative phases.

Finally, we note that the lifetime of an $F=2$
rubidium condensate is limited by hyperfine changing collisions \cite{Schmaljohann04}. Consequently, the instabilities are visible
only if the their growth rate is large enough compared to the lifetime of the condensate. We also remark that
the stability analysis was performed for a homogeneous condensate, whereas in experiments an inhomogeneous trapping potential is used. The stability properties can be sensitive to the shape of this potential \cite{Klempt09}.