Kazhdan's Property T and Spectral Gap

Let $a=x^{\ast }x$. By [F1], $a$ is positive and $\mathrm{\parallel }a\mathrm{\parallel }=\mathrm{\parallel }x{\mathrm{\parallel }}^2>0$. Choose the unital C*-algebra $D$ of [F2], and set $C=C^{\ast }(1,a)\subseteq D$.

The positive calculus of Positive calculus and order estimates in a C star algebra gives $b=a^{1\mathrm{/}2}\in C^{\ast }(1,a)=C$ with $b=b^{\ast }$ and $b^{\ast }b=a$. Thus the Gelfand transform sends $a$ to the nonnegative continuous function $\widehat{a}(\chi )=\mathrm{\mid }\chi (b){\mathrm{\mid }}^2$ on the nonempty compact space $\mathrm{\Delta }(C)$. Its supremum norm is $\mathrm{\parallel }a\mathrm{\parallel }$ by [F4]. By [F5], its maximum is attained at some character ${\chi }_0\in \mathrm{\Delta }(C)$ and equals $\mathrm{\parallel }a\mathrm{\parallel }$.

The character ${\chi }_0$ has norm one and takes $1$ to $1$. Extend it by [F6] to $F\in D^{\ast }$ with $\mathrm{\parallel }F\mathrm{\parallel }=1$, $F(1)=1$, and $F(a)=\mathrm{\parallel }a\mathrm{\parallel }$.

Define $S(D)=\{\psi \in D^{\ast }:\mathrm{\parallel }\psi \mathrm{\parallel }\le 1,\psi (1)=1,\psi (b^{\ast }b)\ge 0\text{ for every }b\in D\}$. Every member is a state: the unit-ball bound and $\psi (1)=1$ give norm one, and the remaining condition is positivity; conversely every state belongs to this set by [F9]. It is convex because its normalization and positivity constraints are preserved by convex combinations, which remain in the unit ball. Within that ball its normalization and positivity conditions are intersections of closed evaluation constraints, so [F7] and the closed-subset compactness clause there make $S(D)$ weak-star compact. The weak-star topology is locally convex and Hausdorff by [F8].

If $c=c^{\ast }\in D$ and $t\in \mathbb{R}$, then $\mathrm{\mid }1+itF(c){\mathrm{\mid }}^2=\mathrm{\mid }F(1+itc){\mathrm{\mid }}^2\le \mathrm{\parallel }1+itc{\mathrm{\parallel }}^2=\mathrm{\parallel }1+t^2{c}^2\mathrm{\parallel }\le 1+t^2\mathrm{\parallel }c{\mathrm{\parallel }}^2$. Expanding the left side and letting $t$ tend to zero through positive and negative values forces $\mathrm{Im}F(c)=0$. If $0\le b\le 1$, then [F3] gives $\mathrm{\parallel }1-b\mathrm{\parallel }\le 1$; since $F(b)$ is real, $\mathrm{\mid }1-F(b)\mathrm{\mid }\le 1$ implies $F(b)\ge 0$. Scaling proves $F(b)\ge 0$ for every positive $b$, so $F$ is a state and $F(a)=\mathrm{\parallel }a\mathrm{\parallel }$.

Let $K=\{\psi \in S(D):\psi (a)=\mathrm{\parallel }a\mathrm{\parallel }\}$. It is nonempty by step 3.1 and is weak-star closed, hence compact. For every $\psi \in S(D)$, positivity and [F3], together with $\psi (1)=1$, give $0\le \psi (a)\le \mathrm{\parallel }a\mathrm{\parallel }$; therefore $K$ is convex and is a face of $S(D)$, since a proper convex combination can attain the upper bound only if both terms attain it.

By [F12], $K$ has an extreme point $\phi$. Because $K$ is a face of $S(D)$, any convex decomposition of $\phi$ in $S(D)$ has both terms in $K$; extremality in $K$ then makes both terms equal $\phi$. Thus $\phi$ is an extreme state of $D$, and $\phi (a)=\mathrm{\parallel }a\mathrm{\parallel }$.

Define $N=\{b\in D:\phi (b^{\ast }b)=0\}$. Cauchy–Schwarz [F9] makes $N$ a linear subspace, makes $⟨[b],[c]⟩=\phi (c^{\ast }b)$ independent of representatives, and makes it positive definite on $D\mathrm{/}N$; this form is linear in its first variable. For $d,b\in D$, [F3] gives $0\le \phi (b^{\ast }{d}^{\ast }db)\le \mathrm{\parallel }d{\mathrm{\parallel }}^2\phi (b^{\ast }b)$, so $N$ is a left ideal and left multiplication is bounded on the quotient with norm at most $\mathrm{\parallel }d\mathrm{\parallel }$. By [F10], the inner-product space $D\mathrm{/}N$ has a Hilbert completion $H$ under AC.

Define $\pi (d)[b]=[db]$ on $D\mathrm{/}N$ and extend it continuously to $H$. Then $\mathrm{\parallel }\pi (d)\mathrm{\parallel }\le \mathrm{\parallel }d\mathrm{\parallel }$, $\pi (de)=\pi (d)\pi (e)$, $\pi (1)=I_H$, and $\pi (d^{\ast })=\pi (d{)}^{\ast }$: the product and unit identities hold on quotient classes, and $⟨\pi (d)[b],[c]⟩=\phi (c^{\ast }db)=⟨[b],\pi (d^{\ast })[c]⟩$ gives the adjoint identity. The Hilbert adjoint has the properties used here by [F11].

The vector $\xi =[1]$ has norm one, the span of $\pi (D)\xi$ is dense by construction, and $\phi (d)=⟨\pi (d)\xi ,\xi ⟩$ for every $d\in D$. In particular, $\mathrm{\parallel }\pi (x)\xi {\mathrm{\parallel }}^2=\phi (x^{\ast }x)=\mathrm{\parallel }x{\mathrm{\parallel }}^2>0$. Thus $H\mathrm{\ne }\{0\}$ and $\pi (x)\mathrm{\ne }0$.

Suppose a nonzero proper closed subspace $M\subset H$ is invariant under every $\pi (d)$. Since $D$ is star-closed, $M^{\perp }$ is invariant too: for $v\in M^{\perp }$, $m\in M$, and $d\in D$, $⟨\pi (d)v,m⟩=⟨v,\pi (d^{\ast })m⟩=0$. By [F10] its orthogonal projection $E$ is a nonzero proper self-adjoint idempotent commuting with every $\pi (d)$. Cyclicity implies $t:=\mathrm{\parallel }E\xi {\mathrm{\parallel }}^2$ satisfies $0<t<1$, since either $E\xi =0$ or $(I-E)\xi =0$ would make $E=0$ or $E=I$ on the dense cyclic span.

Put ${\phi }_1(d)=t^{-1}⟨\pi (d)E\xi ,E\xi ⟩$ and ${\phi }_2(d)=(1-t{)}^{-1}⟨\pi (d)(I-E)\xi ,(I-E)\xi ⟩$. These vector functionals are positive; they have norm one because they take $1$ to $1$ and are bounded by one using $\mathrm{\parallel }\pi (d)\mathrm{\parallel }\le \mathrm{\parallel }d\mathrm{\parallel }$. Since $M$ and $M^{\perp }$ are invariant, the cross terms vanish and $\phi =t{\phi }_1+(1-t){\phi }_2$. The extremality of $\phi$ from step 5.1 gives ${\phi }_1={\phi }_2=\phi$.

For $b,c\in D$, commutation of $E$ with $\pi (D)$ and the orthogonality of $M$ and $M^{\perp }$ give $⟨E\pi (b)\xi ,\pi (c)\xi ⟩=⟨\pi (b)E\xi ,\pi (c)E\xi ⟩=t\phi (c^{\ast }b)=t⟨\pi (b)\xi ,\pi (c)\xi ⟩$. The vectors $\pi (b)\xi$ span a dense subspace, so continuity implies $E=t{I}_H$. This contradicts $E^2=E$ and $0<t<1$. Thus $\pi$ is irreducible.

If $A$ is unital, take $\sigma =\pi$. If $A$ is nonunital, its closed ideal property in [F2] makes $H_0=\stackrel{}{\mathrm{span}}\pi (A)H$ a reducing subspace for $\pi (D)$: for $d\in D$ and $a\in A$, both $da$ and $d^{\ast }a$ lie in $A$, so $\pi (d)$ and its adjoint preserve the dense spanning set. It is nonzero because $\pi (x)\xi \mathrm{\ne }0$. Irreducibility gives $H_0=H$, so $\sigma :=\pi {\mathrm{\mid }}_A$ is nondegenerate. Any closed $A$-invariant subspace is invariant under $\pi (D)=\pi (A)+\mathbb{C}{I}_H$, hence $\sigma$ is irreducible as well. In either case $\sigma$ is nonzero and $\sigma (x)\mathrm{\ne }0$.

We have constructed the required representation for each nonzero $x\in A$, so an element in the intersection of all the stated kernels must be zero. If $A=\{0\}$, there is no nonzero $x$, and the empty intersection inside $A$ is $A=\{0\}$, as stated.

The definitions of Kazhdan pair and Kazhdan set apply to every subset $Q\subseteq G$, including $Q=\mathrm{\varnothing }$. When $Q=\mathrm{\varnothing }$, every unit vector is $(Q,\epsilon )$-invariant, so the pair condition requires every representation with a unit vector to have a nonzero invariant vector.

Kazhdan-pair parameters are downward closed: if $(Q,\delta )$ is a pair and $0<\epsilon \le \delta$, then each $(Q,\epsilon )$-invariant unit vector is also $(Q,\delta )$-invariant, so $(Q,\epsilon )$ is a pair. Thus, with the zero adjoined in the definition, $\kappa (G,Q)$ is a well-defined extended-real threshold, equals $0$ if there is no positive Kazhdan parameter, and equals $+\mathrm{\infty }$ if every positive parameter is Kazhdan.

For a unit vector $\xi$, each displacement is at most $2$ by [F2]. If $Q\mathrm{\ne }\mathrm{\varnothing }$, its displacement values form a nonempty subset of $[0,2]$ and have a real supremum by [F4]; if $Q=\mathrm{\varnothing }$, $d_Q(\pi ,\xi )=0$ by definition. Thus $\kappa (G,Q,\pi )\in [0,2]$ for $H\mathrm{\ne }\{0\}$. If $H=\{0\}$, there are no unit vectors, so the declared extended-real empty-infimum convention gives $\kappa (G,Q,\pi )=+\mathrm{\infty }$.

If $Q$ is compact and nonempty, for every unit vector $\xi$ the function $x\mapsto \mathrm{\parallel }\pi (x)\xi -\xi \mathrm{\parallel }$ is continuous: the orbit map is continuous by [F1], and the norm is continuous by the reverse triangle inequality in [F2]. Its image on $Q$ is compact and therefore has a maximum by [F3]. For $Q=\mathrm{\varnothing }$, the explicitly assigned displacement $0$ serves as its maximum convention.

By the definition of extended-real supremum, every pair parameter $\epsilon$ satisfies $\epsilon \le \kappa (G,Q)$, and if $0<\epsilon <\kappa (G,Q)$ then some pair parameter $\delta$ has $\delta >\epsilon$. Downward closure from step 1.2 makes $(Q,\epsilon )$ a pair.

For compact $Q$, a unit vector is $(Q,\epsilon )$-invariant exactly when $d_Q(\pi ,\xi )<\epsilon$: in the nonempty case this follows because the continuous displacement attains its maximum, and in the empty case both conditions hold for every $\epsilon >0$. Consequently, for a representation on a nonzero Hilbert space with no invariant vector, it has no such unit vector exactly when $\kappa (G,Q,\pi )\ge \epsilon$.

Suppose $\pi$ has no nonzero invariant vector and $(Q,\epsilon )$ is a Kazhdan pair. For every unit vector $\xi$, $d_Q(\pi ,\xi )\ge \epsilon$: otherwise every $x\in Q$ would have displacement $<\epsilon$, making $\xi$ a $(Q,\epsilon )$-invariant unit vector and forcing a nonzero invariant vector. This includes $Q=\mathrm{\varnothing }$, where such a representation cannot exist on a nonzero Hilbert space. Taking the infimum over unit vectors, then the supremum over pair parameters, gives $\kappa (G,Q,\pi )\ge \kappa (G,Q)$.

The zero-space representation has no unit vectors and value $+\mathrm{\infty }$, so it never obstructs a Kazhdan pair. For compact $Q$, step 2.2 therefore says that $(Q,\epsilon )$ is a pair exactly when every representation without nonzero invariant vectors has displacement constant at least $\epsilon$, equivalently when their extended-real infimum is at least $\epsilon$. Taking the supremum of $\{0\}$ together with the admissible pair parameters yields $\kappa (G,Q)={\inf }_{\pi }\kappa (G,Q,\pi )$.

Steps 2.1 and 3.1 establish the threshold statements for arbitrary $Q$; steps 1.4–3.2 establish the endpoint equivalence and the infimum formula for compact $Q$, with empty $Q$ and the zero-space representation handled by the stated conventions.

Let $\xi$ be a unit vector and put $\phi (g)=⟨\pi (g)\xi ,\xi ⟩$. Unitarity and expansion of the squared norm give $\mathrm{\parallel }\pi (g)\xi -\xi {\mathrm{\parallel }}^2=2(1-\mathrm{Re}\phi (g))$, while [F3] gives $\mathrm{\mid }1-\phi (g)\mathrm{\mid }\le \mathrm{\parallel }\pi (g)\xi -\xi \mathrm{\parallel }$. By [F2], $\phi$ is a normalized continuous function of positive type.

If $\pi$ has almost invariant vectors, let $I=\{(Q,\epsilon ):Q\subseteq G\text{ compact},\epsilon >0\}$ and order it by $(Q,\epsilon )⪯(Q^{\mathrm{\prime }},{\epsilon }^{\mathrm{\prime }})$ when $Q\subseteq Q^{\mathrm{\prime }}$ and ${\epsilon }^{\mathrm{\prime }}\le \epsilon$. The index set is nonempty because $\mathrm{\varnothing }$ is compact. It is directed: for two indices, $Q\cup Q^{\mathrm{\prime }}$ is compact by [F5] and $\min (\epsilon ,{\epsilon }^{\mathrm{\prime }})>0$, so $(Q\cup Q^{\mathrm{\prime }},\min (\epsilon ,{\epsilon }^{\mathrm{\prime }}))$ is a common upper bound.

For every $i=(Q,\epsilon )\in I$, the witness set of unit vectors that are $(Q,\epsilon )$-invariant is nonempty by almost invariance. By AC [F8], choose one witness ${\xi }_i$ for each index. For any compact $K$ and $\eta >0$, set $i_0=(K,\eta \mathrm{/}2)$. If $i=(Q,\epsilon )⪰i_0$, then $K\subseteq Q$ and $\epsilon \le \eta \mathrm{/}2$, so for every $x\in K$, [F3] gives $\mathrm{\mid }1-{\phi }_i(x)\mathrm{\mid }\le \mathrm{\parallel }\pi (x){\xi }_i-{\xi }_i\mathrm{\parallel }<\epsilon \le \eta \mathrm{/}2<\eta$. Thus the coefficient net converges to $1$ uniformly on compact subsets.

Conversely, suppose the coefficient net of unit vectors ${\xi }_i$ converges to $1$ uniformly on compact subsets. Fix compact $Q$ and $\epsilon >0$. Uniform convergence with tolerance ${\epsilon }^2\mathrm{/}2$ gives an index $i_0$ such that $\mathrm{\mid }1-{\phi }_i(x)\mathrm{\mid }<{\epsilon }^2\mathrm{/}2$ for every $i⪰i_0$ and $x\in Q$. The identity in step 1.1 yields $\mathrm{\parallel }\pi (x){\xi }_i-{\xi }_i{\mathrm{\parallel }}^2=2(1-\mathrm{Re}{\phi }_i(x))\le 2\mathrm{\mid }1-{\phi }_i(x)\mathrm{\mid }<{\epsilon }^2$, so each such ${\xi }_i$ is $(Q,\epsilon )$-invariant. Taking ${\xi }_{i_0}$ supplies a witness for each given compact set and tolerance; hence $\pi$ has almost invariant vectors.

The estimate in step 2.1 used only eventual $(Q,\epsilon )$-invariance, not how the net was obtained. Therefore the coefficient net of any net of almost-invariant unit vectors also converges to $1$ uniformly on compact subsets.

For the GNS assertion, each normalized ${\phi }_i$ has ${\phi }_i(e)=1$, so [F6] gives a cyclic GNS triple with $\mathrm{\parallel }{\xi }_{{\phi }_i}\mathrm{\parallel }=1$ and coefficient ${\phi }_i$. Applying step 2.2's displacement estimate to the coefficient convergence shows that for each compact $Q$ and $\epsilon >0$, some $i_0$ has every ${\xi }_{{\phi }_i}$ $(Q,\epsilon )$-invariant for all $i⪰i_0$.

Embed ${\xi }_{{\phi }_{i_0}}$ as the vector supported in the $i_0$ coordinate of ${\widehat{⨁}}_{i\in I}{H}_{{\phi }_i}$. By [F7] this is a unit vector, and the direct sum action on that coordinate agrees with ${\pi }_{{\phi }_{i_0}}$. It is therefore $(Q,\epsilon )$-invariant. Since this works for every compact $Q$ and every $\epsilon >0$, the Hilbert direct sum has almost invariant vectors.

AC is used to select all witnesses in step 2.1 and is assumed by the GNS and direct-sum interfaces [F6]–[F8]. The estimates, the reverse implication in step 2.2, and the coordinate embedding use no further choice.

Assume (ii), and let $\pi$ have almost invariant vectors. Choose the compact $Q$ and $\epsilon >0$ from (ii). By [F1], $\pi$ has a $(Q,\epsilon )$-invariant unit vector, so the pair property [F2] supplies a nonzero invariant vector. Hence $G$ has property (T).

Assume (i) and suppose, for contradiction, that no compact Kazhdan pair exists. Let $I$ be the set of pairs $(Q,\eta )$ with $Q$ compact in $G$ and $\eta >0$. For each $i=(Q,\eta )\in I$, let $C_i$ be the subset of the set ${\mathbb{C}}^G$ consisting of normalized diagonal coefficients of unit vectors in strongly continuous representations with no nonzero invariant vector that are $(Q,\eta )$-invariant. Failure of $(Q,\eta )$ to be a pair makes $C_i$ nonempty. AC chooses one function ${\phi }_i\in C_i$ for each $i\in I$; this is a choice from subsets of the set ${\mathbb{C}}^G$, not from the class of all representations.

If (ii) holds for a compact $Q$ and $\epsilon >0$, [F2] gives $\kappa (G,Q)\ge \epsilon >0$, so (iii) holds. Conversely, if (iii) holds for a compact $Q$, choose $\alpha >0$ strictly below $\kappa (G,Q)$ (choose $\alpha =1$ when the threshold is $+\mathrm{\infty }$); [F2] says $(Q,\alpha )$ is a Kazhdan pair. Hence (ii) and (iii) are equivalent.

For the quantitative clause put $M=H^G$. By [F5] it is closed and invariant, its orthogonal complement is invariant, and the restricted representation on $M^{\perp }$ has no nonzero invariant vector. By [F6], write $\xi =P\xi +{\xi }^{\mathrm{\prime }\mathrm{\prime }}$ with ${\xi }^{\mathrm{\prime }\mathrm{\prime }}=\xi -P\xi \in M^{\perp }$. If ${\xi }^{\mathrm{\prime }\mathrm{\prime }}=0$, the claimed estimate is immediate. If ${\xi }^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\ne }0$ and $Q=\mathrm{\varnothing }$, then ${\xi }^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}\parallel {\xi }^{\mathrm{\prime }\mathrm{\prime }}\parallel$ is vacuously $(Q,\epsilon )$-invariant, so the pair would force an invariant vector in the restricted representation, a contradiction.

For each $i$, let $({\rho }_i,K_i,{\zeta }_i)$ be the canonical pointed cyclic GNS representation of ${\phi }_i$ from [F3]. This representation has no nonzero invariant vector: a witness in the definition of $C_i$ has a closed cyclic subspace generated by its unit vector; that subspace has no invariant vector, and its pointed cyclic representation has coefficient ${\phi }_i$, so [F3] identifies it unitarily with the GNS representation. In particular, $\mathrm{\parallel }{\zeta }_i\mathrm{\parallel }=1$ and ${\zeta }_i$ is $(Q,\eta )$-invariant in ${\rho }_i$.

Thus in the remaining case $Q\mathrm{\ne }\mathrm{\varnothing }$ and ${\xi }^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\ne }0$. The unit vector ${\xi }^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}\parallel {\xi }^{\mathrm{\prime }\mathrm{\prime }}\parallel$ cannot be $(Q,\epsilon )$-invariant, because the restricted representation has no nonzero invariant vector and $(Q,\epsilon )$ is a Kazhdan pair. Hence some $q\in Q$ satisfies $\parallel \pi (q){\xi }^{\mathrm{\prime }\mathrm{\prime }}-{\xi }^{\mathrm{\prime }\mathrm{\prime }}\parallel \ge \epsilon \parallel {\xi }^{\mathrm{\prime }\mathrm{\prime }}\parallel$. Since $P\xi \in H^G$, this displacement equals $\parallel \pi (q)\xi -\xi \parallel$, which is strictly less than $\delta \epsilon \parallel \xi \parallel$ by the assumed invariance of $\xi$. Canceling $\epsilon >0$ gives $\parallel \xi -P\xi \parallel =\parallel {\xi }^{\mathrm{\prime }\mathrm{\prime }}\parallel <\delta \parallel \xi \parallel$, and therefore the stated weak inequality.

Form the set-indexed Hilbert direct sum $\rho ={\widehat{⨁}}_{i\in I}{\rho }_i$. Given any compact $Q$ and $ϵ>0$, the coordinate indexed by $(Q,ϵ\mathrm{/}2)$ contains a unit vector that is $(Q,ϵ\mathrm{/}2)$-invariant; its image under the coordinate inclusion is $(Q,ϵ)$-invariant in $\rho$. Thus $\rho$ has almost invariant vectors. By (i) it has a nonzero invariant vector, but each coordinate of an invariant vector is invariant in its summand by [F4], and every ${\rho }_i$ has no such vector by step 2.1. All coordinates must therefore vanish, a contradiction. This proves (i)$\Rightarrow$(ii).

Steps 1.1 and 3.1 prove (i)$⟺$(ii), step 1.3 proves (ii)$⟺$(iii), and steps 1.4 and 2.2 prove the quantitative clause, including the zero vector, zero representation and empty-$Q$ cases. AC is used for the set-indexed coefficient selection, GNS/direct-sum constructions, and through Countable Choice in the orthogonal-decomposition and projection suppliers; no proper-class selection is used.