Properties Preserved under Morita Equivalence of

C ∗-algebras

Dana P. Williams

Dartmouth College

GPOTS 2007

This is joint work with Astrid an Huef and Iain Raeburn.

Proc. Amer. Math. Soc. 135 (2007), no. 5, pp. 1495–1503.

1

Morita Equivalence

Note that we use simply “Morita equivalence” in place ofwhat Rieffel called “strong Morita equivalence”. Ordinary —or what we prefer to call “weak”, or better yet “categorical”Morita equivalence — is of little use, and it seems peculiar toreserve the undecorated term for it.Formally, A and B are Morita equivalent if we can find anA – B-imprimitivity bimodule.

Definition

A Banach A – B-bimodule X is called an A – B-imprimitivitybimodule if it is both a left Hilbert A-module and a right HilbertB-module such that

The idealsA〈X , X〉 and 〈X , X〉

Bare dense in A and B,

respectively.

〈a · x , y〉B

= 〈x , a∗ · y〉B

andA〈x · b , y〉 =

A〈x , y · b∗〉 and

A〈x , y〉 · z = x · 〈y , z〉

B.

2

The Brown-Green-Rieffel Theorem— How not to think of Morita equivalence

Morita equivalence is weaker than isomorphism.

If A and B are separable — or even σ-unital — then theBrown-Green-Rieffel Theorem says that A and B are Moritaequivalent if and only if A and B are stably isomorphic (i.e.,A ⊗K ∼= B ⊗K).

Although much celebrated, the BGR-Theorem is probably nota good way to “understand” Morita theory.

Instead, I encourage you to think in terms of representationtheory.

3

Induced Representations

Let X be a A – B-imprimitivity bimodule, and suppose that πis a representation of B on Hπ.

We can form a Hilbert space X ⊗B Hπ which is thecompletion of (the algebraic tensor product) X ¯Hπ withrespect to the pre-inner product satisfying

(x ⊗ ξ | y ⊗ η) =(

π(

〈y , x〉B

)

ξ | η)

.

We get an induced representation Ind π of A on X ⊗B Hπ via

Ind π(a)[x ⊗ ξ] := [a · x ⊗ ξ].

The map π 7→ Ind π establishes an equivalence between thecategories of representations of A and representations of B.

In particular, π is irreducible if and only if Ind π is irreducibleand every irreducible representation of A is of this form.

4

The Rieffel Correspondence

Suppose that A and B are Morita equivalent via X.

Then there are lattice isomorphisms between the ideals of B,the closed A – B-submodules of X and the ideals of A.

If I is an ideal in B, then the corresponding submodule is X · Iand the corresponding ideal in A is span{

A〈x , x〉 : x ∈ X · I }.

The map sending I in B to its counterpart J := Ind I in A iscalled the Rieffel correspondence.

Since the Rieffel correspondence respects the inductionprocess as well as containment we have J = Ind I if and only if

{π ∈ B : π(I ) 6= 0 } = {π ∈ B : Ind π(J) 6= 0 }.

5

Continuous and Bounded Trace Algebras

Let A be a C ∗-algebra. Then

m(A) := span{ a ∈ A+ : π 7→ tr(

π(a))

is finite and continuous on A }

is a (not necessarily closed, two-sided) ideal in A called theideal of continuous-trace elements.

Similarly,

t(A) := span{ a ∈ A+ : π 7→ tr(

π(a))

is bounded on A }

is an ideal (called the ideal of bounded-trace elements).

We say that A has continuous trace if m(A) is dense.

We say that A has bounded trace if t(A) is dense.

6

Key Lemma

Lemma (Green, Zettl)

Suppose that X is an A – B-imprimitivity bimodule and that

π ∈ RepB. Then for all x ∈ X,

tr(

π(

〈x , x〉B

))

= tr(

Ind π(

A〈x , x〉

))

.

Sketch of the Proof.

Define Tx : Hπ → X ⊗B Hπ by Tx(ξ) := x ⊗ ξ. ThenT ∗

x (y ⊗ η) = π(

〈x , y〉B

)

η and

T ∗

x Tx = π(

〈x , x〉B

)

while Tx T ∗

x = Ind π(

A〈x , x〉

)

.

The result follows from tr(T ∗

x Tx ) = tr(Tx T ∗

x ).

7

The Continuous-Trace and Bounded-Trace Ideals

Proposition (Green,Zettl)

Suppose that X is an A – B-imprimitivity bimodule. Then the

Rieffel correspondence carries m(B) onto m(A) and t(B) onto

t(A). In particular, A has continuous trace if and only if B has

continuous trace, and A has bounded trace if and only if B has

bounded trace.

9

Sketch of the Proof

Sketch of the Proof.

We’ll just consider m(A) as the result for t(A) is proved similarly.Note that it suffices to see that

{π ∈ B : π(

m(B))

6= 0 } = {π ∈ B : Ind π(

m(A))

6= 0 }.

But m(B) = span{ 〈x , x〉B

: x ∈ X · m(B) }. If π(

m(B))

6= 0, then

there is a x ∈ X · m(B) such that π(

〈x , x〉B

)

6= 0. But the “KeyLemma” implies that

A〈x , x〉 ∈ m(A), and that Ind π

A〈x , x〉 6= 0.

Therefore Ind π(

m(A))

6= 0, and the result follows bysymmetry.a

aMorita equivalence is an equivalence relation and therefore symmetric. In

particular, inducing representations from A to B is an inverse — up to

equivalence — for Ind.

10

Multiplicity

One clever way to measure the extent to which an algebrafails to have continuous trace is through the notion, due toArchbold, of the multiplicity of an irreducible representation.

Unfortunately, the actual definition is forbidding and not veryinstructive for a 20 minute talk.

Fortunately, the concept is easily grasped via an example.

11

Archbold’s Example

Example

Let A be the subalgebra of f ∈ C(

[0, 1], M4

)

such that for eachn ≥ 1,

f (1/n) =

(

σn(f ) 00 σn(f )

)

where σn(f ) ∈ M2, and such that f (0) = λ(f )1. We haveA ∼= [0, 1] ∼= {πt }, where πt is evaluation at t if 1

t/∈ N, π 1

n

= σn

and π0 = λ. If tn → 0+ and each tn /∈ { 1m}∞m=1, then

limn

tr(

πtn(f ))

= 4 tr(

π0(f ))

.

On the other hand,

limn

tr(

π 1n

(f ))

= 2 tr(

π0(f ))

.

12

Multiplicity

In general, Archbold assigns both an upper multiplicity MU(π) anda lower multiplicity ML(π) to each π ∈ A.

Example

In the previous example, MU(π0) = 4 while ML(π0) = 2. On theother hand, for n ∈ N, MU(π 1

n

) = 2 = ML(π 1n

).

In this talk, we will use the concept of upper multiplicity relative toa sequence. If (πn) is a sequence in A converging to π ∈ A, thenwe will take the following result of Archbold & Spielberg asdefinition.

(tr) MU(π, (πn)) ≤ k if and only if there is a b ∈ B+

such that π(b) is a nonzero projection and such thattr(

πn(b))

≤ k eventually.

13

Main Result

Theorem

Suppose that X is an A – B-imprimitivity bimodule and that π ∈ B.

If (πn) is a sequence in B converging to π, then

MU

(

π, (πn))

= MU

(

Ind π, (Ind πn))

.

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Sketch of the Proof

Sketch of the Proof.

By symmetry, it suffices to prove that

MU

(

Ind π, (Ind πn))

≤ MU

(

π, (πn))

.

Suppose that k := MU

(

π, (πn))

< ∞. By (tr), there is a b ∈ B+

such that π(b) is a nonzero projection and such that we eventuallyhave tr

(

πn(b))

≤ k. For convenience, assume that b = 〈x , x〉B.

Then, as in the Key Lemma,

π(

〈x , x〉B

)

= T ∗

x Tx

where Tx(ξ) := x ⊗ ξ. Since T ∗

x Tx is a nonzero projection, so is

Tx T ∗

x = Ind π(

A〈x , x〉

)

.

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Proof Continued

Proof Continued.

Our Key Lemma implies that we eventually have

tr(

Ind πn

(

A〈x , x〉

))

= tr(

πn

(

〈x , x〉B

))

≤ k.

Then (tr) implies that MU

(

Ind π, (Ind πn))

≤ k.This proves the result when k < ∞.It also shows that if one upper multiplicity of finite, then so is theother. Thus, if one is infinite, then so is the other. This finishesthe sketch.

17

Ordinary Multiplicities

Corollary

Let X be an A – B-imprimitivity bimoduleand suppose that π ∈ B.

1 MU(π) = MU

(

Ind π)

.

2 If {π } is not open in B, then ML(π) = ML

(

Ind π)

.

Sketch of the Proof.

We’ll just look at the lower multiplicity. Of course, it suffices toprove that ML

(

Ind π)

≤ ML(π). Archbold & Spielberg show that

there is (πn) ∈ B such that ML(π) = MU

(

π, (πn))

. But then,

ML

(

Ind π)

≤ ML

(

Ind π, (Ind πn))

≤ MU

(

Ind π, (Ind πn))

= MU

(

π, (πn))

= ML(π).

18

An Application: Fell Algebras

We say that A is a Fell algebra if given π ∈ A, there is ana ∈ A+ such that σ(a) is a rank-one projection for all σ ∈ A

close to π.

Fell algebras are called type I0 algebras by Pedersen.

Archbold has shown that A is a Fell algebra if and only ifMU(π) = 1 for all π ∈ A.

Corollary

If A and B are Morita equivalent, then A is a Fell algebra if and

only if B is a Fell algebra.

19

Things There Wasn’t Time to Talk About

Recall that A is called liminary if π(A) = K(Hπ) for all π ∈ A,and postliminary if every nonzero quotient of A has a nonzeroliminary ideal.

We have followed the lead of our late friend Gert Pedersen intranslating liminaire as “liminary” in parallel with the obvioustranslation of preliminaire. We have also followed hisASHCEFLC dictum and avoided the acronyms CCR and GCR.

A is called nuclear if the minimal and maximal tensor productnorms agree on A ¯ B for all C ∗-algebras B.

Every C ∗-algebra A has a largest liminary ideal L(A), a largestpostliminary ideal P(A) and a largest nuclear ideal N(A).

20

Last Slide

Theorem

Suppose that X is an A – B-imprimitivity bimodule. Then the

Rieffel correspondence carries

1 L(B) onto L(A),

2 P(B) onto P(A) and

3 N(B) onto N(A).

Therefore

4 A is liminary if and only if B is liminary,

5 A is postliminary if and only if B is postliminary and

6 A is nuclear if and only if B is nuclear.

These results are essentially all well-known. In particular, much ofthis result is due to Zettl and item 6 is also due to Beer. However,we feel we are able to give a more modern treatment, and in somecases, a more elementary proof).

21

Bibliography

Robert J. Archbold and Jack S. Spielberg, Upper and lower

multiplicity for irreducible representations of C ∗-algebras. II, J.Operator Theory 36 (1996), no. 2, 201–231.

Walter Beer, On Morita equivalence of nuclear C ∗-algebras, J.Pure & Appl. Algebra 26 (1982), 249–267.

Astrid an Huef, Iain Raeburn, and Dana P. Williams,Properties preserved under Morita equivalence of C ∗-algebras,Proc. Amer. Math. Soc. 135 (2007), no. 5, 1495–1503.

Heinrich H. Zettl, Ideals in Hilbert modules and invariants

under strong Morita equivalence of C ∗-algebras, Arch. Math.(Basel) 39 (1982), no. 1, 69–77.

, Strong Morita equivalence of C ∗-algebras preserves

nuclearity, Arch. Math. (Basel) 38 (1982), no. 5, 448–452.

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