Cartan MASAs and Exact Sequences of Inverse Semigroups

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1 Cartan MASAs and Exact Sequences of Inverse Semigroups Adam H. Fuller (University of Nebraska - Lincoln) joint work with Allan P. Donsig and David R. Pitts NIFAS Nov. 2014, Des Moines, Iowa

2 Cartan MASAs Let M be a von Neumann algebra. A maximal abelian subalgebra (MASA) D in M is a Cartan MASA if 1 the unitaries U M such that UDU = U DU = D span a weak- dense subset in M; 2 there is a normal, faithful conditional expectation E : M D.

3 Cartan MASAs Let M be a von Neumann algebra. A maximal abelian subalgebra (MASA) D in M is a Cartan MASA if 1 the unitaries U M such that UDU = U DU = D span a weak- dense subset in M; 2 there is a normal, faithful conditional expectation E : M D. Alternatively 1 the partial isometries V M such that V DV, V DV D span a weak- dense subset in M; 2 there is a normal, faithful conditional expectation E : M D.

4 Cartan MASAs Let M be a von Neumann algebra. A maximal abelian subalgebra (MASA) D in M is a Cartan MASA if 1 the unitaries U M such that UDU = U DU = D span a weak- dense subset in M; 2 there is a normal, faithful conditional expectation E : M D. Alternatively 1 the partial isometries V M such that V DV, V DV D span a weak- dense subset in M; 2 there is a normal, faithful conditional expectation E : M D. We will call the pair (M, D) a Cartan pair. We call the normalizing partial isometries groupoid normalizers, written G M (D).

5 Examples of Cartan Pairs Example Let M n be the n n complex matrices, and let D n be the diagonal n n matrices. Then (M n, D n ) is a Cartan pair: 1 the matrix units normalize D n and generate M n ; 2 The map E : [a ij ] diag[a 11,..., a nn ] gives a faithful normal conditional expectation.

6 Examples of Cartan Pairs Example Let M n be the n n complex matrices, and let D n be the diagonal n n matrices. Then (M n, D n ) is a Cartan pair: 1 the matrix units normalize D n and generate M n ; 2 The map E : [a ij ] diag[a 11,..., a nn ] gives a faithful normal conditional expectation. Example Let D = L (T) and let α be an action of Z on T by irrational rotation. Then L (T) is a Cartan MASA in L (T) α Z.

7 Examples of Cartan Pairs Example Let G = {( ) } a b : a, b R, a 0, 0 1 and let H = {( ) } 1 b : b R. 0 1 Then H is a normal subgroup of G and L(H) is Cartan MASA in L(G).

8 Feldman & Moore approach Feldman and Moore (1977) explored Cartan pairs (M, D) where M is separable and D = L (X, µ). They showed: 1 there is a measurable equivalence relation R on X with countable equivalence classes and a 2-cocycle σ on R s.t. M M(R, σ) and D A(R, σ), where M(R, σ) are functions on R and A(R, σ) are the functions supported on diag. {(x, x) : x X }; 2 every sep. acting pair (M, D) arises this way.

9 A simple example Consider the Cartan pair (M 3, D 3 ). Let G = G M3 (D 3 ). E.g., an element of G could look like 0 λ 0 V = µ 0 0, 0 0 γ with λ, µ, γ T. Let P = G D n. And let S = G/P. So elements of S are of the form S = From (M n, D n ) we have 3 semigroups: P, G and S.

10 A simple example: continued Conversely, starting with S, we can construct P: P is all the continuous functions from the idempotents of S into T. From S and P we can construct G, since every element of G is the product of an element in S and an element in P. From G we can construct (M n, D n ) as the span of G.

11 Our Objective: Give an alternative approach using algebraic rather than measure theoretic tools which conceptually simpler; applies to the non-separably acting case.

12 Inverse Semigroups A semigroup S is an inverse semigroup if for each s S there is a unique inverse element s such that ss s = s and s ss = s. We denote the idempotents in an inverse semigroup S by E(S). The idempotents form an abelian semigroup. For any element s S, ss E(S).

13 Inverse Semigroups A semigroup S is an inverse semigroup if for each s S there is a unique inverse element s such that ss s = s and s ss = s. We denote the idempotents in an inverse semigroup S by E(S). The idempotents form an abelian semigroup. For any element s S, ss E(S). An inverse semigroup S has a natural partial order defined by for some idempotent e E(S). s t if and only if s = te

14 Matrix example Example Consider the Cartan pair (M n, D n ) again. Again, let G = G Mn (D n ) = {partial isometries V M n : VD n V D n, V D n V D n }. Then G is an inverse semigroup: if V, W G then (VW )D n (VW ) = V (WD n W )V D n, so VW G;

15 Matrix example Example Consider the Cartan pair (M n, D n ) again. Again, let G = G Mn (D n ) = {partial isometries V M n : VD n V D n, V D n V D n }. Then G is an inverse semigroup: if V, W G then (VW )D n (VW ) = V (WD n W )V D n, so VW G; the inverse of V is V ;

16 Matrix example Example Consider the Cartan pair (M n, D n ) again. Again, let G = G Mn (D n ) = {partial isometries V M n : VD n V D n, V D n V D n }. Then G is an inverse semigroup: if V, W G then (VW )D n (VW ) = V (WD n W )V D n, so VW G; the inverse of V is V ; the idempotents are the projections in D n ;

17 Matrix example Example Consider the Cartan pair (M n, D n ) again. Again, let G = G Mn (D n ) = {partial isometries V M n : VD n V D n, V D n V D n }. Then G is an inverse semigroup: if V, W G then (VW )D n (VW ) = V (WD n W )V D n, so VW G; the inverse of V is V ; the idempotents are the projections in D n ; V W if V = WP for some projection P D n.

18 Bigger Matrix example More generally... Example Let (M, D) be a Cartan pair. Then the groupoid normalizers G M (D) form an inverse semigroup. if V, W G M (D) then so VW G M (D); (VW )D(VW ) = V (W DW )V D, the inverse of V is V ; the idempotents are the projections in D; V W if V = WP for some projection P D.

19 Extensions of Inverse Semigroups Let S and P be inverse semigroups. And let π : P S, be a surjective homomorphism such that π E(P) is an isomorphism from E(P) to E(S). An idempotent separating extension of S by P is an inverse semigroup G with P ι G q S and ι is an injective homomorphism; q is a surjective homomorphism; q(g) E(S) if and only if g = ι(p) for some p P; q ι = π. Note that E(P) = E(G) = E(S).

20 The Munn Congruence Let G be an inverse semigroup. Define an equivalence relation (the Munn congruence) on G by s t if ses = tet for all e E(G).

21 The Munn Congruence Let G be an inverse semigroup. Define an equivalence relation (the Munn congruence) on G by If s t and u v then s t if ses = tet for all e E(G). su tv. Thus S = G/ is an inverse semigroup.

22 The Munn Congruence Let G be an inverse semigroup. Define an equivalence relation (the Munn congruence) on G by If s t and u v then s t if ses = tet for all e E(G). su tv. Thus S = G/ is an inverse semigroup. Let P = {v G : v e for some e E(G)}. Then P is an inverse semigroup. And G is an extension of S by P: P G S.

23 From Cartan Pairs to Extensions of Inverse Semigroups Let (M, D) be a Cartan pair. Let G = G M (D) = {v M a partial isometry: vdv D and v Dv D}.

24 From Cartan Pairs to Extensions of Inverse Semigroups Let (M, D) be a Cartan pair. Let G = G M (D) = {v M a partial isometry: vdv D and v Dv D}. Let S = G/, where is the Munn congruence on G and let P = {V G : V P, P Proj(D)}. Definition We call the extension P G S, the extension associated to the Cartan pair (M, D).

25 Properties of associated extensions Let (M, D) be a Cartan pair, and let P G S, be the associated extension. Then P = G M (D) D, i.e. P is simply the partial isometries in D.

26 Properties of associated extensions Let (M, D) be a Cartan pair, and let P G S, be the associated extension. Then P = G M (D) D, i.e. P is simply the partial isometries in D. The inverse semigroup S has the following properties 1 S is fundamental: E(S) is maximal abelian in S; 2 E(S) is a hyperstonean boolean algebra, i.e. the idempotents are the projection lattice of an abelian W -algebra; 3 S is a meet semilattice under the natural partial order on S; 4 for every pairwise orthogonal family F S, F exists in S. 5 S contains 1 and 0.

27 Properties of associated extensions Let (M, D) be a Cartan pair, and let P G S, be the associated extension. Then P = G M (D) D, i.e. P is simply the partial isometries in D. The inverse semigroup S has the following properties 1 S is fundamental: E(S) is maximal abelian in S; 2 E(S) is a hyperstonean boolean algebra, i.e. the idempotents are the projection lattice of an abelian W -algebra; 3 S is a meet semilattice under the natural partial order on S; 4 for every pairwise orthogonal family F S, F exists in S. 5 S contains 1 and 0. Definition An inverse semigroup S, satisfying the conditions above is called a Cartan inverse monoid.

28 Matrix example Example In the matrix example (M n, D n ), the semigroups P, G and S are the semigroups discussed earlier: 1 G is the partial isometries V such that VD n V, V D n V D n ; 2 P is the partial isometries in D n ; 3 S is the matrices in G with only 0 and 1 entries.

29 Equivalent Extensions of Cartan Inverse monoid Let α: S 1 S 2 be an isomorphism of Cartan inverse monoids. Then E(S i ) is the lattice of projections for a W -algebra, D i = C(Ê(S i)). The isomorphism α induces an isomorphism α from D 1 to D 2.

30 Equivalent Extensions of Cartan Inverse monoid Let α: S 1 S 2 be an isomorphism of Cartan inverse monoids. Then E(S i ) is the lattice of projections for a W -algebra, D i = C(Ê(S i)). The isomorphism α induces an isomorphism α from D 1 to D 2. Definition Let S 1 and S 2 be isomorphic Cartan inverse monoids. Let P i be the partial isometries in D i. Extensions G i of S i by P i are equivalent if there is an isomorphism α: G 1 G 2 such that α P 1 ι 1 q 1 G1 S1 α α commutes. ι P 2 q 2 2 G2 S2.

31 More on Extensions of Inverse Monoids It was shown by Laush (1975) that there is one-to-one correspondence between extensions of S by P and the second cohomology group H 2 (S, P). It is also shown that every extension of S by P is determined by cocycle function σ : S S P.

32 Uniqueness of Extension Theorem Let (M 1, D 1 ) and (M 2, D 2 ) be two Cartan pairs with associated extensions P i G i S i for i = 1, 2. There is a normal isomorphism θ : M 1 M 2 such θ(d 1 ) = D 2 if and only if the two associated extensions are equivalent.

33 Going in the other direction Let S be a Cartan inverse monoid. Let D = C(Ê(S)), and let P be the partial isometries in D. Given an extension P G S we want to construct a Cartan pair (M, D) with associated extension (equivalent to) P G S.

34 A D-valued Reproducing kernel space Let j be an order-preserving map, j : S G such that j q = id. That is j(s) j(t) when s t and j : E(S) E(G) is an isomorphism.

35 A D-valued Reproducing kernel space Let j be an order-preserving map, j : S G such that j q = id. That is j(s) j(t) when s t and j : E(S) E(G) is an isomorphism. Define a map K : S S D by K(s, t) = j(s t 1).

36 A D-valued Reproducing kernel space Let j be an order-preserving map, j : S G such that j q = id. That is j(s) j(t) when s t and j : E(S) E(G) is an isomorphism. Define a map K : S S D by K(s, t) = j(s t 1). The idempotent s t 1 is the minimal idempotent e such that se = te = s t. Thus K(s, t) is the idempotent in G defining j(s) j(t).

37 A D-valued Reproducing kernel space Let j be an order-preserving map, j : S G such that j q = id. That is j(s) j(t) when s t and j : E(S) E(G) is an isomorphism. Define a map K : S S D by K(s, t) = j(s t 1). The idempotent s t 1 is the minimal idempotent e such that se = te = s t. Thus K(s, t) is the idempotent in G defining j(s) j(t). The map K is positive: that is for c 1,..., c k C and s 1,..., s k S c i c j K(s i, s j ) 0. i,j

38 A D-valued Reproducing kernel space For each s S define a kernel-map k s : S D by k s (t) = K(t, s). Let A 0 = span{k s : s S}. The positivity of K shows that the c i k si, d j k tj = i,j c i d j K(s i, t j ) defines a D-valued inner product on A 0. Let A be completion of A 0. Thus A is a reproducing kernel Hilbert D-module of functions from S into D.

39 A left representation of G For g G define an adjointable operator λ(g) on A by λ(g)k s = k q(g)s σ(g, s), where σ : G S P is a cocycle-like function (related to the cocycles of Lausch). This is determined by the equation gj(s) = j(q(g)s)σ(g, s), i.e. elements of the form gj(s) can be factored into the product of an element in j(s) by an element in P.

40 A left representation of G For g G define an adjointable operator λ(g) on A by λ(g)k s = k q(g)s σ(g, s), where σ : G S P is a cocycle-like function (related to the cocycles of Lausch). This is determined by the equation gj(s) = j(q(g)s)σ(g, s), i.e. elements of the form gj(s) can be factored into the product of an element in j(s) by an element in P. The mapping λ: G L(A) is a representation of G by partial isometries.

41 A left representation of G on a Hilbert space Let π be a faithful representation of D on a Hilbert space H. We can form a Hilbert space A π H by completing A H with respect to the inner product a h, b k := h, π( a, b )k.

42 A left representation of G on a Hilbert space Let π be a faithful representation of D on a Hilbert space H. We can form a Hilbert space A π H by completing A H with respect to the inner product a h, b k := h, π( a, b )k. Then π determines a faithful representation ˆπ of L(A) on the Hilbert space A π H by ˆπ(T )(a h) = (Ta) h.

43 A left representation of G on a Hilbert space Let π be a faithful representation of D on a Hilbert space H. We can form a Hilbert space A π H by completing A H with respect to the inner product a h, b k := h, π( a, b )k. Then π determines a faithful representation ˆπ of L(A) on the Hilbert space A π H by ˆπ(T )(a h) = (Ta) h. Thus, we have a faithful representation of G on the hilbert space A π H by λ π : g ˆπ(λ(g)).

44 Creating Cartan pairs Let M q = λ(g), and D q = λ(e(s)). Then (M q, D q ) is a Cartan pair such that 1 The pair (M q, D q ) is independent of choice of j and π;

45 Creating Cartan pairs Let M q = λ(g), and D q = λ(e(s)). Then (M q, D q ) is a Cartan pair such that 1 The pair (M q, D q ) is independent of choice of j and π; 2 D q is isomorphic to D = C(Ê(S));

46 Creating Cartan pairs Let M q = λ(g), and D q = λ(e(s)). Then (M q, D q ) is a Cartan pair such that 1 The pair (M q, D q ) is independent of choice of j and π; 2 D q is isomorphic to D = C(Ê(S)); 3 The conditional expectation E : M q D q is induced from the map S E(S) s s 1.

47 Creating Cartan pairs Let M q = λ(g), and D q = λ(e(s)). Then (M q, D q ) is a Cartan pair such that 1 The pair (M q, D q ) is independent of choice of j and π; 2 D q is isomorphic to D = C(Ê(S)); 3 The conditional expectation E : M q D q is induced from the map S E(S) s s 1. 4 The extension associated to (M q, D q ) is equivalent to P G q S (the extension we started with).

48 Main Theorem Theorem (Feldman-Moore; Donsig-F-Pitts) If S is a Cartan inverse monoid and P G q S is an extension of S by P := p.i.(c (E(S)), then the extension determines a Cartan pair (M, D) which is unique up to isomorphism. Equivalent extensions determine isomorphic Cartan pairs. Every Cartan pair (M, D) determines uniquely an extension of a Cartan inverse semigroup S by P, P G q S.

VON NEUMANN ALGEBRAS AND EXTENSIONS OF INVERSE SEMIGROUPS

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 2017 VON NEUMANN ALGEBRAS AND EXTENSIONS OF INVERSE