CUBO A Mathematical Journal
Vol.16, No¯ 02, (49–52). June 2014

K-theory for the group C∗-algebras of a residually finite
discrete group with Kazhdan property T

Takahiro Sudo

Department of Mathematical Sciences,

Faculty of Science, University of the Ryukyus,

Senbaru 1, Nishihara, Okinawa 903-0213, Japan.

sudo@math.u-ryukyu.ac.jp

ABSTRACT

We compute the K-theory groups for the full and reduced group C∗-algebras of a

residually finite, finitely generated discrete group with Kazhdan property T.

RESUMEN

Calculamos los grupos de la K-teoŕıa para grupo de C∗-algebras de reducido y completo

de un grupo discreto generado finitamente y residualmente finito con la propiedad T

de Kazhdan.

Keywords and Phrases: Group C*-algebra, K-theory, discrete group, projection.

2010 AMS Mathematics Subject Classification: 46L05, 46L80, 19K14.



50 Takahiro Sudo CUBO
16, 2 (2014)

1 Introduction

In this paper, first of all, we compute the K-theory groups for the full group C∗-algebra of a

residually finite, finitely generated discrete group with Kazhdan property T, such as SLn(Z) (n ≥ 3)

the n × n special linear groups over the integers. The highly non-trivial and interesting problem

to compute the K-theory groups has been considered by the author [6], but it was found to be not

completed, and to be corrected as [7] (perhaps partly). This time we obtain a sort of solution for

this problem to settle the issue, without using the results of [5] used in [6], but more precisely K0
only, with a mysterious part left. We next compute the K-theory groups for the reduced group

C∗-algebra of a residually finite, finitely generated discrete group with Kazhdan property T, by

using the six-term exact sequence of K-groups and the results obtained in the full case.

2 The main result

Proposition 1. Let Γ be a residually finite, finitely generated discrete group with Kazhdan prop-

erty T and C∗(Γ) its full group C∗-algebra. Then the K0-group K0(C
∗(Γ)) has a direct summand

isomorphic to the group generated by the infinite direct sum of copies of Z and one copy of Z

corresponding to the unit.

Proof. Since Γ is residully finite, then there is a (separable) family of finite dimensional irreducible

representations πλ of Γ such that the intersection of their kernels is trivial (see [4, p. 480]). Denote

also by πλ the corresponding finite dimensional irreducible representations of C
∗(Γ). Then C∗(Γ)

has a ∗-homomorphism (which can not be injective in general, see [2]) into the direct product

C∗-algebra ΠλMnλ(C) of the nλ × nλ matrix algebras Mnλ(C) over C, where nλ = dim πλ,

by the direct product representation Πλπλ of C
∗(Γ). The representation implies the K-theory

homomorphism:

K∗(C
∗(Γ))

(Πλπλ)∗
−−−−−−→ K∗(ΠλMnλ(C))

for ∗ = 0, 1, and K∗(ΠλMnλ(C))
∼= ΠλK∗(Mnλ(C)) with K0(Mnλ(C))

∼= Z and K1(Mnλ(C))
∼= 0.

Note that since Γ is discrete, C∗(Γ) has the unit and that the map (Πλπλ)∗ is unital.

On the other hand, since Γ has Kazhdan propery T, then C∗(Γ) has Mnλ(C) as a direct

summand (see [9]). Hence K∗(C
∗(Γ)) has K∗(Mnλ(C)) as a direct summand. Since K∗(Mnλ(C))

is mapped injectively under the induced map (Πλπλ)∗, it follows that both K∗(C
∗(Γ)) and the

image of K∗(C
∗(Γ)) contain the infinite direct sum ⊕λK∗(Mnλ(C)). Furthermore, all or nothing

principle tells us that the image of K0(C
∗(Γ)) does not contain other classes corresponding to other

non-trivial (infinite) projections in ΠλMnλ(C) except projections in the group generated by the

direct sum ⊕λZ and Z of the unit class, because if it does contain, the principle implies that the

image must be equal to ΠλK0(Mnλ(C)), so that C
∗(Γ) has ΠλMnλ(C) as a quotient, but the direct

product is non-separable, while C∗(Γ) is separable, a contradiction. Indeed, we can not find the

difference among those extra infinite projections in ΠλMnλ(C). Hence the proof is completed.



CUBO
16, 2 (2014)

K-theory for the group C∗-algebras of a residually finite discrete . . . 51

Remark. Unfortunately, we do not know about the mysterious kernel Ker(Πλπλ)∗ of the K-theory

homomorphism in the K0 and K1-groups K∗(C
∗(Γ)) which may not be trivial in general, so that

we could not determine the K0 and K1-group.

Corollary 1. For n ≥ 3, the abelian group K0(C
∗(SLn(Z))) has a direct summand isomorphic to

the group generated by an infinite direct sum of copies of Z and one copy of Z.

Proof. Note that SLn(Z) for n ≥ 3 are residually finite, finitely generated groups with Kazhdan

property T. Indeed, it is known that every finitely generated subgroup of SLn(C) is residually finite

(see [1] and also [4]) and that SLn(Z) have Kazhdan property T (see [3, p. 34]).

Theorem 1. Let Γ be a non-amenable, residually finite, finitely generated discrete group with

Kazhdan property T and C∗r(Γ) its reduced group C
∗-algebra. Then

K0(C
∗

r(Γ))
∼= Z ⊕ q∗[Ker(Πλπλ)∗]

and K1(C
∗

r(Γ)) is a quotient of K1(C
∗(Γ)), where this quotient and q∗ are induced from the canonical

quotient map q : C∗(G) → C∗r(G).

Proof. Denote by IΓ the kernel of q. Then we have the following six-term diagram:

K0(IΓ )
i∗

−−−−→ K0(C
∗(Γ))

q∗
−−−−→ K0(C

∗

r(Γ))
!
⏐
⏐

⏐
⏐
#

K1(C
∗

r(Γ))
q∗

←−−−− K1(C
∗(Γ))

i∗
←−−−− K1(IΓ )

where i∗ is induced by the inclusion i : IΓ → C
∗(Γ). Note that the infinite direct sum of Z in

K0(C
∗(Γ)) is mapped to zero by q∗ since Γ is non-amenable, so that C

∗

r(Γ) has no finite dimensional

representation (a fact of the representation theory for Γ), and the other copy of Z in K0(C
∗(Γ)) is

mapped injectively. It follows that K0(IΓ ) is isomorphic to the direct sum ⊕Z. Since i∗ on K0 is

injective, the index map from K1(C
∗

r(Γ)) is zero, so that q∗ on K1(C
∗(Γ)) is surjective. Since the

class of the unit in K0(C
∗

r(Γ)) is mapped to zero by the exactness of the diagram, it follows that

K0(C
∗

r(Γ))
∼= Z ⊕ q∗[Ker(Πλπλ)∗] and i∗ on K1(IΓ ) is injective.

Corollary 2. For n ≥ 3, we have

K0(C
∗

r(SLn(Z)))
∼= Z ⊕ q∗[Ker(Πλπλ)∗],

and K1(C
∗

r(SLn(Z))) is a quotient of K1(C
∗(SLn(Z))).

Remark. Note that q∗[Ker(Πλπλ)∗] is not trivial. Because if it is zero, then K0(C
∗

r(SLn(Z)))
∼= Z,

which implies that C∗r(SLn(Z)) does not contain non-trivial projections. But SLn(Z) has torsion

since it contains SL2(Z) ∼= Z4 ∗Z2 Z6 as a subgroup, so that C
∗

r(SLn(Z)) has non-trivial projections,

a contradiction.



52 Takahiro Sudo CUBO
16, 2 (2014)

The Kadison-Kaplansky conjecture is that if Γ is a torsion free, discrete group, then C∗r(Γ) has

no non-trivial projections. See [8] about the conjecture.

Received: March 2013. Revised: September 2013.

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